WO2026010844A1 - System and method for generalized brunauer-emmett-teller isotherm for single and mixed-gas multilayer adsorption equilibria - Google Patents

System and method for generalized brunauer-emmett-teller isotherm for single and mixed-gas multilayer adsorption equilibria

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WO2026010844A1
WO2026010844A1 PCT/US2025/035855 US2025035855W WO2026010844A1 WO 2026010844 A1 WO2026010844 A1 WO 2026010844A1 US 2025035855 W US2025035855 W US 2025035855W WO 2026010844 A1 WO2026010844 A1 WO 2026010844A1
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adsorption
adsorbate
gas
component
mixed
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Chau-Chyun Chen
Usman HAMID
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Texas Tech University TTU
Texas Tech University System
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    • G01—MEASURING; TESTING
    • G01N—INVESTIGATING OR ANALYSING MATERIALS BY DETERMINING THEIR CHEMICAL OR PHYSICAL PROPERTIES
    • G01N25/00—Investigating or analyzing materials by the use of thermal means
    • G01N25/20—Investigating or analyzing materials by the use of thermal means by investigating the development of heat, i.e. calorimetry, e.g. by measuring specific heat, by measuring thermal conductivity
    • G01N25/48—Investigating or analyzing materials by the use of thermal means by investigating the development of heat, i.e. calorimetry, e.g. by measuring specific heat, by measuring thermal conductivity on solution, sorption, or a chemical reaction not involving combustion or catalytic oxidation
    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B01—PHYSICAL OR CHEMICAL PROCESSES OR APPARATUS IN GENERAL
    • B01D—SEPARATION
    • B01D53/00—Separation of gases or vapours; Recovering vapours of volatile solvents from gases; Chemical or biological purification of waste gases, e.g. engine exhaust gases, smoke, fumes, flue gases, aerosols
    • B01D53/02—Separation of gases or vapours; Recovering vapours of volatile solvents from gases; Chemical or biological purification of waste gases, e.g. engine exhaust gases, smoke, fumes, flue gases, aerosols by adsorption, e.g. preparative gas chromatography
    • G—PHYSICS
    • G06—COMPUTING OR CALCULATING; COUNTING
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    • G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
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    • G16C—COMPUTATIONAL CHEMISTRY; CHEMOINFORMATICS; COMPUTATIONAL MATERIALS SCIENCE
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    • G16C20/10—Analysis or design of chemical reactions, syntheses or processes
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    • G16C—COMPUTATIONAL CHEMISTRY; CHEMOINFORMATICS; COMPUTATIONAL MATERIALS SCIENCE
    • G16C20/00—Chemoinformatics, i.e. ICT specially adapted for the handling of physicochemical or structural data of chemical particles, elements, compounds or mixtures
    • G16C20/30—Prediction of properties of chemical compounds, compositions or mixtures
    • G—PHYSICS
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    • G16C—COMPUTATIONAL CHEMISTRY; CHEMOINFORMATICS; COMPUTATIONAL MATERIALS SCIENCE
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    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B01—PHYSICAL OR CHEMICAL PROCESSES OR APPARATUS IN GENERAL
    • B01D—SEPARATION
    • B01D2253/00—Adsorbents used in seperation treatment of gases and vapours
    • B01D2253/10—Inorganic adsorbents
    • B01D2253/102—Carbon
    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B01—PHYSICAL OR CHEMICAL PROCESSES OR APPARATUS IN GENERAL
    • B01D—SEPARATION
    • B01D2253/00—Adsorbents used in seperation treatment of gases and vapours
    • B01D2253/10—Inorganic adsorbents
    • B01D2253/104—Alumina
    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B01—PHYSICAL OR CHEMICAL PROCESSES OR APPARATUS IN GENERAL
    • B01D—SEPARATION
    • B01D2253/00—Adsorbents used in seperation treatment of gases and vapours
    • B01D2253/10—Inorganic adsorbents
    • B01D2253/112—Metals or metal compounds not provided for in B01D2253/104 or B01D2253/106
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    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B01—PHYSICAL OR CHEMICAL PROCESSES OR APPARATUS IN GENERAL
    • B01D—SEPARATION
    • B01D2256/00—Main component in the product gas stream after treatment
    • B01D2256/10—Nitrogen
    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B01—PHYSICAL OR CHEMICAL PROCESSES OR APPARATUS IN GENERAL
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    • B01D2256/12—Oxygen
    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B01—PHYSICAL OR CHEMICAL PROCESSES OR APPARATUS IN GENERAL
    • B01D—SEPARATION
    • B01D2256/00—Main component in the product gas stream after treatment
    • B01D2256/24—Hydrocarbons
    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B01—PHYSICAL OR CHEMICAL PROCESSES OR APPARATUS IN GENERAL
    • B01D—SEPARATION
    • B01D2257/00—Components to be removed
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    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B01—PHYSICAL OR CHEMICAL PROCESSES OR APPARATUS IN GENERAL
    • B01D—SEPARATION
    • B01D2257/00—Components to be removed
    • B01D2257/10—Single element gases other than halogens
    • B01D2257/104—Oxygen
    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B01—PHYSICAL OR CHEMICAL PROCESSES OR APPARATUS IN GENERAL
    • B01D—SEPARATION
    • B01D2257/00—Components to be removed
    • B01D2257/50—Carbon oxides
    • B01D2257/504—Carbon dioxide
    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B01—PHYSICAL OR CHEMICAL PROCESSES OR APPARATUS IN GENERAL
    • B01D—SEPARATION
    • B01D2257/00—Components to be removed
    • B01D2257/70—Organic compounds not provided for in groups B01D2257/00 - B01D2257/602
    • B01D2257/702—Hydrocarbons
    • B01D2257/7022—Aliphatic hydrocarbons
    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B01—PHYSICAL OR CHEMICAL PROCESSES OR APPARATUS IN GENERAL
    • B01D—SEPARATION
    • B01D2257/00—Components to be removed
    • B01D2257/70—Organic compounds not provided for in groups B01D2257/00 - B01D2257/602
    • B01D2257/702—Hydrocarbons
    • B01D2257/7027—Aromatic hydrocarbons
    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B01—PHYSICAL OR CHEMICAL PROCESSES OR APPARATUS IN GENERAL
    • B01D—SEPARATION
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    • B01D2257/80—Water
    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B01—PHYSICAL OR CHEMICAL PROCESSES OR APPARATUS IN GENERAL
    • B01D—SEPARATION
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    • B01D2258/06—Polluted air
    • Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
    • Y02—TECHNOLOGIES OR APPLICATIONS FOR MITIGATION OR ADAPTATION AGAINST CLIMATE CHANGE
    • Y02C—CAPTURE, STORAGE, SEQUESTRATION OR DISPOSAL OF GREENHOUSE GASES [GHG]
    • Y02C20/00—Capture or disposal of greenhouse gases
    • Y02C20/40—Capture or disposal of greenhouse gases of CO2

Definitions

  • the present invention relates in general to non-ideal mixed-gas multilayer adsorption equilibria.
  • the present invention relates to a system and method for generalized Brunauer-Emmett-Teller isotherm for single and mixed-gas multilayer adsorption equilibria.
  • STATEMENT OF FEDERALLY FUNDED RESEARCH [0003] This invention was made with government support under DE-EE0009768 awarded by the U.S. Department of Energy’s Office of Energy Efficiency and Renewable Energy (EERE) under the Bioenergy Technologies Office. The government has certain rights in the invention.
  • Adsorptive separation is an energy efficient and sustainable alternative to conventional thermal separation.
  • Example adsorptive separation applications include CO 2 capture, hydrocarbons processing, air separation, direct air capture, trace elements and heavy metals removal, aqueous organic acids separation, among others.
  • adsorption isotherms express the adsorbate loading as a function of pressure and temperature.
  • Type I isotherms are monolayer adsorption and Type II and Type III isotherms are multilayer adsorption. In the case of monolayer adsorption, both single and mixed-gas adsorption equilibria data are relatively abundant.
  • thermodynamically consistent models such as Adsorbed Solution Theory [13] and generalized Langmuir Isotherm [14] provide powerful and rigorous thermodynamic frameworks to reliably correlate and predict mixed-gas adsorption equilibria [15-18].
  • thermodynamic models have been reported in the literature for multilayer adsorption equilibria.
  • the classical Brunauer-Emmett-Teller (BET) isotherm [20] for single component multilayer adsorption is the most prominent model that considers first layer as adsorption and second and higher layers as condensation-evaporation phenomena.
  • the BET isotherm has been widely practiced for decades, it is valid only for ideal adsorption on homogeneous adsorbent surface for single adsorbate and the calculated surface areas should be considered first approximation [21,22].
  • Guggenheim-Anderson-de Boer (GAB) isotherm enables improved representation of single component isotherm data [23].
  • the relative pressure term in the GAB model was later further scaled exponentially with the heterogeneity parameter of Freundlich and Sips isotherms [24].
  • the three-parameter BET equation also has been extended for multicomponent multilayer adsorption assuming the monolayer adsorption is controlled by the adsorbate-specific monolayer adsorption equilibrium constants and the subsequent layer adsorption is controlled by the adsorbate-specific condensation-evaporation equilibrium constants [25].
  • the extended BET equation was successfully applied to several ideal binary multilayer adsorption systems [25]. Apart from various extensions of the BET isotherm, Sircar and Myers revised Ideal Adsorbed Solution Theory (IAST) [13] for multilayer adsorption [19].
  • the generalized Langmuir (gL) isotherm further provides a rigorous thermodynamic framework for multicomponent monolayer adsorption equilibria and outperforms various empirical Langmuirian models in current practices [14,18,31]. (see also U.S. Patent Application Serial Number 18/291,981 filed on January 25, 2024).
  • gL enables reliable predictions of isosteric heat of adsorption, [32] elucidates the spreading pressure dependence in Adsorbed Solution Theory [33], and identifies the thermodynamic condition for adsorption azeotrope [34]. Looking beyond monolayer adsorption, Vyawahare et al.
  • tBET thermodynamic Brunauer–Emmett–Teller
  • the activity-based tBET isotherm provides a refined model over the concentration-based classical BET isotherm in correlating Type II and Type III isotherms for estimating adsorbent surface area.
  • a natural extension of tBET is to generalize it for multicomponent multilayer adsorption equilibria.
  • This disclosure presents a rigorous thermodynamic framework to calculate mixed-gas multilayer adsorption equilibria from single component isotherms and vapor-liquid equilibria.
  • gBET Brunauer-Emmett-Teller
  • the newly formulated isotherm considers adsorbent surface heterogeneity, competitive adsorption on the monolayer, condensation-evaporation on the subsequent layers, and adsorbed phase nonideality for the monolayer and the subsequent layers.
  • the monolayer adsorbed phase nonideality is tracked using an area-based adsorption nonrandom two-liquid activity coefficient model.
  • the adsorbed phase composition and corresponding nonideality in the subsequent layers are calculated at the dew point condition of the mixed-gas with either an equation of state or an activity coefficient model for the vapor-liquid equilibria.
  • the disclosed model is validated with six single and three binary multilayer adsorption equilibrium systems, and the model results are compared against those from classical BET isotherm for single component adsorption and Ideal Adsorbed Solution Theory for mixed-gas adsorption equilibria.
  • a computerized method for identifying an adsorption equilibrium constant and a condensation-evaporation equilibrium constant for an adsorbate component in a gas adsorption system includes providing one or more processors, a memory communicably coupled to the one or more processors and an input/output interface communicably coupled to the one or more processors.
  • the adsorption equilibrium constant and the condensation-evaporation equilibrium constant are provided to the input/output interface, and the gas adsorption system is configured using the adsorption equilibrium constant and the condensation-evaporation equilibrium constant.
  • the method produces a product using the gas adsorption system.
  • the gas adsorption system further uses a saturation adsorption loading for monolayer adsorption of the adsorbate component ( ⁇ 0 ⁇ ⁇ ), a correction factor ( ⁇ ⁇ ) and a binary interaction parameter ( ⁇ ) for the species ⁇ – species ⁇ pair.
  • the input/output interface comprises an interface to the gas adsorption system.
  • the gas absorption system comprises a CO 2 capture system, a hydrocarbon processing system, an air separation system, a direct air capture system, a trace elements and heavy metals removal system, or an aqueous organic acids separation system.
  • the gas absorption system comprises a liquid adsorption system mathematically formulated as a pseudo gas adsorption system.
  • the liquid adsorption system comprises a trace elements removal system, a heavy metals removal system, an aqueous organic separation system, or an ethanol-water separation system.
  • the adsorption equilibrium constant and the condensation-evaporation equilibrium constant are calculated using a regression analysis.
  • the gas absorption system comprises: an adsorbent comprising titanium dioxide anatase; and the adsorbate component comprises oxygen or nitrogen.
  • the gas absorption system comprises: an adsorbent comprising activated charcoal; and the adsorbate component comprises benzene or cyclohexane.
  • the gas absorption system comprises: an adsorbent comprising activated alumina; and the adsorbate component comprises water or carbon dioxide.
  • a computerized method for identifying a total amount of two or more adsorbate components absorbed for a mixed-gas multilayer adsorption system includes providing one or more processors, a memory communicably coupled to the one or more processors and an input/output interface communicably coupled to the one or more processors.
  • the one or more processors provide the total of the two or more adsorbate components (i) ⁇ ⁇ absorbed ( ⁇ ) to the input/output the mixed-gas multilayer adsorption system configured using the total amount of the two or more adsorbate components (i) absorbed ( ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ).
  • a product is produced using the mixed-gas multilayer
  • the mixed-gas multilayer absorption system comprises a CO 2 capture system, a hydrocarbons processing system, an air separation system, a direct air capture system, a trace elements and heavy metals removal system, or an aqueous organic acids separation system.
  • the mixed-gas multilayer adsorption system comprises a liquid adsorption system mathematically formulated as a pseudo gas adsorption system.
  • the liquid adsorption system comprises a trace elements removal system, a heavy metals removal system, an aqueous organic separation system, or an ethanol-water separation system.
  • the input/output interface comprises an interface to the mixed-gas multilayer adsorption system.
  • the mixed-gas multilayer adsorption system is further configured using a binary interaction parameter ( ⁇ ⁇ ), a correction factor ( ⁇ ) and a pressure (P).
  • the total amount of each of the or more adsorbate components (i) absorbed for the mixed-gas multilayer adsorption system ( ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ) are calculated using a regression analysis.
  • the one or more processors calculate the total amount of each of the two or more adsorbate components (i) absorbed for the mixed-gas multilayer adsorption system ( ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ) by: providing input data comprising a pressure (P), a gas phase mole fraction for each adsorbate component ( ⁇ ⁇ ), the total amount of the two or more adsorbate components (i) absorbed ( ⁇ ⁇ ⁇ ⁇ ), a saturation adsorption loading for monolayer adsorption of each adsorbate component an adsorption equilibrium constant each adsorbate component ( ⁇ ⁇ ⁇ ⁇ ), a condensation-evaporation equilibrium constant for each adsorbate component ( ⁇ ⁇ , ⁇ ), a binary interaction parameter ( ⁇ ⁇ ) for the species ⁇ – species ⁇ pair, an adsorbate area for each adsorbate component ( ⁇ ⁇ ), an adsorbate area for a reference
  • the mixed-gas multilayer adsorption system comprises: an adsorbent comprising titanium dioxide anatase; and the adsorbate component comprises oxygen and nitrogen.
  • the mixed-gas multilayer adsorption system comprises: an adsorbent comprising activated charcoal; and the adsorbate component comprises benzene and cyclohexane.
  • the mixed-gas multilayer adsorption system comprises: an adsorbent comprising activated alumina; and the adsorbate component comprises water and carbon dioxide.
  • a system for identifying an adsorption equilibrium constant and a condensation-evaporation equilibrium constant for an adsorbate component in a gas adsorption system includes a memory, an input/output interface, and one or more processors communicably coupled to the memory and the input/output interface.
  • the gas adsorption system is configured using the adsorption equilibrium constant and the condensation-evaporation equilibrium constant.
  • a product is produced using the gas adsorption system.
  • the gas adsorption system is further configured using a saturation adsorption loading for monolayer adsorption of the adsorbate component ( ⁇ 0 ⁇ ⁇ ), a correction factor ( ⁇ ⁇ ) and a binary interaction parameter ( ⁇ ) for the species ⁇ – species ⁇ pair.
  • the input/output interface comprises an interface to the gas adsorption system.
  • the gas absorption system comprises a CO 2 capture system, a hydrocarbon processing system, an air separation system, a direct air capture system, a trace elements and heavy metals removal system, or an aqueous organic acids separation system.
  • the gas absorption system comprises a liquid adsorption system mathematically formulated as a pseudo gas adsorption system.
  • the liquid adsorption system comprises a trace elements removal system, a heavy metals removal system, an aqueous organic separation system, or an ethanol-water separation system.
  • the adsorption equilibrium constant and the condensation-evaporation equilibrium constant are calculated using a regression analysis.
  • the adsorption equilibrium the condensation-evaporation equilibrium constant are calculated for multiple temperatures.
  • the gas absorption system comprises: an adsorbent comprising titanium dioxide anatase; and the adsorbate component comprises oxygen or nitrogen.
  • the gas absorption system comprises: an adsorbent comprising activated charcoal; and the adsorbate component comprises benzene or cyclohexane.
  • the gas absorption system comprises: an adsorbent comprising activated alumina; and the adsorbate component comprises water or carbon dioxide.
  • a system for identifying a total amount of two or more adsorbate components absorbed for a mixed-gas multilayer adsorption system includes a memory, an input/output interface, and one or more processors communicably coupled to the memory and the input/output interface.
  • the mixed-gas multilayer adsorption system is configured using the total amount of two or more adsorbate components absorbed.
  • a product is produced using the mixed-gas multilayer adsorption system.
  • the mixed-gas multilayer absorption system comprises a CO 2 capture system, a hydrocarbons processing system, an air separation system, a direct air capture system, a trace elements and heavy metals removal system, or an aqueous organic acids separation system.
  • the mixed-gas multilayer adsorption system comprises a liquid adsorption system mathematically formulated as a pseudo gas adsorption system.
  • the liquid adsorption system comprises a trace elements removal system, a heavy metals removal system, an aqueous organic separation system, or an ethanol-water separation system.
  • the input/output interface comprises an interface to the mixed-gas multilayer adsorption system.
  • the mixed-gas multilayer adsorption system is further configured using a binary interaction parameter ( ⁇ ⁇ ), a correction factor ( ⁇ ) and a pressure (P).
  • ⁇ ⁇ binary interaction parameter
  • ⁇ correction factor
  • P a pressure
  • the total amount of each of more adsorbate components (i) absorbed for the mixed-gas multilayer adsorption system are calculated using a regression analysis.
  • the one or more processors calculate the total amount of each of the two or more adsorbate components (i) absorbed for the mixed-gas multilayer adsorption system ( ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ) by: providing input data comprising a pressure (P), a gas phase mole fraction for each adsorbate component ( ⁇ ⁇ ), the total amount of the two or more adsorbate components (i) absorbed ( ⁇ ⁇ ⁇ ⁇ ), a saturation adsorption loading for monolayer adsorption of each adsorbate component an adsorption equilibrium constant for each adsorbate component ( ⁇ ⁇ ⁇ ⁇ ), a condensation-evaporation equilibrium constant for each adsorbate component ( ⁇ ⁇ , ⁇ ), a binary interaction parameter ( ⁇ ⁇ ) for the species ⁇ – species ⁇ pair, an adsorbate area for adsorbate component ( ⁇ ⁇ ), an area for a reference molecule ( ⁇ ⁇ ⁇
  • the mixed-gas multilayer adsorption system comprises: an adsorbent comprising titanium dioxide anatase; and the adsorbate component comprises oxygen and nitrogen.
  • the mixed-gas multilayer adsorption system comprises: an adsorbent comprising activated charcoal; and the adsorbate component comprises benzene and cyclohexane.
  • the mixed-gas multilayer adsorption system comprises: an adsorbent comprising activated alumina; and the adsorbate component comprises water and carbon dioxide.
  • FIG.1 is a flow chart of an algorithm of generalized Brunauer-Emmett-Teller isotherm for pure component adsorption in accordance with one embodiment of the present disclosure
  • FIG.2 depicts a schematic diagram for generalized Brunauer-Emmett-Taller isotherm for multilayer mixed-gas adsorption equilibria in accordance with one embodiment of the present disclosure
  • FIG.3 is a flow chart of an algorithm of generalized Brunauer-Emmett-Teller isotherm for multicomponent adsorption equilibria in accordance with one embodiment of the present disclosure
  • FIGS.4A-4C depict single component adsorption isotherm representation of (a) N 2 and O 2 adsorption on Ti
  • FIG. 6A-6C depict vapor-liquid equilibria of (a) O 2 (1) + N 2 (2) using Peng-Robinson equation of state at 78.2 K and experimental data at 74.70 K and 79.07 K [41,42], (b) benzene (1) + cyclohexane (2) using ideal gas equation of state and NRTL activity coefficient model at 303.15 K [44,45], and (c) H 2 O (1) + CO 2 (2) using PC-SAFT equation of state at 303.15 K [47] in accordance with one embodiment of the present disclosure; [0028] FIG.
  • FIG.10 depicts binary adsorption equilibria of H 2 O (1) + CO 2 (2) on activated alumina F- 200 at constant CO 2 partial pressure of 1.021 bar and varying relative humidity of H 2 O [26] using IAST and gBET isotherm at 303.15
  • FIG. 14A-14C depict a sensitivity analysis of the adsorbate ⁇ adsorbent interaction parameter on the isosteric enthalpy of adsorption at (FIG. 14A) ⁇ i ⁇ from ⁇ 3 to 0, (FIG. 14B) ⁇ i ⁇ from 0 to +3, and (FIG.14C) data and model estimates of isosteric enthalpy of adsorption for CH 4 at 297.0 K on BPL activated carbon, C 2 H 6 at 297.0 K on BPL activated carbon, and CO 2 at 305.95 K on NaX using gL isotherm in accordance with one embodiment of the present disclosure; [0036] FIG.
  • FIG. 15 depicts single-component adsorption isotherm of CO 2 on zeolite H-mordenite at 283.15 ⁇ 323.15 K using cL, Sips, DPL, and gL isotherms in accordance with one embodiment of the present disclosure; [0037] FIG.
  • FIGS.17A-17C depict adsorption selectivity of (FIG.17A) H 2 S (1) ⁇ CO 2 (2) at 0.156 bar, (FIG.17B) C 3 H 8 (1) ⁇ H 2 S (2) at 0.081 bar, and (FIG.
  • FIG. 18 depicts Table 7 with regressed single-component parameters for the generalized Langmuir (gL) isotherm, Table 8 with a summary of binary mixed-gas adsorption equilibrium estimation using models in current practice and the generalized Langmuir isotherm, and Table 9 with single-component generalized Langmuir isotherm parameters in accordance with one embodiment of the present disclosure;
  • FIG.19 depict binary adsorption equilibrium data and model results for (1) H 2 S (1) ⁇ CO 2 (2) at 0.156 bar, (2) C 3 H 8 (1) ⁇ H 2 S (2) at 0.081 bar, and (3) C 3 H 8 (1) ⁇ CO 2 (2) at 0.405 bar on zeolite H-mordenite at 303.15 K using the IAST, eL, Sips-LRC
  • 20A-20C depict binary adsorption equilibrium experimental data and gL model results for the adsorption of C 2 H 4 (1) ⁇ CO 2 (2) binary on molecular sieves 5A at (FIG.20A) 0.003 bar, (FIG. 20B) 0.02 bar, and (FIG. 20C) ⁇ ⁇ x′y phase diagram at 0.003, 0.011, 0.02, and 0.06 bar.
  • FIG.21B binary adsorption equilibrium of N 2 (1) ⁇ O 2 (2) at 0.1013,
  • gBET Brunauer–Emmett–Teller
  • the condensation- evaporation phenomenon for the adsorption beyond monolayer is determined at the dew point condition of the mixed gas with either an equation of state or an activity coefficient model for the vapor-liquid equilibria.
  • the disclosed model is validated with six single and three binary multilayer adsorption equilibrium systems, and the model results are compared against those from classical BET isotherm for single component adsorption and Ideal Adsorbed Solution Theory for mixed-gas adsorption equilibria.
  • gBET Brunauer-Emmett-Teller
  • BET Brunauer-Emmett-Teller
  • gBET further treats adsorption vacant sites as an integral part of the thermodynamic system for modeling and substitutes the first layer concentrations with activities to address adsorbent surface heterogeneity and adsorbate-adsorbent interactions at the first layer.
  • the difference between gBET and tBET resides in the treatment of adsorbate molecular size. The gBET derivation follows.
  • the adsorption and desorption equilibrium reaction of adsorbate component ⁇ with the vacant adsorbing site ⁇ is: ⁇ ⁇ ( ⁇ ) + ⁇ ⁇ ⁇ ⁇ ⁇ (1) where ⁇ ⁇ ⁇ represents the occupied the rates of adsorption and desorption reactions on the adsorbent surface are equal.
  • ⁇ ⁇ ⁇ ⁇ is the adsorption equilibrium constant
  • ⁇ 1 and ⁇ ⁇ 1 are the adsorption and desorption rate constants, respectively
  • ⁇ is the pressure
  • ⁇ 1 ⁇ and ⁇ 1 ⁇ are the activities of adsorbate component ⁇ and reference molecule ⁇ representative of the vacant sites on the adsorption first layer
  • ⁇ 1 ⁇ and ⁇ 1 ⁇ are the amounts adsorbed in the adsorption first layer
  • ⁇ 1 ⁇ and ⁇ 1 ⁇ are the corresponding activity coefficients calculated based on the monolayer loadings of ⁇ ⁇ and ⁇ ⁇ to be discussed later
  • ⁇ ⁇ are the molecular areas
  • ⁇ ⁇ denotes the ratio of adsorbate component i and reference molecule ⁇ .
  • the vacant site surface area with first layer ⁇ 1 ⁇ ⁇ ⁇ is the adsorption zeroth layer ⁇ 0 .
  • the adsorbed molecules are the layers.
  • the combined rate of adsorption on the vacant adsorbing sites and evaporation from the second layer must equal the combined rate of desorption from the first layer and condensation on the first layer.
  • ⁇ and ⁇ ⁇ 2 are the for second layer, respectively;
  • ⁇ 1 and ⁇ 2 denote the adsorbent surface areas covered with one layer and two layers, respectively.
  • ⁇ is the ⁇ , and should correspond to the reciprocal of the saturation pressure of the adsorbate component ⁇ , ⁇ ⁇ ⁇ ⁇ , or equivalently, the dew point pressure, ⁇ ⁇ ⁇ ⁇ , of adsorbate component ⁇ at the system temperature.
  • ⁇ is the adsorption properties of infinite series
  • the simplified relation is: 1 ⁇ ⁇ ⁇ 1 ⁇ ⁇ ⁇ ⁇ ⁇ , ⁇ + ⁇ ⁇ ⁇ 1 ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇
  • simplification of summation property gives Eq. (12).
  • the adsorption constant are functions of temperature, as shown below.
  • ⁇ ⁇ ⁇ , ⁇ 1 1 ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ exp ⁇ (18) ⁇ ⁇ ⁇ ⁇ ⁇
  • ⁇ ⁇ and ⁇ ⁇ ⁇ ⁇ , ⁇ are at reference temperature, ⁇ ⁇ ;
  • ⁇ represents the gas constant;
  • ⁇ denotes the system temperature;
  • ⁇ ⁇ , ⁇ is the enthalpy of adsorption;
  • ⁇ ⁇ , ⁇ refers to the enthalpy of condensation.
  • Input data is provided in block 102, which includes adsorption data (i.e., a pressure (P) and an adsorption isotherm ( ⁇ ⁇ ⁇ ⁇ )), adsorbate areas (i.e., an adsorbate area for the adsorbate component ( ⁇ ⁇ ) and an adsorbate area for a reference molecule ( ⁇ ⁇ )), and a saturation pressure ( ⁇ ⁇ ⁇ ⁇ ⁇ ).
  • adsorption data i.e., a pressure (P) and an adsorption isotherm ( ⁇ ⁇ ⁇ ⁇ )
  • adsorbate areas i.e., an adsorbate area for the adsorbate component ( ⁇ ⁇ ) and an adsorbate area for a reference molecule ( ⁇ ⁇ )
  • saturation pressure ⁇ ⁇ ⁇ ⁇ ⁇
  • a saturation adsorption loading for monolayer adsorption of the adsorbate s calculated in block 104 from an experimental adsorbate surface area or ⁇ ⁇ 0 i 0 ⁇ ⁇ ⁇ ⁇ ⁇ where ⁇ 0 adsorbent surface area.
  • Initial values or guesses for the gBET are provided in block 106.
  • the gBET parameters include the adsorption equilibrium constant , a correction factor ( ⁇ ⁇ ) and a binary interaction parameter ( ⁇ ⁇ ) for the species ⁇ – species ⁇ pair.
  • Initial values or guesses for an amount adsorbed of the component due to monolayer adsorption ( ⁇ ⁇ ), an amount adsorbed of the reference due to monolayer adsorption ( ⁇ ⁇ ) and a total of amounts adsorbed of the adsorbate component due to condensation on layers ( ⁇ ⁇ ⁇ ⁇ ⁇ ) are provided in block 108.
  • an adsorbed phase mole fraction the adsorbate component ( ⁇ ⁇ ) and the reference molecule ( ⁇ ⁇ ) in the monolayer only are calculated using Eqs. (39)-(40) in block 108.
  • a gas phase mole fraction for the adsorbate component ( ⁇ ⁇ ) and the reference molecule ( ⁇ ⁇ ) are calculated from area-based aNRTL activity coefficient model using Eq. (35) in block 108.
  • Eqs. (14)-(16) are solved for new values for ⁇ ⁇ , ⁇ ⁇ and ⁇ ⁇ ⁇ ⁇ ⁇ in decision block 110.
  • a total amount of adsorbate component ⁇ due to monolayer adsorption and condensation on subsequent layers ( ⁇ ⁇ ⁇ ⁇ ) is calculated in decision block 110. If the guessed values for ⁇ ⁇ , ⁇ ⁇ and ⁇ ⁇ ⁇ ⁇ ⁇ from block 106 do not match the calculated values from Eqs.
  • the process loops back to block 106 where the new values for the adsorption equilibrium constant ( ⁇ ⁇ ⁇ ⁇ ), the correction factor ( ⁇ ⁇ ) and the binary interaction parameter ( ⁇ ⁇ ) are used. If, however, Eq. (41) is minimized, as determined in decision block 112, the parameters (i.e., saturation adsorption loading for monolayer adsorption of the adsorbate component ( ⁇ 0 ⁇ ⁇ ), adsorption equilibrium constant ( ⁇ ⁇ ⁇ ⁇ ⁇ ), condensation-evaporation equilibrium constant ( ⁇ ⁇ , ⁇ ), the correction factor ( ⁇ ⁇ ) and binary interaction parameter ( ⁇ ⁇ )) are output in block 114.
  • the parameters i.e., saturation adsorption loading for monolayer adsorption of the adsorbate component ( ⁇ 0 ⁇ ⁇ ), adsorption equilibrium constant ( ⁇ ⁇ ⁇ ⁇ ⁇ ), condensation-evaporation equilibrium constant ( ⁇ ⁇ , ⁇ ), the correction factor ( ⁇ ⁇
  • the algorithm can be to model adsorption isotherms for multiple temperatures instead of a fixed system temperature as shown in FIG.1.
  • temperature (T) would be included in block 102, and new values for the adsorption equilibrium constant ( ⁇ ⁇ ⁇ ⁇ ⁇ ) and the condensation-evaporation equilibrium constant ( ⁇ ⁇ , ⁇ ) would be calculated using Eqs. (18)- (19) in block 108.
  • gBET Isotherm for Multicomponent Adsorption Equilibria [0058] Referring now to FIG. 2, the gBET isotherm formulation can be further generalized for multicomponent multilayer adsorption equilibria as shown.
  • the total occupied adsorbent surface area due to the multicomponent adsorption in the first layer 212 is: ⁇ ⁇ ⁇ 1 ⁇ ⁇ ⁇ ⁇ ⁇ )
  • ⁇ denotes the number [0061]
  • the subsequent adsorption layers 214, 216 due to condensation- evaporation of the mixed-gas can be expressed as follows: ⁇ ⁇ ( ⁇ ) + ⁇ 1 ⁇ ⁇ 2 (22)
  • ⁇ denotes the gas mixture with specific gas phase composition, ⁇ , dew point pressure, ⁇ ⁇ , and dew point liquid phase composition, ⁇ ⁇ .
  • gBET makes no assumptions on the nonideality of the condensed liquid above the monolayer since it is up to the thermodynamic model appropriately chosen for the dew point calculations.
  • one Eq. (30), ⁇ number of Eq. (31), and one Eq. (32) coupled with the aNRTL activity coefficient model need to be solved simultaneously.
  • FIG.3 a calculation algorithm of the gBET isotherm for multicomponent adsorption equilibria at a fixed system temperature is shown in accordance with one embodiment of the present disclosure.
  • Input data is provided in block 302, which includes mixed-gas adsorption data (i.e., pressure (P), gas phase mole fraction for each adsorbate component ( ⁇ ⁇ ) and adsorption isotherm for each adsorbate component ( ⁇ ⁇ ⁇ ⁇ )), pure component gBET parameters (see e.g., FIG.
  • Initial values or guesses for the gBET binary parameters are provided in block 304.
  • the gBET binary parameters include a correction factor ( ⁇ ⁇ ) and a binary interaction parameter ( ⁇ ⁇ ) for the species ⁇ – species ⁇ pair.
  • Initial values or guesses for an amount adsorbed of each adsorbate component due to monolayer adsorption ( ⁇ ⁇ ), an amount adsorbed of the reference molecule due to monolayer adsorption ( ⁇ ⁇ ) and a total of amounts adsorbed of each adsorbate component due to condensation on subsequent layers ( ⁇ ⁇ ⁇ ⁇ ⁇ ) at a given gas phase mole fraction for each adsorbate component ( ⁇ ⁇ ) and pressure (P) are provided in block 306.
  • an adsorbed phase mole fraction for each adsorbate component ( ⁇ ⁇ ) and the reference molecule ( ⁇ ⁇ ) in the monolayer only are calculated using Eqs. (39)-(40) in block 306.
  • a gas phase mole fraction for each adsorbate component ( ⁇ ⁇ ) and the reference molecule ( ⁇ ⁇ ) are calculated from area-based aNRTL activity coefficient model using Eq. (35) in block 306.
  • Eqs. (30)-(32) are solved for new values for ⁇ ⁇ , ⁇ ⁇ and ⁇ ⁇ ⁇ ⁇ in decision block 308.
  • a total amount of adsorbate component ⁇ due to monolayer adsorption and condensation on subsequent layers ( ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ) is calculated using Eq. (33) in decision block 308. If the guessed values for ⁇ ⁇ , ⁇ ⁇ and ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ from block 306 do not match the calculated values from Eqs. (30)-(32) in 308, as determined in decision block 308 (False), the process loops back to block 306 where the new values for ⁇ ⁇ , ⁇ ⁇ and ⁇ ⁇ ⁇ ⁇ ⁇ are used.
  • the binary parameters i.e., binary interaction parameter ( ⁇ ⁇ ) and correction factor ( ⁇ )
  • mixed-gas adsorption equilibria i.e., pressure (P) and adsorption isotherm for each adsorbate component ( ⁇ ⁇ ⁇ ⁇ ) at gas phase mole fraction for each adsorbate component ( ⁇ ⁇ )
  • P pressure
  • adsorption isotherm for each adsorbate component ( ⁇ ⁇ ⁇ ⁇ ) at gas phase mole fraction for each adsorbate component ( ⁇ ⁇ )
  • the algorithm can be modified to model adsorption isotherms for multiple temperatures instead of a fixed system temperature as shown in FIG.3.
  • ⁇ ⁇ and ⁇ ⁇ are the adsorbed are the interaction energies
  • ⁇ is the nonrandomness factor set to 0.3 per the NRTL convention [35]
  • ⁇ ⁇ is the binary interaction parameter for the species ⁇ – species ⁇ pair with the temperature dependence containing an entropic contribution term, ⁇ ⁇ , and an enthalpic contribution term, ⁇ h ⁇ .
  • the aNRTL model uses a symmetric reference state, i.e., ⁇ ⁇ 1 at ⁇ ⁇ 1 and ⁇ ⁇ 1 at ⁇ ⁇ 1, to address the nonideality of the monolayer.
  • the binary mixtures include O 2 (1) + N 2 (2) adsorption on TiO 2 anatase at five different compositions and 78.2 K, [36] benzene (1) + cyclohexane (2) adsorption on activated charcoal at four different compositions and 303.15 K, [37] and H 2 O (1) + CO 2 (2) adsorption on activated alumina F-200 at varying relative humidity of H 2 O and constant partial pressure of CO2 at 303.15 K [26].
  • the BET and gBET representations for the single component multilayer adsorption of the six adsorbates on the three adsorbents are shown in accordance with one embodiment of the present disclosure.
  • the regressed parameters for BET and gBET using the experimental data are reported in Table 1 and Table 2, respectively in FIG.5.
  • the BET isotherm has three adjustable parameters ( ⁇ 0 ⁇ ⁇ , ⁇ ⁇ ⁇ ⁇ , and ⁇ ⁇ , ⁇ ) while the gBET isotherm has four model parameters ( ⁇ 0 ⁇ ⁇ , ⁇ ⁇ ⁇ ⁇ , ⁇ ⁇ , ⁇ , and ⁇ ⁇ ).
  • FIG.4A shows the adsorption of both O 2 and N 2 on TiO 2 anatase at 78.2 K exhibits a rapid increase in adsorption loading as the pressure approaches the corresponding saturation pressures, typical of multilayer adsorption isotherm behavior.
  • Both BET and gBET give similar representations and capture the experimental data above ⁇ 0.01 bar. At pressures below ⁇ 0.01 bar, the BET isotherm significantly deviates from the experimental data while the gBET isotherm accurately represents the data, although both the BET and gBET isotherms obey Henry’s law in the low-pressure region with slope of unity in the log-log scale.
  • Table 1 and Table 2 in FIG. 5 show the ARD%’s from gBET are less than 9% and 6% for O 2 and N 2 , respectively, while the ARD%’s from BET can be as high as 23% and 13% for O 2 and N 2 , respectively.
  • the BET and gBET model parameters are nearly identical for the saturation loading, ⁇ 0 ⁇ ⁇ , and the condensation- evaporation constant, ⁇ ⁇ , ⁇ , while the BET and gBET model parameters are significantly different for the adsorption equilibrium constant, ⁇ ⁇ ⁇ ⁇ ⁇ . This difference is due to the fact that the ⁇ ⁇ in gBET together with the adsorbate-adsorbent interaction parameter, ⁇ ⁇ , take into account the adsorbent surface heterogeneity and the monolayer adsorbed phase nonideality. Table 1 and Table 2 in FIG.
  • FIG.4B shows BET qualitatively represents the benzene and cyclohexane isotherm data with the ARD%’s larger than 6% while gBET quantitively matches the experimental data for the complete range of pressure with the ARD%’s less than 1.4%.
  • Table 1 and Table 2 in FIG. 5 show the ⁇ ⁇ ⁇ ⁇ ⁇ parameter values from BET are significantly higher than those for the ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ parameters from gBET.
  • the ⁇ 0 ⁇ ⁇ parameter and the ⁇ ⁇ , ⁇ parameter values from both BET and gBET are similar.
  • the ⁇ is found to be ⁇ 3 for BET and ⁇ 4 to 5 for gBET.
  • FIG. 4C shows the adsorption isotherms of water and carbon dioxide on activated alumina F-200 at 303.15 K together with the BET and gBET model results.
  • the model parameters are reported in Table 1 and Table 2 in FIG.5.
  • FIG. 4C shows the Type II multilayer adsorption isotherm data for H 2 O where the adsorption loading rapidly increases with the increase in H 2 O partial pressure.
  • the adsorption data for CO 2 suggest monolayer adsorption where the isotherm exhibits concave down shape.
  • Both BET and gBET result in satisfactory fitting for H 2 O adsorption isotherm data with the ARD%’s around 5%.
  • the ⁇ ⁇ parameter value for gBET is zero and the ⁇ 0 and ⁇ value ⁇ ⁇ ⁇ s for both BET and gBET are
  • the ⁇ ⁇ parameter value from gBET is the ⁇ ⁇ ⁇ ⁇ parameter from BET due to the inclusion of adsorbate size in the gBET isotherm.
  • the corresponding ⁇ value is ⁇ 1.2 for both BET and gBET.
  • gBET provides better representation of the data than BET.
  • the ARD% is 2.4% for gBET and 5.0% for BET.
  • the ⁇ ⁇ value for gBET is –2.49 indicative of nonideal monolayer adsorbed phase.
  • the ⁇ 0 value BET is almost half of the ⁇ 0 ⁇ ⁇ value from while the ⁇ ⁇ ⁇ ⁇ value from BET is ten times of the ⁇ ⁇ ⁇ ⁇ value from gBET. Due to the absence of multilayer adsorption behavior for CO 2 , the factor ⁇ is set to unity for both BET and gBET. [0076] Dew Point Calculations [0077] Referring now to FIGS.6A-6C, the vapor-liquid equilibria of the three pairs of adsorbates are shown in accordance with one embodiment of the present disclosure.
  • the gBET isotherm for multicomponent adsorption requires information about the dew point pressure, ⁇ ⁇ , and the dew point liquid phase composition, ⁇ ⁇ , of the mixed-gas at the system temperature.
  • the dew point calculations for the three binary mixtures have been carried out using Aspen Properties V14 39 .
  • the choice of thermodynamic models depends on the type of components in the multicomponent mixtures and the thermodynamic conditions. These thermodynamic models are also used to calculate the ⁇ ⁇ ⁇ ⁇ ⁇ ’s required for the single component adsorption isotherm calculations previously described.
  • the benzene (1) + cyclohexane (2) binary mixture is modeled using the ideal gas equation of state for the vapor phase and the nonrandom two-liquid (NRTL) activity coefficient 35, 39 model for the liquid phase solution nonideality.
  • the model representation with experimental data is presented in FIG. 6B.
  • the binary mixture of benzene (1) + cyclohexane (2) shows a maximum pressure azeotrope at the vapor and liquid phase mole fractions of benzene around 0.5. It suggests that the selectivity shall be reversed as the vapor phase mole fraction of benzene crosses the azeotropic composition.
  • the H 2 O (1) + CO 2 (2) binary system is modeled using Perturbed-Chain Statistical Associating Fluid Theory (PC-SAFT) of Gross and Sadowski 39, 46 .
  • Yan and Chen 47 accurately regressed and validated the PC-SAFT model parameters for the binary mixture that remain valid at high CO 2 pressures.
  • FIG.6C shows the VLE results based on the PC-SAFT model parameters from Yan and Chen 47 .
  • Mixed-gas Adsorption Equilibria [0082] The mixed-gas adsorption equilibria for the three binary adsorption systems with varying gas phase compositions are calculated with both IAST and the gBET isotherm. The IAST results were calculated from the BET isotherm with the BET parameters reported in Table 1 in FIG.5 for single component adsorption. The gBET results were calculated with the gBET parameters reported in Table 2 in FIG.5 for single component adsorption.
  • the factor ⁇ relating ⁇ ⁇ , ⁇ and ⁇ ⁇ ⁇ ⁇ has been treated as an adjustable parameter in fitting the single component adsorption isotherm data. Therefore, the factor ⁇ relating ⁇ ⁇ and ⁇ ⁇ for the mixed-gas adsorption gBET isotherm is also treated as an adjustable parameter and regressed from the binary adsorption equilibria data along with the binary aNRTL interaction parameter ⁇ 12 .
  • the factor ⁇ determined from the binary adsorption equilibria data is same or close to that determined from single component adsorption isotherm data for the same adsorbent.
  • FIG.8A to FIG.8E show that IAST consistently overpredicts the O2 loading and underpredicts the N2 loading as ⁇ 1 increases from 0.149 to 0.853.
  • Table 2 in FIG. 5 and Table 3 in FIG.7 show the factor ⁇ remains the same for the same adsorbent whether for single or multicomponent adsorption. Table 3 in FIG.
  • FIGS. 9A-9D the model results for the adsorption of benzene (1) + cyclohexane (2) on activated charcoal at 303.15 K up to 0.16 bar and at four different gas phase compositions are shown in accordance with one embodiment of the present disclosure. Note that the literature 37 reported the total adsorption loading in mass-basis and the gas phase composition in mole-basis. No individual component loadings data were reported. FIG. 9A to FIG.
  • IAST predicts an adsorption selectivity switching from cyclohexane to benzene at ⁇ 1 ⁇ 0.3 while gBET predicts the adsorption selectivity switch at around the azeotrope point, i.e., ⁇ 1 ⁇ 0.5.
  • Table 3 in FIG.7 shows that the ARD% are 3.9% for IAST, 1.9% for gBET with ⁇ 12 set to 0, and 1.5% for gBET with ⁇ 12 adjusted.
  • the adsorption of CO 2 from the binary mixture of H 2 O (1) + CO 2 (2) is of high interest due to the immense interest in adsorption for CO 2 capture applications.
  • FIG.8 shows the experimental data 26 and the modeling results for the binary adsorption of H 2 O (1) + CO 2 (2) on activated alumina F-200 at 303.15 K, constant CO2 partial pressure of 1.012 bar, and variable H2O relative humidity in accordance with one embodiment of the present disclosure.
  • the experimental data shows a rapid increase in the H 2 O loading consistent with multilayer adsorption while the CO 2 loading remains relatively constant and independent of the relative humidity.
  • FIG. 10 shows the IAST results are available for only up to 37% relative humidity. Beyond this range, the IAST calculations fail because the constant spreading pressure constraint for the adsorbates cannot be satisfied due to the mismatch in the high spreading pressure for H 2 O and the low spreading pressure for CO 2 for the system. Previous studies have reported similar failures and the related challenges in applying IAST for multilayer adsorption. 26 Furthermore, inconsistent with the experimental data, IAST predicts rapid increase in the H 2 O loading and rapid drop in the CO 2 loading with increasing relative humidity.
  • the ARD% for gBET is 44.0% if ⁇ 12 is set to 0. Note that ⁇ was manually adjusted to 1.13 to optimize the fit and ⁇ 12 was regressed from the CO 2 loading data alone. The high ⁇ 12 value of 6.30 suggests strong nonideality in the monolayer adsorbed phase.
  • the ARD% for IAST is not reported in Table 3 in FIG. 7 due to the failure in computing the binary adsorption above 37% relative humidity. [0086] Now referring to FIG.
  • the system 1100 for identifying an adsorption equilibrium constant and a condensation-evaporation equilibrium constant for an adsorbate component in a gas adsorption system 1102 includes a memory 1104, an input/output interface 1106, and one or more processors 1108 communicably coupled to the memory 1104 and the input/output interface 1106.
  • ⁇ ⁇ is the adsorption constant at a reference temperature ( ⁇ ⁇ )
  • ⁇ ⁇ ⁇ ⁇ , ⁇ the condensation-evaporation equilibrium constant the reference temperature
  • ⁇ is a gas constant
  • ⁇ is a system temperature
  • ⁇ ⁇ , ⁇ is an enthalpy of adsorption
  • ⁇ ⁇ , ⁇ is an enthalpy of condensation
  • the gas adsorption system is configured using the adsorption equilibrium constant and the condensation-evaporation equilibrium constant.
  • the one or more processors 1108 can be part of one or more controller, computers, servers or other devices suitable for performing the method.
  • the memory 1104 can be any type of data storage.
  • the input/output interface 1106 can be any component capable of interfacing with the one or more processors.
  • the components can be local, remote or a combination thereof.
  • the components can be part of a distributed computing architecture, design system or control system.
  • the gas absorption system 1102 comprises a CO 2 capture system, a hydrocarbon processing system, an air separation system, a direct air capture system, a trace elements and heavy metals removal system, or an aqueous organic acids separation system.
  • the gas absorption system 1102 comprises a liquid adsorption system mathematically formulated as a pseudo gas adsorption system.
  • the liquid adsorption system comprises a trace elements removal system, a heavy metals removal system, an aqueous organic separation system, or an ethanol-water separation system.
  • the input/output interface 1106 comprises an interface directly or indirectly to the gas adsorption system 1102. [0087] In one aspect, a product is produced using the gas adsorption system 1102.
  • the gas adsorption system 1102 is further configured using a saturation adsorption loading for monolayer adsorption of the adsorbate component ( ⁇ 0 ⁇ ⁇ ), a correction factor ( ⁇ ⁇ ) and a binary interaction parameter ( ⁇ ⁇ ) for the species ⁇ – species ⁇ pair.
  • the adsorption equilibrium constant condensation-evaporation equilibrium constant are calculated using a regression analysis.
  • the adsorption constant and the condensation-evaporation equilibrium constant are calculated for multiple temperatures.
  • the gas absorption system 1102 comprises: an adsorbent comprising titanium dioxide anatase; and the adsorbate component comprises oxygen or nitrogen.
  • the gas absorption system 1102 comprises: an adsorbent comprising activated charcoal; and the adsorbate component comprises benzene or cyclohexane.
  • the gas absorption system 1102 comprises: an adsorbent comprising activated alumina; and the adsorbate component comprises water or carbon dioxide.
  • the system 1150 for identifying a total amount of two or more adsorbate components absorbed for a mixed-gas multilayer adsorption system 1102 includes a memory 1104, an input/output interface 1106, and one or more processors 1108 communicably coupled to the memory 1104 and the input/output interface 1106.
  • ⁇ is an amount adsorbed of each adsorbate component ( ⁇ ) due to monolayer adsorption
  • ⁇ ⁇ is a total of amounts adsorbed due to condensation on subsequent layers
  • ⁇ ⁇ ⁇ ⁇ ⁇ is a dew point liquid phase composition
  • the mixed-gas multilayer adsorption system 1102 is configured using the total amount of two or more adsorbate components absorbed.
  • the one or more processors 1108 can be part of one or more controller, computers, servers or other devices suitable for performing the method.
  • the memory 1104 can be any type of data storage.
  • the input/output interface 1106 can be any component capable of interfacing with the one or more processors.
  • the components can be local, remote or a combination thereof. Moreover, the components can be part of a distributed computing architecture, design system or control system.
  • the mixed-gas multilayer adsorption system 1102 comprises a CO 2 capture system, a hydrocarbon processing system, an air separation system, a direct air capture system, a trace elements and heavy metals removal system, or an aqueous organic acids separation system.
  • the mixed-gas multilayer adsorption system 1102 comprises a liquid adsorption system mathematically formulated as a pseudo gas adsorption system.
  • the liquid adsorption system comprises a trace elements removal system, a heavy metals removal system, an aqueous organic separation system, or an ethanol-water separation system.
  • the input/output interface 1106 comprises an interface directly or indirectly to the mixed-gas multilayer adsorption system 1102.
  • a product is produced using the mixed-gas multilayer adsorption system 1102.
  • the mixed-gas multilayer adsorption system 1102 is further configured using a binary interaction parameter ( ⁇ ⁇ ), a correction factor ( ⁇ ) and a pressure (P).
  • the total amount of each of two or more adsorbate components (i) absorbed for the mixed-gas multilayer adsorption system ( ⁇ ⁇ ⁇ ⁇ ) are calculated using a regression analysis.
  • the one or more processors 1108 calculate the total amount of each of the two or more adsorbate components (i) absorbed for the mixed-gas multilayer adsorption system ( ⁇ ⁇ ⁇ ⁇ ) by: providing input data comprising a pressure (P), a gas phase mole fraction for each adsorbate component ( ⁇ ⁇ ), the total amount of the two or more adsorbate components (i) absorbed ( ⁇ ⁇ ⁇ ⁇ ), a saturation adsorption loading for monolayer adsorption of each adsorbate , an adsorption equilibrium constant for each adsorbate component ( ⁇ ⁇ ⁇ ⁇ ⁇ ), a condensation-evaporation equilibrium constant for each adsorbate component ( ⁇ ⁇ , ⁇ ), a binary interaction
  • the total amount of more adsorbate components (i) absorbed for the mixed-gas multilayer adsorption system ( ⁇ ⁇ ⁇ ⁇ ) are calculated for multiple temperatures.
  • the mixed-gas multilayer adsorption system 1102 comprises: an adsorbent comprising titanium dioxide anatase; and the adsorbate component comprises oxygen and nitrogen.
  • the mixed-gas multilayer adsorption system 1102 comprises: an adsorbent comprising activated charcoal; and the adsorbate component comprises benzene and cyclohexane.
  • the mixed-gas multilayer adsorption system 1102 comprises: an adsorbent comprising activated alumina; and the adsorbate component comprises water and carbon dioxide.
  • FIG.12 a flowchart of a computerized method 1200 for identifying an adsorption equilibrium constant and a condensation-evaporation equilibrium constant for an adsorbate component in a gas adsorption system is shown in accordance with one embodiment of the present disclosure.
  • the computerized method 1200 for identifying an adsorption equilibrium constant and a condensation-evaporation equilibrium constant for an adsorbate component in a gas adsorption system includes providing one or more processors, a memory communicably coupled to the one or more processors and an input/output interface communicably coupled to the one or more processors in block 1202.
  • the one or more processors equilibrium constant ( ⁇ ) for the adsorbate compo ⁇ ⁇ , ⁇ 1 1 ⁇ , ⁇ nent (i) using ⁇ , ⁇ ⁇ , ⁇ exp ⁇ ⁇ ⁇ ⁇ ⁇ (see Eq. ⁇ (19)), wherein: ⁇ is the adsorption constant ⁇ ⁇ , ⁇ is condensation- equilibrium constant the reference temperature ( ⁇ ), ⁇ is a gas constant, ⁇ is a system temperature, ⁇ , ⁇ is an enthalpy of adsorption, and ⁇ , ⁇ is an enthalpy of condensation in block 1206.
  • the adsorption equilibrium constant and the condensation-evaporation equilibrium constant are provided to the input/output interface in block 1208, and the gas adsorption system is configured using the adsorption equilibrium constant and the condensation-evaporation equilibrium constant in block 1210.
  • the one or more processors can be part of one or more controller, computers, servers or other devices suitable for performing the method.
  • the memory can be any type of data storage.
  • the input/output interface can be any component capable of interfacing with the one or more processors.
  • the components can be local, remote or a combination thereof.
  • the components can be part of a distributed computing architecture, design system or control system.
  • the gas absorption system comprises a CO 2 capture system, a hydrocarbon processing system, an air separation system, a direct air capture system, a trace elements and heavy metals removal system, or an aqueous organic acids separation system.
  • the gas absorption system comprises a liquid adsorption system mathematically formulated as a pseudo gas adsorption system.
  • the liquid adsorption system comprises a trace elements removal system, a heavy metals removal system, an aqueous organic separation system, or an ethanol-water separation system.
  • the input/output interface comprises an interface directly or indirectly to the gas adsorption system. [0091] In one aspect, the method produces a product using the gas adsorption system.
  • the gas adsorption system further uses a saturation adsorption loading for monolayer adsorption of the adsorbate component ( ⁇ 0 ⁇ ⁇ ), a correction factor ( ⁇ ⁇ ) and a binary interaction parameter ( ⁇ ⁇ ) for the species ⁇ – species ⁇ pair.
  • a saturation adsorption loading for monolayer adsorption of the adsorbate component ⁇ 0 ⁇ ⁇
  • a correction factor ⁇ ⁇
  • ⁇ ⁇ binary interaction parameter
  • the gas absorption system comprises: an adsorbent comprising titanium dioxide anatase; and the adsorbate component comprises oxygen or nitrogen.
  • the gas absorption system comprises: an adsorbent comprising activated charcoal; and the adsorbate component comprises benzene or cyclohexane.
  • the gas absorption system comprises: an adsorbent comprising activated alumina; and the adsorbate component comprises water or carbon dioxide.
  • the foregoing computerized method 1200 can be implemented as a non-transitory computer readable medium containing program instructions that cause the one or more processors to perform the foregoing computerized method 1200.
  • FIG. 13 a flowchart of a computerized method 1300 for identifying a total amount of two or more adsorbate components absorbed for a mixed-gas multilayer adsorption system is shown in accordance with one embodiment of the present disclosure.
  • the computerized method 1300 includes providing one or more processors, a memory communicably coupled to the one or more processors and an input/output interface communicably coupled to the one or more processors in block 1302.
  • the one or more processors provide the total amount of the two or more adsorbate components (i) absorbed ( ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ) to the input/output interface in block 1306, and the mixed-gas multilayer adsorption system is configured using the total amount of the two or more adsorbate components (i) absorbed ( ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ) in block 1308.
  • the one or more processors can be part of one or more controller, computers, servers or other devices suitable for performing the method.
  • the memory can be any type of data storage.
  • the input/output interface can be any component capable of interfacing with the one or more processors.
  • the components can be local, remote or a combination thereof.
  • the components can be part of a distributed computing architecture, design system or control system.
  • the mixed-gas multilayer adsorption system comprises a CO 2 capture system, a hydrocarbon processing system, an air separation system, a direct air capture system, a trace elements and heavy metals removal system, or an aqueous organic acids separation system.
  • the mixed-gas multilayer adsorption system comprises a liquid adsorption system mathematically formulated as a pseudo gas adsorption system.
  • the liquid adsorption system comprises a trace elements removal system, a heavy metals removal system, an aqueous organic separation system, or an ethanol-water separation system.
  • the input/output interface comprises an interface directly or indirectly to the mixed-gas multilayer adsorption system.
  • a product is produced using the mixed-gas multilayer adsorption system.
  • the mixed-gas multilayer adsorption system is further configured using a binary interaction parameter ( ⁇ ⁇ ), a correction factor ( ⁇ ) and a pressure (P).
  • ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ) the total amount of each of more adsorbate components (i) absorbed for the mixed-gas multilayer adsorption system ( ⁇ ⁇ ⁇ ⁇ ) are calculated using a regression analysis.
  • the one or more processors calculate the total amount of each of the two or more adsorbate components (i) absorbed for the mixed-gas multilayer adsorption system ( ⁇ ⁇ ⁇ ⁇ ) by: providing input data comprising a pressure (P), a gas phase mole fraction for each adsorbate component ( ⁇ ⁇ ), the total amount of the two or more adsorbate components (i) absorbed ( ⁇ ⁇ ⁇ ⁇ ), a saturation adsorption loading for monolayer adsorption of each adsorbate component ( ⁇ ⁇ 0 ⁇ ), an adsorption equilibrium constant for each adsorbate component ( ⁇ ⁇ ⁇ ⁇ ⁇ ), a condensation-evaporation equilibrium constant for each adsorbate component ( ⁇ ⁇ , ⁇ ), a interaction parameter ( ⁇ ⁇ ) for the species ⁇ – species ⁇ pair, an adsorbate area for each component ( ⁇ ⁇ ), an adsorbate area for a reference
  • the total amount of adsorbate components (i) absorbed for the mixed-gas multilayer are calculated for multiple temperatures.
  • the mixed- multilayer adsorption system comprises: an adsorbent comprising titanium dioxide anatase; and the adsorbate component comprises oxygen and nitrogen.
  • the mixed-gas multilayer adsorption system comprises: an adsorbent comprising activated charcoal; and the adsorbate component comprises benzene and cyclohexane.
  • the mixed-gas multilayer adsorption system comprises: an adsorbent comprising activated alumina; and the adsorbate component comprises water and carbon dioxide.
  • the foregoing computerized method 1300 can be implemented as a non-transitory computer readable medium containing program instructions that cause the one or more processors to perform the foregoing computerized method 1300.
  • the thermodynamic Brunauer-Emmett-Teller isotherm for single component multilayer adsorption isotherm has been extended for multicomponent multilayer adsorption equilibria.
  • the generalized Brunauer- Emmett-Teller isotherm provides the first rigorous and comprehensive thermodynamic framework for multicomponent multilayer adsorption equilibria.
  • the gBET isotherm model parameters include the three adsorbate-specific Langmuir isotherm parameters for the monolayer adsorption: the saturation loading ⁇ 0 ⁇ ⁇ , the adsorption equilibrium constant ⁇ ⁇ ⁇ ⁇ , and the adsorbate-adsorbent binary interaction parameter ⁇ ⁇ .
  • the adsorbate-specific condensation-evaporation equilibrium constant ⁇ from adsorbate saturation pressure and a correction factor ⁇ is required for the condensation on the subsequent layers.
  • the mixed-gas condensation- evaporation equilibrium constant ⁇ ⁇ is calculated from the dew point pressure of the mixed-gas and ⁇ .
  • One adsorbate-adsorbate binary interaction parameter ⁇ per ⁇ ⁇ ⁇ pair may be introduced to further account for the adsorbed monolayer phase nonideality.
  • the resulting gBET isotherm further enables tracking of adsorbent vacant sites, loading and composition of monolayer adsorbed phase, and loading and composition of adsorbed phase beyond monolayer. Future studies should explore applications of gBET for various industrial multicomponent multilayer adsorption systems.
  • the generalized Langmuir isotherm for monolayer adsorption and the generalized Brunauer ⁇ Emmett ⁇ Teller isotherm for multilayer adsorption address various thermodynamic modeling challenges including adsorbent surface heterogeneity, isosteric enthalpies of adsorption, BET surface areas, adsorbed phase nonideality, adsorption azeotrope formation, and multilayer adsorption. Also discussed below is the importance of quality adsorption data that cover sufficient temperature, pressure, and composition ranges for reliable determination of the model parameters to support adsorption process simulation, design, and optimization.
  • Isosteric Enthalpies of Adsorption play an integral part in the mass and energy balances of adsorption units and, together with adsorption isotherms, determine process performance and product purity. Isosteric enthalpies of adsorption for homogeneous adsorbents do not change with adsorption loading, while isosteric enthalpies of adsorption for heterogeneous adsorbents decrease with increase in adsorption loading.
  • gL expresses the loading dependence through the binary interaction parameter ⁇ i ⁇ for absorbed phase activity coefficients.
  • the gL expression for the isosteric enthalpy of adsorption of adsorbate component i is shown in Eq. (43).
  • ⁇ ⁇ ( ⁇ , ⁇ ) ⁇ ⁇ ln ⁇ ⁇ ln ⁇ ⁇ , ⁇ + ⁇ 2 ⁇ ⁇ , ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ (43) ⁇ ⁇
  • ⁇ ln ⁇ ⁇ ln ⁇ ⁇ and ⁇ equals zero, the adsorbent surface is homogeneous while nonzero ⁇ i ⁇ adsorbent surface heterogeneity.
  • FIGS.14A-14C are adapted from [32].
  • FIG.14A shows how the predicted isosteric enthalpy of adsorption varies with ⁇ i ⁇ .
  • FIG.14A shows ⁇ ⁇ , ⁇ is a strong concave-up function of loading at low
  • FIG.14B shows ⁇ ⁇ , ⁇ is a strong concave down function of loading at high loadings.
  • BET Surface Areas BET surface areas of specific adsorbents are widely determined from the two-parameter BET equation for various physical chemistry and materials science applications. Consistent with concerns on adsorption isotherm reproducibility in the literature, [51] reported BET surface areas often suffer reproducibility issues.
  • the source of discrepancies in reported BET surface areas can be attributed to inability of the BET isotherm to represent the complete adsorption isotherm and reliance solely on a linear fit to data over the P/P sat range from 0.05 to 0.3 for adsorbent surface area calculations.
  • a recent study prepared jointly by more than one hundred adsorption experts reported that multiple linear fit solutions to BET are possible and can dramatically alter the calculated adsorbent surface area.
  • One such example is the adsorbent surface areas for NU-1104 MOF reported by 61 different laboratories, which vary from the lowest estimate of 1757 m 2 /g to the highest estimate of 9341 m 2 /g, or a factor of ⁇ 5.
  • FIG. 15 Single-component isotherm data for CO 2 at 283.15, 303.15, and 323.15 K are shown in FIG. 15 along with model results from cL, Sips, DPL, and gL.
  • FIG.15 is adapted from [18] and shows that cL fails to represent CO 2 isotherm data, Sips fails to follow Henry’s Law behavior at low pressures, and both DPL and gL provide reasonable representations of the isotherm data and fidelity to Henry’s Law at low pressures.
  • the regressed isotherm parameters for CO 2 , H 2 S, and C 3 H 8 are reported in Tables 4 through Table 6 in FIG.16 and Table 7 in FIG.18. [00105] FIGS.
  • FIG.17A-17C are adapted from [14] where symbols are data and lines are model results.
  • FIG.17A shows that for the H 2 S (1) ⁇ CO 2 (2) binary adsorption at 0.156 bar, IAST and eL qualitatively represent the adsorption equilibrium data, [52] although both slightly underestimate ⁇ 1 ′ with ⁇ 1 ⁇ 0.2 and overestimate ⁇ 1 ′ with ⁇ 1 > 0.2.
  • Both the DPL and Sips-LRC predictions deviate significantly from experimental data while Sips-LRC also incorrectly predicts an adsorption azeotrope.
  • FIG.17B and FIG.17C further show that, for the C 3 H 8 (1) ⁇ H 2 S (2) binary at 0.081 bar and the C 3 H 8 (1) ⁇ CO 2 (2) binary at 0.405 bar, IAST and eL both predict ideal adsorption, inconsistent with the experimental data.
  • Sips-LRC and DPL provide qualitative representations of the data and correctly predict azeotrope formation.
  • the gL isotherm accurately correlates the C 3 H 8 (1) ⁇ H 2 S (2) and the C 3 H 8 (1) ⁇ CO 2 (2) binary adsorption data including azeotropes with ⁇ 12 adjusted to 10.96 and 14.24, respectively.
  • Adsorption Azeotropes form when the equilibrium gas phase and adsorbed phase compositions, ⁇ ⁇ and ⁇ ⁇ ′ ⁇ ⁇ , respectively, become equal, causing the equilibrium line to cross the 45° line on the ⁇ ⁇ ′ ⁇ ⁇ - ⁇ ⁇ plot. Although adsorption azeotropes can be predicted or correlated with various isotherm models, the exact thermodynamic condition for adsorption azeotropes remained elusive.
  • thermodynamic azeotropes is analogous to the thermodynamic condition for azeotrope formation in vapor ⁇ liquid equilibrium shown below.
  • ⁇ ⁇ ⁇ sat ⁇ ⁇ sat 1 1 2 ⁇ 2 (48)
  • ⁇ 1 sat and ⁇ 2 sat are the saturation pressures of the two components (1) and (2) in vapor ⁇ liquid equilibrium.
  • FIGS.20A-20C are adapted from [14] where symbols are data and lines are model results.
  • FIG. 20C further shows the corresponding ⁇ ⁇ x′ y phase diagram and, the azeotropes at 0.02 and 0.06 bar are maximum surface coverage azeotropes.
  • the single-component isotherm parameters are reported in Table 9 in FIG. 18. A comprehensive study on the thermodynamic condition for azeotrope formation is available elsewhere. [34] [00113] Generalized Langmuir Isotherm and Real Adsorbed Solution Theory.
  • both gL and RAST have been designed to model nonideal mixed-gas adsorption equilibrium.
  • gL treats the adsorbed phase as a ternary system consisting of occupied sites with adsorbate component (1), occupied sites with adsorbate component (2), and phantom molecule ⁇ for vacant sites.
  • the reference states for the three components are chosen to be saturated occupied sites with adsorbate component (1), saturated occupied sites with adsorbate component (2), and fully vacant sites, respectively.
  • the adsorbed phase nonideality is then characterized with three aNRTL binary interaction parameters: ⁇ 1 ⁇ , ⁇ 2 ⁇ , and ⁇ 12 .
  • ⁇ 1 ⁇ and ⁇ 2 ⁇ are to be determined from single-component adsorption isotherms, and ⁇ 12 is the only parameter to be identified from binary gas adsorption equilibrium data.
  • RAST treats the adsorbed phase as a binary system consist of occupied sites with adsorbate component (1) and occupied sites with adsorbate component (2).
  • the reference states are the single-component adsorption systems of adsorbate component (1) and adsorbate component (2) at the same system “spreading pressure” as that of the binary system. As shown in Eq. (49), “spreading pressure” can be viewed as a measure of vacant sites surface fraction ⁇ ⁇ as shown below.
  • FIGS.21A-21C are adapted from [33].
  • FIG.21A shows that the single-component isotherms of N 2 and O 2 at 303.15 K [33,43,[54] are accurately represented by the gL isotherm.
  • the same single-component gL isotherms of N 2 and O 2 were then used in both the gL and RAST thermodynamic frameworks for mixed-gas adsorption equilibrium.
  • FIG.21B shows that the gL correlation at 6.078 bar and the gL predictions at 0.1013, 1.013, and 10.0 bar for the binary adsorption system accurately match the RAST-SPD-aNRTL results for both adsorbate components.
  • FIG. 21C further shows that the SPD-aNRTL binary interaction parameter ⁇ 1 ′ 2 approaches zero at system pressures of 0.1013 and 1.013 bar, indicative of an ideal adsorbed phase at low spreading pressures, i.e., ⁇ ⁇ 1 mmol/g. Note that, at a given system pressure, there is a range of spreading pressure reflecting the variation in the mixed-gas composition and the loading.
  • Multicomponent Multilayer Adsorption Equilibrium Multicomponent Multilayer adsorption equilibrium may happen with mixed-gas adsorption systems containing condensable adsorbate species.
  • FIGS. 22A-22F are adapted from [55] where symbols are data and lines are model results.
  • FIG.22A shows the single component Type II adsorption isotherms of O 2 and N 2 on anatase at 78.2 K 51 with BET and gBET isotherm representations.
  • both the BET and gBET isotherms provide consistent representation at high pressures, the BET isotherm deviates from the isotherm data at pressures below 10 ⁇ 2 bar yielding average relative deviation of 23.4% for O 2 and 12.9% for N 2 .
  • the gBET isotherm accurately represents the isotherm data at all pressures, and results in average relative deviation of 8.5% and 5.5% for O 2 and N 2 , respectively.
  • IAST consistently overpredicts the O 2 adsorption loading and underpredicts the N 2 loading for all mixed-gas phase compositions.
  • the overall average relative deviation for the IAST predictions is 61.3%.
  • gBET reliably predicts the experimental adsorption loading data with an overall average relative deviation of 13.9%.
  • the Peng ⁇ Robinson equation of state predictions for the dew point liquid phase composition are shown in FIG.23, which is adapted from [55] where symbols are data and lines are model results.
  • the corresponding single-component BET and gBET isotherm parameters are given in Table 1 and Table 2.
  • Adsorption Equilibrium Data Successful development and validation of rigorous and predictive thermodynamic models for multicomponent adsorption equilibrium, monolayer or multilayer, ultimately depend on the availability of quality experimental adsorption equilibrium data. Ideally, data for single-component adsorption isotherms for the complete range of pressure at multiple temperatures are needed for proper identification of gL and gBET model parameters.
  • Type I adsorption isotherms the single-component adsorption isotherms must have data points at low pressures, i.e., the Henry’s Law region, medium pressures, i.e., the concave down region, and high pressures to properly determine the adsorption equilibrium constant, the aNRTL binary parameter for adsorbate ⁇ adsorbent interactions, and the saturation loading.
  • additional isotherm data for the concave up region showing the condensation-evaporation behavior are required to identify the condensation- evaporation equilibrium constant.
  • Data for adsorption isotherms at multiple temperatures, although relatively rare, are exceedingly useful to help validate the isotherm data at individual temperatures, and from the Clausius ⁇ Clapeyron equation, to estimate the isosteric enthalpy of adsorption as a function of loading.
  • data on isosteric enthalpy of adsorption as a function of loading would facilitate reliable estimation of adsorption isotherms over a range of temperature.
  • data for binary gas adsorption equilibrium over reasonable ranges of pressure and gas composition are essential for validation of gL/gBET predictions for ideal mixed-gas adsorption equilibrium, and for proper determination of the aNRTL binary interaction parameters for nonideal mixed-gas adsorption equilibrium.
  • thermodynamic consistency of data While treating the experimental data, one must properly consider data uncertainty, consistency of data from different sources, and thermodynamic consistency of data for different properties. With availability of quality data for single and binary gas adsorption equilibrium, reliable predictions for multicomponent adsorption equilibrium, monolayer or multilayer, are possible. [00119] Accurate, rigorous, and predictive thermodynamic models serve as key scientific foundations for simulation, design, and optimization of industrial chemical processes. For decades, advances in adsorptive separations have been hampered by a lack of rigorous adsorption thermodynamic models for multicomponent adsorption equilibrium.
  • thermodynamic modeling challenges such as adsorbent surface heterogeneity and adsorption isotherm, isosteric enthalpies of adsorption vs loadings, BET surface areas for heterogeneous adsorbents, adsorbed phase nonideality and multicomponent competitive adsorption, thermodynamic condition for adsorption azeotrope formation, and monolayer vs condensed layer adsorption in multilayer adsorption.
  • the words “comprising” (and any form of comprising, such as “comprise” and “comprises”), “having” (and any form of having, such as “have” and “has”), “including” (and any form of including, such as “includes” and “include”) or “containing” (and any form of containing, such as “contains” and “contain”) are inclusive or open- ended and do not exclude additional, unrecited features, elements, components, groups, integers, and/or steps, but do not exclude the presence of other unstated features, elements, components, groups, integers and/or steps.
  • compositions and methods comprising or may be replaced with “consisting essentially of” or “consisting of”.
  • the term “consisting” is used to indicate the presence of the recited integer (e.g., a feature, an element, a characteristic, a property, a method/process step or a limitation) or group of integers (e.g., feature(s), element(s), characteristic(s), property(ies), method/process steps or limitation(s)) only.
  • the phrase “consisting essentially of” requires the specified features, elements, components, groups, integers, and/or steps, but do not exclude the presence of other unstated features, elements, components, groups, integers and/or steps as well as those that do not materially affect the basic and novel characteristic(s) and/or function of the claimed invention. [00125]
  • the term “or combinations thereof” as used herein refers to all permutations and combinations of the listed items preceding the term.
  • A, B, C, or combinations thereof is intended to include at least one of: A, B, C, AB, AC, BC, or ABC, and if order is important in a particular context, also BA, CA, CB, CBA, BCA, ACB, BAC, or CAB.
  • expressly included are combinations that contain repeats of one or more item or term, such as BB, AAA, AB, BBC, AAABCCCC, CBBAAA, CABABB, and so forth.
  • BB BB
  • AAA AAA
  • AB BBC
  • AAABCCCCCC CBBAAA
  • CABABB CABABB
  • words of approximation such as, without limitation, “about”, “substantial” or “substantially” refers to a condition that when so modified is understood to not necessarily be absolute or perfect but would be considered close enough to those of ordinary skill in the art to warrant designating the condition as being present. The extent to which the description may vary will depend on how great a change can be instituted and still have one of ordinary skill in the art recognize the modified feature as still having the required characteristics and capabilities of the unmodified feature. In general, but subject to the preceding discussion, a numerical value herein that is modified by a word of approximation such as “about” may vary from the stated value by at least ⁇ 1, 2, 3, 4, 5, 6, 7, 10, 12 or 15%.
  • compositions and/or methods disclosed and claimed herein can be made and executed without undue experimentation in light of the present disclosure. While the compositions and methods of this invention have been described in terms of preferred embodiments, it will be apparent to those of skill in the art that variations may be applied to the compositions and/or methods and in the steps or in the sequence of steps of the method described herein without departing from the concept, spirit and scope of the invention. All such similar substitutes and modifications apparent to those skilled in the art are deemed to be within the spirit, scope and concept of the invention as defined by the appended claims.

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Abstract

A system and computerized method identify an adsorption equilibrium constant and a condensation-evaporation equilibrium constant for an adsorbate component in a gas adsorption system. In addition, a system and computerized method identify a total amount of two or more adsorbate components absorbed for a mixed-gas multilayer adsorption system.

Description

SYSTEM AND METHOD FOR GENERALIZED BRUNAUER-EMMETT-TELLER ISOTHERM FOR SINGLE AND MIXED-GAS MULTILAYER ADSORPTION EQUILIBRIA CROSS-REFERENCE TO RELATED APPLICATIONS [0001] This application claims priority to U.S. Provisional Application Serial No. 63/666,190, filed June 30, 2024. This application is related to: (1) U.S. Patent Application Serial Number 18/291,981 filed on January 25, 2024 and entitled “Generalization of Thermodynamic Langmuir Isotherms for Mixed-gas Adsorption Equilibria”, which claims priority to PCT Application Serial No. PCT/US20/45586, filed August 10, 2020, which claims priority to U.S. Provisional Application Serial No. 62/860,319, filed June 12, 2019; and (2) U.S. Patent Application Serial Number 17/286,720 filed on April 19, 2021 and entitled “Method and System for Adsorbed Phase Activity Coefficients for Mixed-Gas Adsorption”, which claims priority to PCT Application Serial No. PCT/US19/057165, filed October 21, 2019, which claims priority to U.S. Provisional Application Serial No. 62/750,165, filed October 24, 2018. The contents of the foregoing applications are hereby incorporated by reference in their entirety. TECHNICAL FIELD OF THE INVENTION [0002] The present invention relates in general to non-ideal mixed-gas multilayer adsorption equilibria. In particular, the present invention relates to a system and method for generalized Brunauer-Emmett-Teller isotherm for single and mixed-gas multilayer adsorption equilibria. STATEMENT OF FEDERALLY FUNDED RESEARCH [0003] This invention was made with government support under DE-EE0009768 awarded by the U.S. Department of Energy’s Office of Energy Efficiency and Renewable Energy (EERE) under the Bioenergy Technologies Office. The government has certain rights in the invention. BACKGROUND OF THE INVENTION [0004] Without limiting the scope of the invention, its background is described in connection with adsorption separation. [0005] Adsorptive separation is an energy efficient and sustainable alternative to conventional thermal separation. [1-3] The development of tailored adsorbents for selective separation of particular adsorbate components from mixtures has yielded numerous applications. [1,3] Example adsorptive separation applications include CO2 capture, hydrocarbons processing, air separation, direct air capture, trace elements and heavy metals removal, aqueous organic acids separation, among others. [4-9] Reflecting the characteristics of interactions between adsorbate molecules and adsorbent surface, adsorption isotherms express the adsorbate loading as a function of pressure and temperature. [10] Within the six types of adsorption isotherms according to the International Union of Pure and Applied Chemistry, [11] Type I isotherms are monolayer adsorption and Type II and Type III isotherms are multilayer adsorption. In the case of monolayer adsorption, both single and mixed-gas adsorption equilibria data are relatively abundant. [12] In addition, thermodynamically consistent models such as Adsorbed Solution Theory [13] and generalized Langmuir Isotherm [14] provide powerful and rigorous thermodynamic frameworks to reliably correlate and predict mixed-gas adsorption equilibria [15-18]. However, adsorption equilibria data and isotherms for multilayer adsorption are relatively rare in the literature. [19] It is imperative that rigorous thermodynamic frameworks be developed to accurately correlate and predict mixed- gas multilayer adsorption equilibria with minimal reliance on experimental data. [0006] A few thermodynamic models have been reported in the literature for multilayer adsorption equilibria. The classical Brunauer-Emmett-Teller (BET) isotherm [20] for single component multilayer adsorption is the most prominent model that considers first layer as adsorption and second and higher layers as condensation-evaporation phenomena. Though the BET isotherm has been widely practiced for decades, it is valid only for ideal adsorption on homogeneous adsorbent surface for single adsorbate and the calculated surface areas should be considered first approximation [21,22]. Introducing an additional parameter to the two-parameter BET equation, the Guggenheim-Anderson-de Boer (GAB) isotherm enables improved representation of single component isotherm data [23]. The relative pressure term in the GAB model was later further scaled exponentially with the heterogeneity parameter of Freundlich and Sips isotherms [24]. The three-parameter BET equation also has been extended for multicomponent multilayer adsorption assuming the monolayer adsorption is controlled by the adsorbate-specific monolayer adsorption equilibrium constants and the subsequent layer adsorption is controlled by the adsorbate-specific condensation-evaporation equilibrium constants [25]. The extended BET equation was successfully applied to several ideal binary multilayer adsorption systems [25]. Apart from various extensions of the BET isotherm, Sircar and Myers revised Ideal Adsorbed Solution Theory (IAST) [13] for multilayer adsorption [19]. However, challenges in applying Adsorbed Solution Theory for mixed-gas adsorption equilibria remain due to the mismatch in single component spreading pressures and the resulting undefined reference states for one or more adsorbates involved in adsorption. [26] In short, the current state-of-the-art in adsorption thermodynamics is incapable of rigorously addressing mixed-gas multilayer adsorption equilibria. [0007] Recently, Chang et al. [27] transformed the concentration-based classical Langmuir isotherm [28] for single component monolayer adsorption into an activity-based thermodynamic Langmuir (tL) isotherm, and they accounted for the adsorbent surface heterogeneity and the corresponding adsorbed phase nonideality with the adsorption nonrandom two-liquid (aNRTL) activity coefficient model [29]. The tL isotherm has shown accurate representation of single component adsorption isotherms and reliable prediction of isosteric heat of adsorption [16,30]. A generalization of the tL isotherm, the generalized Langmuir (gL) isotherm further provides a rigorous thermodynamic framework for multicomponent monolayer adsorption equilibria and outperforms various empirical Langmuirian models in current practices [14,18,31]. (see also U.S. Patent Application Serial Number 18/291,981 filed on January 25, 2024). Moreover, gL enables reliable predictions of isosteric heat of adsorption, [32] elucidates the spreading pressure dependence in Adsorbed Solution Theory [33], and identifies the thermodynamic condition for adsorption azeotrope [34]. Looking beyond monolayer adsorption, Vyawahare et al. [21] extended the tL isotherm and proposed thermodynamic Brunauer–Emmett–Teller (tBET) isotherm for single component multilayer adsorption. The activity-based tBET isotherm provides a refined model over the concentration-based classical BET isotherm in correlating Type II and Type III isotherms for estimating adsorbent surface area. [21] A natural extension of tBET is to generalize it for multicomponent multilayer adsorption equilibria. [0008] Accordingly, there is a need for a system and method for generalized Brunauer-Emmett- Teller isotherm for single and mixed-gas multilayer adsorption equilibria. SUMMARY OF THE INVENTION [0009] This disclosure presents a rigorous thermodynamic framework to calculate mixed-gas multilayer adsorption equilibria from single component isotherms and vapor-liquid equilibria. Named generalized Brunauer-Emmett-Teller (gBET) isotherm, the newly formulated isotherm considers adsorbent surface heterogeneity, competitive adsorption on the monolayer, condensation-evaporation on the subsequent layers, and adsorbed phase nonideality for the monolayer and the subsequent layers. The monolayer adsorbed phase nonideality is tracked using an area-based adsorption nonrandom two-liquid activity coefficient model. The adsorbed phase composition and corresponding nonideality in the subsequent layers are calculated at the dew point condition of the mixed-gas with either an equation of state or an activity coefficient model for the vapor-liquid equilibria. The disclosed model is validated with six single and three binary multilayer adsorption equilibrium systems, and the model results are compared against those from classical BET isotherm for single component adsorption and Ideal Adsorbed Solution Theory for mixed-gas adsorption equilibria. [0010] In one embodiment in accordance with the present disclosure, a computerized method for identifying an adsorption equilibrium constant and a condensation-evaporation equilibrium constant for an adsorbate component in a gas adsorption system includes providing one or more processors, a memory communicably coupled to the one or more processors and an input/output interface communicably coupled to the one or more processors. The one or more processors calculate an adsorption equilibrium constant (^^^^^^^^ ^^^^ ) for the adsorbate component (i) using ^^^^^^^^ ^^^^ = ^^^^^^^^ ^^^^^^^^^^^^ ^^^^ exp�−∆^^^^^^^^^^^^^^^^,^^^^ ^^^^ �1 ^^^^ − 1 ^^^^^^^^^^^^^^^^ �� . The one or more processors calculate a condensation- for the adsorbate component (i) using ^^^^^^^^,^^^^ = ^^^^ ^^^^^^^^^^^^ ^^^^,^^^^ exp�−∆^^^^^^^^^^^^^^^^^^^^,^^^^ ^^^^ �1 ^^^^ − 1 ^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^��, wherein: ^^^^^^^ ^ ^^^^ is the adsorption constant at a reference evaporation equilibrium constant at the reference temperature (^^^^^^^^^^^^^^^^ a gas constant, ^^^^ is a system temperature, ∆^^^^^^^^^^^^^^^^,^^^^ is an enthalpy of adsorption, and ∆^^^^^^^^^^^^^^^^^^^^,^^^^ is an enthalpy of condensation. The adsorption equilibrium constant and the condensation-evaporation equilibrium constant are provided to the input/output interface, and the gas adsorption system is configured using the adsorption equilibrium constant and the condensation-evaporation equilibrium constant. [0011] In one aspect, the method produces a product using the gas adsorption system. In another aspect, the gas adsorption system further uses a saturation adsorption loading for monolayer adsorption of the adsorbate component (^^^^0 ^^^^ ), a correction factor (^^^^^^^^ ) and a binary interaction parameter (^^^^^^^^^^^^ ) for the species ^^^^ – species ^^^^ pair. In another aspect, the input/output interface comprises an interface to the gas adsorption system. In another aspect, the gas absorption system comprises a CO2 capture system, a hydrocarbon processing system, an air separation system, a direct air capture system, a trace elements and heavy metals removal system, or an aqueous organic acids separation system. In another aspect, the gas absorption system comprises a liquid adsorption system mathematically formulated as a pseudo gas adsorption system. In another aspect, the liquid adsorption system comprises a trace elements removal system, a heavy metals removal system, an aqueous organic separation system, or an ethanol-water separation system. In another aspect, the adsorption equilibrium constant and the condensation-evaporation equilibrium constant are calculated using a regression analysis. In another aspect, calculating the adsorption equilibrium constant and the condensation-evaporation equilibrium constant comprises: providing input data comprising a pressure (P), an adsorption isotherm (^^^^^^^^ ^^^^ ), an adsorbate area for the adsorbate component (^^^^^^^^), an adsorbate area for a reference molecule (^^^^^^^^), and a saturation pressure (^^^^^^^ ^ ^^^^^^^^^^^^); calculating a saturation adsorption loading for monolayer adsorption of the adsorbate component 0 ^^^^ om an experimental adsorbate surface area or ^^^^ ^^^^0 (^^^^ ) fr 0 ^^^^ = 0 ^^^^^^^^ where ^^^^ is a total adsorbent surface providing initial values for the adsorption ^^^^ constant (^^^^^^^^ ), a correction factor (^^^^^^^^) and a binary interaction parameter (^^^^ ) for the species ^^^^ – species providing initial for an amount adsorbed of the component due to monolayer adsorption (^^^^^^^^), an amount adsorbed of the reference molecule due to monolayer adsorption (^^^^^^^^) and a total of amounts adsorbed of the adsorbate component due to condensation on layers (^^^^^ ∗ ^^^); calculating an adsorbed phase mole fraction for the adsorbate component in the monolayer only (^^^^^^^^) using ^^^^^^^^ = ^^^^^^^^ ^^^^ ^^^^ ; calculating an adsorbed phase mole fraction for the reference molecule in the ^^^^ (^^^^^^^^) using ^^^^^^^^ = ^^^^ ∑^^^^ ^^^^=1 ^^^^^^^^ +^^^^^^^^; calculating a gas phase mole fraction for the adsorbate component ( ^^^^^^^^ ) using ln ^^^^^^^^ = ^^^^^^^^ ∑ ^^^^+1 ^^^^^ 2 ^^^^^^^^ 2 ^^^^^^^^^^^^^^^�Ω^^^^^^^^−1� ^^^^=1�∑^^^^ ^^^ �2 , wherein: ^^^^^^^^^^^^ = ∆^^^^^^^^^^^^ + ∆ℎ^^^^^^^^ ^^^^ , Ω^^^^^^^^ = ^^^^=1 ^^^^^^^^^^^^^Ω^^^^^^^^ exp�−^^^^^^^^ � and ^^^^ is a about 0.3 per the NRTL phase molecule ( ^^^^^^^^ ) using ln ^^^^^^^^ = ^^^^^^^^ ∑ ^^^^+1 ^^^^ ^^^^ ^^^^^^^^ ^^^^=1�∑^^^^ ; calculating a new value for ^^^^ , ^^ ∗ ^^^^=1 ^^^^ 2 ^^^^ ^^^^^^ and ^^^^^^^^ using ^^^^^^^^^^^^Ω^^^^^^^^� ∗�^^^^ ^ ^^^^ ^^ ^^^^ ^^^ ^^^^ ^^ ^^^^ ^ ^^ = = ^^^ ^^ ^^^^^^^^ � 1−^^^^^^^^,^^^^^^^^ � � ^^^^ 1^^^^ ^^^^ (1−^^^^^^^^,^^^^^^^^)+^^^^^^ ^ ^^ ^^^^^^^� calculating a due to monolayer adsorption and condensation on subsequent layers (^^^^^^^^) us ^^^^ ∗ ^^^^ ing ^^^^^^^^ = ^^^^^^^^ + ^^^^^^^^; repeating steps (a)-(h) whenever the initial values for ^^^^ , ^^^^^^^^ and ^^^^∗ do not match the values for ^^^^^^^^, ^^^^ and ^^^^^ ∗ ^^^; and ^^^^^^^^^^^^^^^^ 2 teps (a)-(i) ^^^^^^^^^^^^ = ^^^^ ^^^^ ^^^^ ^^^^^^^^^^^^^^^^ s ∑ ^^^^ ^^^^ ^^^^ −^^^^^^^^ ^^^^ ^^^^=1� ^^^^^^^^^^^^^^^^^^^^ � is not In another aspect, the adsorption equilibrium constant and the condensation-evaporation equilibrium constant are calculated for multiple temperatures. In another aspect, the gas absorption system comprises: an adsorbent comprising titanium dioxide anatase; and the adsorbate component comprises oxygen or nitrogen. In another aspect, the gas absorption system comprises: an adsorbent comprising activated charcoal; and the adsorbate component comprises benzene or cyclohexane. In another aspect, the gas absorption system comprises: an adsorbent comprising activated alumina; and the adsorbate component comprises water or carbon dioxide. [0012] In another embodiment in accordance with the present disclosure, a non-transitory computer readable medium containing program instructions cause the one or more processors to perform the method described in the previous paragraphs. [0013] In one embodiment in accordance with the present disclosure, a computerized method for identifying a total amount of two or more adsorbate components absorbed for a mixed-gas multilayer adsorption system includes providing one or more processors, a memory communicably coupled to the one or more processors and an input/output interface communicably coupled to the one or more processors. The one or more processors calculate the total amount of the two or more adsorbate components (i) absorbed (^^^^^^^^ ^^^^) using ^^^^^^^^ ^^^^ = ^^^^^^^^ + ^^^^∗^^^^∗ ^^^^ , wherein: ^^^^^^^^ is an amount adsorbed of each adsorbate component (^^^^) due to adsorption, ^^^^∗ is a total of amounts adsorbed due to condensation on subsequent layers, ^^^^^ ∗ ^^^ is a dew point liquid phase composition. The one or more processors provide the total of the two or more adsorbate components (i) ^^^^ absorbed (^^^^ ) to the input/output the mixed-gas multilayer adsorption system configured using the total amount of the two or more adsorbate components (i) absorbed (^^^^^ ^ ^^ ^^ ^^). [0014] In one aspect, a product is produced using the mixed-gas multilayer In another aspect, the mixed-gas multilayer absorption system comprises a CO2 capture system, a hydrocarbons processing system, an air separation system, a direct air capture system, a trace elements and heavy metals removal system, or an aqueous organic acids separation system. In another aspect, the mixed-gas multilayer adsorption system comprises a liquid adsorption system mathematically formulated as a pseudo gas adsorption system. In another aspect, the liquid adsorption system comprises a trace elements removal system, a heavy metals removal system, an aqueous organic separation system, or an ethanol-water separation system. In another aspect, the input/output interface comprises an interface to the mixed-gas multilayer adsorption system. In another aspect, the mixed-gas multilayer adsorption system is further configured using a binary interaction parameter (^^^^^^^^^^^^), a correction factor (^^^^) and a pressure (P). In another aspect, the total amount of each of the or more adsorbate components (i) absorbed for the mixed-gas multilayer adsorption system (^^^^^ ^ ^^ ^^ ^^) are calculated using a regression analysis. In another aspect, the one or more processors calculate the total amount of each of the two or more adsorbate components (i) absorbed for the mixed-gas multilayer adsorption system (^^^^^ ^ ^^ ^^ ^^) by: providing input data comprising a pressure (P), a gas phase mole fraction for each adsorbate component (^^^^^^^^), the total amount of the two or more adsorbate components (i) absorbed (^^^^^^^^ ^^^^ ), a saturation adsorption loading for monolayer adsorption of each adsorbate component an adsorption equilibrium constant each adsorbate component (^^^^^^^^ ^^^^ ), a condensation-evaporation equilibrium constant for each adsorbate component (^^^^^^^^,^^^^), a binary interaction parameter (^^^^^^^^^^^^) for the species ^^^^ – species ^^^^ pair, an adsorbate area for each adsorbate component (^^^^^^^^), an adsorbate area for a reference molecule (^^^^^^^^), a dew point pressure for each adsorbate component (^^^^^^^^^^^^^^^^ ^^^^ ), and a dew point liquid phase composition (^^^^^ ∗ ^^^); providing initial values for the binary interaction parameter (^^^^^^^^^^^^) and a correction factor (^^^^); providing initial values for an amount adsorbed of each adsorbate component due to monolayer adsorption (^^^^^^^^ ), an amount adsorbed of the reference molecule due to monolayer adsorption (^^^^^^^^) and a total of amounts adsorbed of each adsorbate component due to condensation on subsequent layers (^^^^^ ∗ ^^^) at a given gas phase mole fraction for each adsorbate component (^^^^^^^^) and pressure (P); calculating an adsorbed phase mole fraction for each adsorbate component in the monolayer only (^^^^^^^^) using ^^^^^^^^ = ^^^^^^^^ ∑^^^^ ^^^^=1 ^^^^^^^^ +^^^^^^^^; calculating an adsorbed phase mole fraction for the ^^^^ reference molecule in the ^^^^ ^^^^) using ^^^^ = ; calculating a gas phase mole ^^^^ ∑^^^^ ^^^^=1 ^^^^^^^^ +^^^^^^^^ fraction for each adsorbate component (^^^^ ) us ^^^^^ 2 ^^^^^^^^ 2 ^^^^^^^^^^^^^^^�Ω^^^^^^^^−1� ^^^^ ing ln ^^^^=1 ^^^^ ^^^^ Ω ^�2 , wherein: ^^^^^^^^^^^^ = ^^^^ ^^^^ ^^^^^^^ ∆ℎ ∆^^^^ + ^^^^^^^^ , Ω^^^^ = exp�−^^^^^^^^ � and ^^^^ is a 0.3; ^^^^ ^^^^ ^^^^^^^^ gas reference molecule ^^^ ^^^^ ∑ ^^^^+1 ^^^^^ 2 ^^^^^^^^ 2 ^^^^^^^^^^^^^^^�Ω^^^^^^^^−1� (^^^^^^^^) using ln ^^^^^ = ^^^^ ^^^^=1 ^^^^ 2 ; ^^^^^^^^^^^^Ω � calculating a new value for ^^^^^^^^, ^^^^^^^^ and ^^^^^ ∗ ^^^ using ^^^^ �^^^^ +^^^^�^^^^^^^^ ^^^^ ^^^^ = ^^^^ ^^^^ = ^^^^ ^^^^ ^^^^ ^^^^ ^^^^0 ^^^^ = ^^^^ 1^^^^ ^^^^ , calculating (i) absorbed (^^^^^^^^ ^^^^ ^^^^) using ^^^^^^^^ = ^^^^ + ^ ∗ ∗ ^^^^ ^^^^^^^; repeating steps (a)-(h) whenever the initial values for ^^^^^^^^, ^^^^^^^^ and ^^^^^^^^ do not match the calculated values for ^^^^ ∗ ^^^^ , ^^^^^^^^ and ^^^^^^^^ ; and repeating step (a)-(i) whenever ^^^^^^^^^^^^ = ^^ ^^^^^^^^^^^^^^^^ ^^^^ ^^^^^^^^^^^^^^^^ 2 ^^^^ ^^^^^^^ ^ ^^ ^^^^ −^^^^^^^^ ^^^^ is not In another aspect, the total amount of each of the two or (i) absorbed for the mixed-gas multilayer adsorption system (^^^^^^^^ ^^^^) are calculated for multiple temperatures. In another aspect, the mixed-gas multilayer adsorption system comprises: an adsorbent comprising titanium dioxide anatase; and the adsorbate component comprises oxygen and nitrogen. In another aspect, the mixed-gas multilayer adsorption system comprises: an adsorbent comprising activated charcoal; and the adsorbate component comprises benzene and cyclohexane. In another aspect, the mixed-gas multilayer adsorption system comprises: an adsorbent comprising activated alumina; and the adsorbate component comprises water and carbon dioxide. [0015] In another embodiment in accordance with the present disclosure, a non-transitory computer readable medium containing program instructions cause the one or more processors to perform the method described in the previous paragraphs. [0016] In another embodiment in accordance with the present disclosure, a system for identifying an adsorption equilibrium constant and a condensation-evaporation equilibrium constant for an adsorbate component in a gas adsorption system includes a memory, an input/output interface, and one or more processors communicably coupled to the memory and the input/output interface. The one or more processors: calculate an adsorption equilibrium constant (^^^^^^ ^ ^^ ^^^ ) for the adsorbate component (i) using ^^^^^^^^ ^^^^ = ^^^^^^^^ ^^^^^^^^^^^^ ^^^^ exp�−∆^^^^^^^^^^^^^^^^,^^^^ ^^^^ �1 ^^^^ − 1 ^^^^^^^^^^^^^^^^��, calculate a condensation-evaporation equilibrium constant (i) using ^^^^ ^^^^^^^^^^^^ −∆^^^^^^^^^^^^^^^^^^^^,^^^^ 1 ^^^^,^^^^ = ^^^^^^^^,^^^^ exp� ^^^^ � ^^^^ − 1 ^^^^^^^^^^^^ ^^^^ ��, wherein: ^^^^^^^^ is the adsorption constant at a reference condensation- equilibrium constant the reference temperature (^^^^^^^^^^^^^^^^ ), ^^^^ is a gas constant, ^^^^ is a system temperature, ∆^^^^^^^^^^^^^^^^,^^^^ is an enthalpy of adsorption, ∆^^^^^^^^^^^^^^^^^^^^,^^^^ is an enthalpy of condensation, and provide the adsorption equilibrium constant and the condensation- evaporation equilibrium constant to the input/output interface. The gas adsorption system is configured using the adsorption equilibrium constant and the condensation-evaporation equilibrium constant. [0017] In one aspect, a product is produced using the gas adsorption system. In another aspect, the gas adsorption system is further configured using a saturation adsorption loading for monolayer adsorption of the adsorbate component (^^^^0 ^^^^ ), a correction factor (^^^^^^^^ ) and a binary interaction parameter (^^^^^^^^^^^^ ) for the species ^^^^ – species ^^^^ pair. In another aspect, the input/output interface comprises an interface to the gas adsorption system. In another aspect, the gas absorption system comprises a CO2 capture system, a hydrocarbon processing system, an air separation system, a direct air capture system, a trace elements and heavy metals removal system, or an aqueous organic acids separation system. In another aspect, the gas absorption system comprises a liquid adsorption system mathematically formulated as a pseudo gas adsorption system. In another aspect, the liquid adsorption system comprises a trace elements removal system, a heavy metals removal system, an aqueous organic separation system, or an ethanol-water separation system. In another aspect, the adsorption equilibrium constant and the condensation-evaporation equilibrium constant are calculated using a regression analysis. In another aspect, the one or more processors calculate the adsorption equilibrium constant and the condensation-evaporation equilibrium constant by: providing input data comprising a pressure (P), an adsorption isotherm (^^^^^^^^ ^^^^), an adsorbate area for the adsorbate component (^^^^^^^^), an adsorbate area for a reference molecule (^^^^^^^^), and a saturation pressure (^^^^^^^ ^ ^^^^^^^^^^^^ ); calculating a saturation adsorption loading for monolayer adsorption of the nent (^^^^ ^^^0 adsorbate compo 0 0 ^ ^^^^) from an experimental adsorbate surface area or ^^^^^^^^ = ^^^^^^^^ where ^^^^0 is a total adsorbent surface area; providing initial values for the adsorption constant (^^^^^^^^ ^ ), ^^^ a correction factor (^^^^^^^^) and a binary interaction parameter (^^^^^^^^^^^^) for the species ^^^^ – species ^^^^ pair; providing initial values for an amount adsorbed of the adsorbate component due to monolayer adsorption (^^^^^^^^), an amount adsorbed of the reference molecule due to monolayer adsorption (^^^^^^^^) and a total of amounts adsorbed of the adsorbate component due to condensation on subsequent layers (^^^^^ ∗ ^^^ ); calculating an adsorbed phase mole fraction for the adsorbate component in the monolayer only (^^^^ ^^^^^^^^ ^^^^) using ^^^^^^^^ = ∑^^^^ ^^^^=1 ^^^^^^^^ +^^^^^^^^; calculating an adsorbed phase mole fraction for the ^^^^ reference molecule in the ^^^^) using ^^^^^^^^ = ^^^^ ∑^^^^ ^^^^=1 ^^^^^^^^ +^^^^^^^^; calculating a gas phase mole ^ ) using ln ^^^^ ^^^^2^^^^2 fraction for the adsorbate component (^^^ ^^^^ ^^^^^^^^^^^^^^^^�Ω^^^^^^^^−1� � , wherein: ^^^^^^^^^^^^ ∆ℎ^^^^^ ∆^^^^ + ^^^ ^^^^ , Ω^^^^^^^^ = exp�−^^^^^^^^^^^^^^^^� and ^^^^ is a 0.3 per the mole fraction for the reference molecule (^^^^^^^^) using ln ^^^^^^^^ = ^^^^^ 2 ^^^^^^^^ 2^^^^^^^^^^^^�Ω^^^^^^^^−1� ^^^^ ∑ ^^^^+1 ^^^ ^^^^=1 ; calculating a new value for ^^^^^^^^, ^^^^^^^^ and ^^^^∗ ^^^^^^^^ �^^^^^^^^+^^^^ ∗ ^^^^�^^^^^^^^ ^^^^ ^^^^ ^^^^ ^^^^^ ^^^^^^^^ ^^^^^^^^ = ^^^^^^^^ = ^^^^0 ^^^^ = ^^^ �1−^^^^^^^^,^^^^^^^^�� ^^^^ 1^^^^ , ^^^^ (1−^^^^ ^^^^ 1^^^^ ^^^^ ^^^^ ^^^^,^^^^^^^^)+^^^^^^^^ ^^^^� calculating a due to monolayer adsorption and condensation on subsequent layers (^^^^^^^^ ^^^^ ∗ ^^^^) using ^^^^^^^^ = ^^^^^^^^ + ^^^^^^^^; and repeating steps (a)-(h) whenever the initial values for ^^^^^^^^, ^^^^^^^^ and ^^^^^ ∗ ^^^ do not match the calculated values for ^^^^^^^^, ^^^^^^^^ and ^^^^^ ∗ ^^^; repeating ^^ ^^^^^^^^^^^^^^^^ ^^^^ ^^^^^^^^^^^^^^^^ 2 steps (a)-(i) = ∑ ^^^^ ^^^^=1�^^^^^^^ ^ ^^ ^^^^ −^^^^^^^^ ^^^^ ^^^^^^^^^^^^^^^^^^^^ � is not minimized. another aspect, the adsorption equilibrium the condensation-evaporation equilibrium constant are calculated for multiple temperatures. In another aspect, the gas absorption system comprises: an adsorbent comprising titanium dioxide anatase; and the adsorbate component comprises oxygen or nitrogen. In another aspect, the gas absorption system comprises: an adsorbent comprising activated charcoal; and the adsorbate component comprises benzene or cyclohexane. In another aspect, the gas absorption system comprises: an adsorbent comprising activated alumina; and the adsorbate component comprises water or carbon dioxide. [0018] In another embodiment in accordance with the present disclosure, a system for identifying a total amount of two or more adsorbate components absorbed for a mixed-gas multilayer adsorption system includes a memory, an input/output interface, and one or more processors communicably coupled to the memory and the input/output interface. The one or more processors: calculate the total amount of the two or more adsorbate components (i) absorbed (^^^^^ ^ ^^ ^^ ^^) using ^^^^^ ^ ^^ ^^ ^^ = ^^^^ + ^^^^∗^^^^∗ ^^^^ , wherein: ^^^^^^^^ is an amount adsorbed of each adsorbate component (^^^^) due to adsorption, ^^^^∗ is a total of amounts adsorbed due to condensation on subsequent layers, ^^^^^ ∗ ^^^ is a dew point liquid phase composition, and provide the total amount of the two or more adsorbate components (i) absorbed (^^^^^ ^ ^^ ^^ ^^) to the input/output interface. The mixed-gas multilayer adsorption system is configured using the total amount of two or more adsorbate components absorbed. [0019] In one aspect, a product is produced using the mixed-gas multilayer adsorption system. In another aspect, the mixed-gas multilayer absorption system comprises a CO2 capture system, a hydrocarbons processing system, an air separation system, a direct air capture system, a trace elements and heavy metals removal system, or an aqueous organic acids separation system. In another aspect, the mixed-gas multilayer adsorption system comprises a liquid adsorption system mathematically formulated as a pseudo gas adsorption system. In another aspect, the liquid adsorption system comprises a trace elements removal system, a heavy metals removal system, an aqueous organic separation system, or an ethanol-water separation system. In another aspect, the input/output interface comprises an interface to the mixed-gas multilayer adsorption system. In another aspect, the mixed-gas multilayer adsorption system is further configured using a binary interaction parameter (^^^^^^^^^^^^), a correction factor (^^^^) and a pressure (P). In another aspect, the total amount of each of more adsorbate components (i) absorbed for the mixed-gas multilayer adsorption system are calculated using a regression analysis. In another aspect, the one or more processors calculate the total amount of each of the two or more adsorbate components (i) absorbed for the mixed-gas multilayer adsorption system (^^^^^ ^ ^^ ^^ ^^) by: providing input data comprising a pressure (P), a gas phase mole fraction for each adsorbate component (^^^^^^^^), the total amount of the two or more adsorbate components (i) absorbed (^^^^^^^^ ^^^^ ), a saturation adsorption loading for monolayer adsorption of each adsorbate component an adsorption equilibrium constant for each adsorbate component (^^^^^^^^ ^^^^ ), a condensation-evaporation equilibrium constant for each adsorbate component (^^^^^^^^,^^^^), a binary interaction parameter (^^^^^^^^^^^^) for the species ^^^^ – species ^^^^ pair, an adsorbate area for adsorbate component (^^^^^^^^), an area for a reference molecule (^^^^^^^^), a dew point pressure for each adsorbate component (^^^^^^^ ^ ^^^^^^^^^^^^), and a dew point liquid phase composition (^^^^^ ∗ ^^^); providing initial values for the binary interaction parameter (^^^^^^^^^^^^) and a correction factor (^^^^); providing initial values for an amount adsorbed of each component due to monolayer adsorption (^^^^^^^^ ), an amount adsorbed of the reference molecule due to monolayer adsorption (^^^^^^^^) and a total of amounts adsorbed of each adsorbate component due to condensation on subsequent layers (^^^^^ ∗ ^^^) at a given gas phase mole fraction for each adsorbate component (^^^^^^^^) and pressure (P); calculating an adsorbed phase mole fraction for each adsorbate component in the monolayer only (^^^^^^^^) using ^^^^^^^^ = ^^^^^^^^ ∑^^^^ ^^^^=1 ^^^^^^^^ +^^^^^^^^; calculating an adsorbed phase mole fraction for the ^^^^ reference molecule in the monolayer only (^^^^ ) usi ^^^^ ^^^^ ng ^^^^^^^^ = ∑^^^^ ^^^^=1 ^^^^^^^^ +^^^^^^^^; calculating a gas phase mole fraction for each adsorbate component (^^^^ ) using ln ^^^^ ^^^^^ 2 ^^^^^^^^ 2 ^^^ ^^^^^^^^^^^^ � Ω^^^^^^^^−1 � ^^^^ = ^^^^^^^^ ^^^^=1�∑^^^^ ^ � , wherein: ^^^^^^^^^^^^ ^^^^=1 ^^^^^^^^^^^^^^^Ω^^^^^^^^ ∆ℎ ∆^^^^ + ^^^^^^^^ ^^^^ , Ω^^^^^^^^ = exp�−^^^^^^^^^^^^^^^^� and ^^^^ is a 0.3; ^^^^2^ 2 gas molecul ∑ ^^^^+1 ^^^^ ^^^^^^^^^^^^^^^^^^^�Ω^^^^^^^^−1� e (^^^^^^^^) using ln ^^^^^^^^ = ^^^^^^^^ ^^^^=1�∑^^^^ ^ �2 ; ^^^^=1 ^^^^^^^^^^^^^^^Ω^^^^^^^^ calculating a new value for ^^^^^^^^, ^^^^^^^^ and ^^^^^ ∗ ^^^ using ^^^^^^^^ ^^^^ =�^^^^^^^^+^^^^ ∗ ^^^^�^^^^ ^^^^ ^^^^ ^^^^^^^^ ^^^^^^ ^ ^^ ^^^^^^^ ^^^^^^^^ = ^^^^ = �1−^^^^ ^^^^�� ^^^^ 1^^^^ (1−^^^^ ^^^^)+ ^^^^ ^^^^ ^^^^^^^^ calculating (i) absorbed (^^^^^ ^ ^^ ^^ ^^) using ^^^^^ ^ ^^ ^^ ^^ = ^^^^^ + ^^^^∗ ^^^^; repeating steps (a)-(h) whenev ∗ ^^^ er the initial values for ^^^^^^^^, ^^^^^^^^ and ^^^^^^^^ do not match the calculated values for ^^^^ , ^^^^ and ^^^∗ ^^^^ ^^^^^ ; and (a)-(i) whenever ^^^^^^^^^^^^ ^^^^^^^^^^^ 2 ∑ ^^^^=1� ^ ^^^^^ ^^^^ ^^^^^^^^^^^^^^^^ ^^^^ ^^^^^^ ^ ^^ ^^ ^^^^ −^^^^^^^^ ^^^^ ^^^^^^^^^^^^^^^^^^^^ � is not In another aspect, the total amount of each of the two or more adsorbate components (i) absorbed for the mixed-gas multilayer adsorption system (^^^^^ ^ ^^ ^^ ^^) are calculated for multiple temperatures. In another aspect, the mixed-gas multilayer adsorption system comprises: an adsorbent comprising titanium dioxide anatase; and the adsorbate component comprises oxygen and nitrogen. In another aspect, the mixed-gas multilayer adsorption system comprises: an adsorbent comprising activated charcoal; and the adsorbate component comprises benzene and cyclohexane. In another aspect, the mixed-gas multilayer adsorption system comprises: an adsorbent comprising activated alumina; and the adsorbate component comprises water and carbon dioxide. [0020] Note that the invention is not limited to the embodiments described herein, instead it has the applicability beyond the embodiments described herein. The brief and detailed descriptions of this disclosure are given in the following. BRIEF DESCRIPTION OF THE DRAWINGS [0021] For a more complete understanding of the features and advantages of the present invention, reference is now made to the detailed description of the invention along with the accompanying figures and in which: [0022] FIG.1 is a flow chart of an algorithm of generalized Brunauer-Emmett-Teller isotherm for pure component adsorption in accordance with one embodiment of the present disclosure; [0023] FIG.2 depicts a schematic diagram for generalized Brunauer-Emmett-Taller isotherm for multilayer mixed-gas adsorption equilibria in accordance with one embodiment of the present disclosure; [0024] FIG.3 is a flow chart of an algorithm of generalized Brunauer-Emmett-Teller isotherm for multicomponent adsorption equilibria in accordance with one embodiment of the present disclosure; [0025] FIGS.4A-4C depict single component adsorption isotherm representation of (a) N2 and O2 adsorption on TiO2 anatase at 78.2 K, [36] (b) benzene and cyclohexane adsorption on activated charcoal at 303.15 K, [37] and (c) H2O and CO2 adsorption on activated alumina F-200 at 303.15 K [26] using BET and gBET isotherms in accordance with one embodiment of the present disclosure; [0026] FIG.5 depicts Table 1 with single component BET isotherm parameters and Table 2 with single component gBET isotherm parameters in accordance with one embodiment of the present disclosure; [0027] FIGS. 6A-6C depict vapor-liquid equilibria of (a) O2 (1) + N2 (2) using Peng-Robinson equation of state at 78.2 K and experimental data at 74.70 K and 79.07 K [41,42], (b) benzene (1) + cyclohexane (2) using ideal gas equation of state and NRTL activity coefficient model at 303.15 K [44,45], and (c) H2O (1) + CO2 (2) using PC-SAFT equation of state at 303.15 K [47] in accordance with one embodiment of the present disclosure; [0028] FIG. 7 depicts Table 3 with binary component results from the gBET isotherm in accordance with one embodiment of the present disclosure; [0029] FIGS.8A-8E depict Binary adsorption equilibria of O2 (1) + N2 (2) on TiO2 anatase at (a) ^^^^1 = 0.149 and ^^^^2 = 0.851, (b) ^^^^1 = 0.298 and ^^^^2 = 0.702, (c) ^^^^1 = 0.502 and ^^^^2 = 0.498, (d) ^^^^1 = 0.702 and ^^^^2 = 0.298 , and (e) ^^^^1 = 0.853 and ^^^^2 = 0.147 [36] using IAST and gBET isotherm at 78.2 K in accordance with one embodiment of the present disclosure; [0030] FIGS. 9A-9D depict binary adsorption equilibria of benzene (1) + cyclohexane (2) on activated charcoal at (a) ^^^^1 = 0.2063 and ^^^^2 = 0.7937, (b) ^^^^1 = 0.3021 and ^^^^2 = 0.6979, (c) ^^^^1 = 0.5018 and ^^^^2 = 0.4982, and (d) ^^^^1 = 0.7988 and ^^^^2 = 0.2012 [37] using IAST (dashed line) and gBET isotherm (solid line) at 78.2 K for benzene (red), cyclohexane (blue), and total (black) in accordance with one embodiment of the present disclosure; [0031] FIG.10 depicts binary adsorption equilibria of H2O (1) + CO2 (2) on activated alumina F- 200 at constant CO2 partial pressure of 1.021 bar and varying relative humidity of H2O [26] using IAST and gBET isotherm at 303.15 K in accordance with one embodiment of the present disclosure; [0032] FIG.11 is a block diagram of a system in accordance with one embodiment of the present disclosure; [0033] FIG.12 is a flowchart of a computerized method for identifying an adsorption equilibrium constant and a condensation-evaporation equilibrium constant for an adsorbate component in a gas adsorption system in accordance with one embodiment of the present disclosure; [0034] FIG.13 is a flowchart of a computerized method for identifying a total amount of each of two or more adsorbate components absorbed for a mixed-gas multilayer adsorption system in accordance with one embodiment of the present disclosure; [0035] FIGS. 14A-14C depict a sensitivity analysis of the adsorbate−adsorbent interaction parameter on the isosteric enthalpy of adsorption at (FIG. 14A) τiϕ from −3 to 0, (FIG. 14B) τiϕ from 0 to +3, and (FIG.14C) data and model estimates of isosteric enthalpy of adsorption for CH4 at 297.0 K on BPL activated carbon, C2H6 at 297.0 K on BPL activated carbon, and CO2 at 305.95 K on NaX using gL isotherm in accordance with one embodiment of the present disclosure; [0036] FIG. 15 depicts single-component adsorption isotherm of CO2 on zeolite H-mordenite at 283.15−323.15 K using cL, Sips, DPL, and gL isotherms in accordance with one embodiment of the present disclosure; [0037] FIG. 16 depicts Table 4 with regressed single-component parameters for the Langmuir isotherm, Table 5 with regressed single-component parameters for the Sips isotherm, and Table 6 with regressed single-component parameters for the Dual Process Langmuir (DPL) isotherm in accordance with one embodiment of the present disclosure; [0038] FIGS.17A-17C depict adsorption selectivity of (FIG.17A) H2S (1) − CO2 (2) at 0.156 bar, (FIG.17B) C3H8 (1) − H2S (2) at 0.081 bar, and (FIG. 17C) C3H8 (1) − CO2 (2) at 0.405 bar on zeolite H-mordenite at 303.15 K using IAST and gL in accordance with one embodiment of the present disclosure; [0039] FIG. 18 depicts Table 7 with regressed single-component parameters for the generalized Langmuir (gL) isotherm, Table 8 with a summary of binary mixed-gas adsorption equilibrium estimation using models in current practice and the generalized Langmuir isotherm, and Table 9 with single-component generalized Langmuir isotherm parameters in accordance with one embodiment of the present disclosure; [0040] FIG.19 depict binary adsorption equilibrium data and model results for (1) H2S (1) − CO2 (2) at 0.156 bar, (2) C3H8 (1) − H2S (2) at 0.081 bar, and (3) C3H8 (1) − CO2 (2) at 0.405 bar on zeolite H-mordenite at 303.15 K using the IAST, eL, Sips-LRC, DPL, and gL isotherms in accordance with one embodiment of the present disclosure; [0041] FIGS. 20A-20C depict binary adsorption equilibrium experimental data and gL model results for the adsorption of C2H4 (1) − CO2 (2) binary on molecular sieves 5A at (FIG.20A) 0.003 bar, (FIG. 20B) 0.02 bar, and (FIG. 20C) θ − x′y phase diagram at 0.003, 0.011, 0.02, and 0.06 bar. θ12 is the total occupied area fraction in accordance with one embodiment of the present disclosure; [0042] FIGS.21A-21C depict (FIG.21A) single-component adsorption isotherm of N2 and O2 on LiLSX at 303.15 K using the gL isotherm, (FIG.21B) binary adsorption equilibrium of N2 (1) − O2 (2) at 0.1013, 1.013, 6.078, and 10.0 bar generated from spreading pressure-dependent RAST and gL isotherms, and (FIG.21C) spreading pressure-dependent τ’12 parameter of the SPD-aNRTL activity coefficient in real adsorbed solution theory for binary adsorption of N2 (1) − O2 (2) on LiLSX at 303.15 K in accordance with one embodiment of the present disclosure; [0043] FIGS. 22A-22F depict (FIG. 22A) single-component adsorption of O2 and N2 using the BET and gBET isotherms, binary adsorption equilibrium for O2 (1) − N2 (2) at gas phase composition of (FIG.22B) y1 = 14.9% and y2 = 85.1%, (FIG.22C) y1 = 29.8% and y2 = 70.2%, (FIG.22D) y1 = 50.2% and y1 = 49.8%, (FIG.22E) y1 = 70.2% and y2 = 29.8%, and (FIG.22F) y1 = 85.3% and y1 = 14.7% on anatase at 78.2 K using ideal adsorbed solution theory and the gBET isotherm in accordance with one embodiment of the present disclosure; and [0044] FIG. 23 depicts vapor−liquid equilibrium of O2 (1) − N2 (2) using the Peng−Robinson equation of state at 74.70−79.07 K in accordance with one embodiment of the present disclosure. DETAILED DESCRIPTION OF THE INVENTION [0045] While the making and using of various embodiments of the present invention are discussed in detail below, it should be appreciated that the present invention provides many applicable inventive concepts that can be embodied in a wide variety of specific contexts. The specific embodiments discussed herein are merely illustrative of specific ways to make and use the invention and do not delimit the scope of the invention. [0046] To facilitate the understanding of this invention, a number of terms are defined below. Terms defined herein have meanings as commonly understood by a person of ordinary skill in the areas relevant to the present invention. Terms such as “a”, “an” and “the” are not intended to refer to only a singular entity but include the general class of which a specific example may be used for illustration. The terminology herein is used to describe specific embodiments of the invention, but their usage does not limit the invention, except as outlined in the claims. [0047] Various methods are described below to provide an example of each claimed embodiment. They do not limit any claimed embodiment. Any claimed embodiment may cover methods that are different from those described above and below. The drawings and descriptions are for illustrative, rather than restrictive, purposes. [0048] In this disclosure, a generalized Brunauer–Emmett–Teller (gBET) isotherm is presented, a simple and yet rigorous thermodynamic framework for multicomponent multilayer adsorption equilibria. Treating the adsorbent vacant sites as a part of the thermodynamic system, gBET considers adsorbent surface heterogeneity, competitive adsorption on the monolayer, condensation-evaporation on the subsequent layers, and adsorbed phase nonideality for the monolayer and the subsequent layers. The monolayer adsorbed phase nonideality is tracked using an area-based adsorption nonrandom two-liquid activity coefficient model. The condensation- evaporation phenomenon for the adsorption beyond monolayer is determined at the dew point condition of the mixed gas with either an equation of state or an activity coefficient model for the vapor-liquid equilibria. The disclosed model is validated with six single and three binary multilayer adsorption equilibrium systems, and the model results are compared against those from classical BET isotherm for single component adsorption and Ideal Adsorbed Solution Theory for mixed-gas adsorption equilibria. [0049] gBET Isotherm for Single Component Adsorption [0050] The generalized Brunauer-Emmett-Teller (gBET) isotherm retains the fundamental assumptions of classical Brunauer-Emmett-Teller (BET) isotherm, i.e., first adsorption layer due to adsorption, subsequent layers due to condensation, adsorbed molecules acting as adsorbing sites for next adsorption layer, and potential for an infinite number of adsorption layers [20]. Same as the activity-based thermodynamic Brunauer-Emmett-Teller (tBET) isotherm, gBET further treats adsorption vacant sites as an integral part of the thermodynamic system for modeling and substitutes the first layer concentrations with activities to address adsorbent surface heterogeneity and adsorbate-adsorbent interactions at the first layer. The difference between gBET and tBET resides in the treatment of adsorbate molecular size. The gBET derivation follows. [0051] On the adsorbent surface, the adsorption and desorption equilibrium reaction of adsorbate component ^^^^ with the vacant adsorbing site ^^^^ is: ^^^^(^^^^) + ^^^^ ↔ ^^^^ ∙ ^^^^ (1) where ^^^^ ∙ ^^^^ represents the occupied the rates of adsorption and desorption reactions on the adsorbent surface are equal. ^^^^1 ^^^^1^^^^∙^^^^ ^^^^1^^^^^^^^1^^^ ^^^^ ^^^^ ^^^^ ^^^^ ^ 1^^^^ 1^^^^ ^^^^ ^^ ^ ^^ ^^^ = = = = Here, ^^^^^^^ ^ ^^^^ is the adsorption equilibrium constant; ^^^^1 and ^^^^−1 are the adsorption and desorption rate constants, respectively; ^^^^ is the pressure; ^^^^1^^^^∙^^^^ and ^^^^1^^^^ are the activities of adsorbate component ^^^^ and reference molecule ^^^^ representative of the vacant sites on the adsorption first layer; ^^^^1^^^^ and ^^^^1^^^^ are the amounts adsorbed in the adsorption first layer; ^^^^1^^^^ and ^^^^1^^^^ are the corresponding activity coefficients calculated based on the monolayer loadings of ^^^^^^^^ and ^^^^^^^^ to be discussed later; ^^^^^^^^ are the molecular areas; ^^^^^^^^ denotes the ratio of adsorbate component i and reference molecule ^^^^. The vacant site surface area with first layer ^^^^1^^^^^^^^^^^^ is the adsorption zeroth layer Θ0. The occupied surface area with the adsorption first layer ^^^^1^^^^^^^^^^^^ is the adsorption first layer Θ1: ^^^^1^^^^^^^^^^^^ ^^^^1 ^^^^ Θ1 = ^^^^1^^^^^^^^^^^^ = ^^^^^^^^ ^^^^ ^^^^ ^^^^ ^^^^ ^^^^ ^^^^^^^^1^^^^^^^^^^^^ = ^ ^^^^ ^^^^ 1^^^^ ^^^1^^^^ ^^^^Θ 0 (3) [0052] The adsorbed molecules are the layers. To achieve dynamic equilibrium for the first layer, the combined rate of adsorption on the vacant adsorbing sites and evaporation from the second layer must equal the combined rate of desorption from the first layer and condensation on the first layer. Thus, the dynamic equilibrium for the first layer can be established as follow: ^^^^−2Θ2 + ^^^^1^^^^^^^^1^^^^Θ0^^^^^^^^ = ^^^^2^^^^Θ1 + ^^^^−1^^^^1^^^^Θ1 (4) where ^^^^ and ^^^^−2 are the for second layer, respectively; Θ1 and Θ2 denote the adsorbent surface areas covered with one layer and two layers, respectively. Subtracting Eq. (2) from (4) results in the following equation for the second layer. ^^^^−2Θ2 = ^^^^2^^^^Θ1 (5) Similarly, for layers ^^^^ > 2 is: ^^^^−^^^^Θ^^^^ = ^^^^^^^^^^^^Θ^^^^−1 (6) [0053] Like the surface area for the , with two or ^^^^ number of layers is: ^^^^ Θ2 = ^^^^^^^^,^^^^^^^^Θ1 =�^^^^^^^^,^^^^^^^^� 1^^^^^^^^^^^^ ^^^^ ^^^^^^^^ ^^^^ ^^^^Θ 0 (7) 1^^^^ Here ^^^^ is the ^^^^, and should correspond to the reciprocal of the saturation pressure of the adsorbate component ^^^^, ^^^^^^^^^^^^^^^^ ^^^^ , or equivalently, the dew point pressure, ^^^^^^^^^^^^^^^^ ^^^^ , of adsorbate component ^^^^ at the system temperature. The total adsorbent surface area, ^^^^^^^^, shall be the summation of the vacant surface area Θ0 and the surface area involving the first and the subsequent layers, presented as follows: ∞ ^^^^^^^^ =� Θ^^^^ = Θ0 + Θ1 + Θ2 + ⋯ (9) Here ^^^^ is the adsorption properties of infinite series, the simplified relation is: 1 − ^^^^ ^^^^1^^^^^^^ ^^^^,^^^^^^^^ + ^ ^^^^ ^^^^1^^^ ^^^^^^^^ ^^^^ ^^^^ ^^^^^^^^ ^ Similarly, the total adsorbed formation excluding the vacant surface area is: ∞ ^^^^^^^^ = ^^^^ = + 2Θ2 + 3Θ3 + ⋯ (11) Again, simplification of summation property gives Eq. (12). Θ ^^^^^^^^ = 0 ^^^^ ∙ 1^^^^^^^^^^^^ ^^^^^^^^^^^^ For single component ^^^^^^^^ = ^^^^ ^^^^ + ^^^^∗ ^^^ ^ ^^^^ ^^^^ ^^^^ ^ ^^^^^^^ = ^^^^^^^^^^^^^^^^ (13) Here ^^^^ and ^^^^∗ are the amo ^^^^ unts due to condensation on subsequent layers, respectively; ^^^^^^^^ ^^^^ is the total amount adsorbed for the multilayer adsorption. Thus, combining Eqs. (10), (12), and (13) results in the gBET equation for single component adsorption equilibria. ^^^^^^^^ (^^^^^^^^ + ^^^^∗)^^^^^ ^^^^^^^^ ^^^^^^^^^^^^ = ^^^^ ^^^ = ^^^^ = ^^^^ ^^^^^^^^ ^^^^^^^^ ^^^^0 ^^^^ − ^^^^ ^^^^1^^^^ − ^^^^^^^^ (14) Combining component ^^^^ and ^^^^ on the ^^^^ ^^^^ ^^^^^^^^^^^^^^^^ ^^^^^^^^ ^^^^^ ^^^^^^^^ ^^^^^^^^ = ^^^ ^^^^0 = ^^^^ ^^^^1^^^^ − ^^^^ ^^^^^^^^ (15) ^^^^ ^^^^ Here,^^^^ = ^^^^ ; ^^^0 0 1^^^^ ^^^^^ and ^^^^^^^^ three- can be recovered by setting the molecular area ratio and the activity coefficients of adsorbate component and reference molecule to unity, given as follows. ^^^^^^^^ ^^^^ ^^^^ ^^^^^^^ ^^^^ = ^ ^^^ ^^^^ ^^^^^^^ (17) ^^^^ ^^^^ ^ ^ [0054] The adsorption constant are functions of temperature, as shown below. ^^^^^^^^^^^^ −∆^^^^^^^^^^^^^^^^,^^ 1 1 ^^^^^^^^ ^^^^ ^^ ^^^^ = ^^^^^^^^ exp� � − �� (18) ^^^^ ^^^^ ^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^ Here, ^^^^^^^^ and ^^^^ ^^^^^^^^^^^^ ^^^^,^^^^ are at reference temperature, ^^^^^^^^^^^^^^^^; ^^^^ represents the gas constant; ^^^^ denotes the system temperature; ∆^^^^^^^^^^^^^^^^,^^^^ is the enthalpy of adsorption; ∆^^^^^^^^^^^^^^^^^^^^,^^^^ refers to the enthalpy of condensation. [0055] Now referring to FIG.1, a calculation algorithm of the generalized gBET isotherm for pure component adsorption at a fixed system temperature is shown in accordance with one embodiment of the present disclosure. Input data is provided in block 102, which includes adsorption data (i.e., a pressure (P) and an adsorption isotherm (^^^^^^^^ ^^^^)), adsorbate areas (i.e., an adsorbate area for the adsorbate component (^^^^^^^^) and an adsorbate area for a reference molecule (^^^^^^^^)), and a saturation pressure (^^^^^^^ ^ ^^^^^^^^^^^^ ). A saturation adsorption loading for monolayer adsorption of the adsorbate s calculated in block 104 from an experimental adsorbate surface area or ^^^^ ^ 0 i 0 ^^^ ^^^^ = ^^^^^^^^ where ^^^^0 adsorbent surface area. Initial values or guesses for the gBET are provided in block 106. The gBET parameters include the adsorption equilibrium constant , a correction factor (^^^^^^^^) and a binary interaction parameter (^^^^^^^^^^^^) for the species ^^^^ – species ^^^^ pair. Initial values or guesses for an amount adsorbed of the component due to monolayer adsorption (^^^^^^^^), an amount adsorbed of the reference due to monolayer adsorption (^^^^^^^^) and a total of amounts adsorbed of the adsorbate component due to condensation on layers (^^^^^ ∗ ^^^) are provided in block 108. In addition, an adsorbed phase mole fraction the adsorbate component (^^^^^^^^) and the reference molecule (^^^^^^^^) in the monolayer only are calculated using Eqs. (39)-(40) in block 108. Finally, a gas phase mole fraction for the adsorbate component (^^^^^^^^) and the reference molecule (^^^^^^^^) are calculated from area-based aNRTL activity coefficient model using Eq. (35) in block 108. Eqs. (14)-(16) are solved for new values for ^^^^^^^^, ^^^^^^^^ and ^^^^^ ∗ ^^^ in decision block 110. In addition, a total amount of adsorbate component ^^^^ due to monolayer adsorption and condensation on subsequent layers (^^^^^^^^ ^^^^) is calculated in decision block 110. If the guessed values for ^^^^^^^^, ^^^^^^^^ and ^^^^^ ∗ ^^^ from block 106 do not match the calculated values from Eqs. (14)-(16) in decision 110, as determined in decision block 110 (False), the process loops back to block 108 where new values or guesses are used for ^^^^^^^^, ^^^^^^^^ and ^^^^^ ∗ ^^^. If, however, the guessed values for ^^^^^^^^, ^^^^^^^^ and ^^^^^ ∗ ^^^ from block 106 match the values from Eqs. (14)-(16) in decision block 110, determined in decision block 110 (True), the process proceeds to decision block 112. If Eq. (41) is not minimized, as determined in decision block 112, the process loops back to block 106 where the new values for the adsorption equilibrium constant (^^^^^^^^ ^^^^ ), the correction factor (^^^^^^^^) and the binary interaction parameter (^^^^^^^^^^^^) are used. If, however, Eq. (41) is minimized, as determined in decision block 112, the parameters (i.e., saturation adsorption loading for monolayer adsorption of the adsorbate component (^^^^0 ^^^^ ), adsorption equilibrium constant (^^^^^^^ ^ ^^^^), condensation-evaporation equilibrium constant (^^^^^^^^,^^^^), the correction factor (^^^^^^^^) and binary interaction parameter (^^^^^^^^^^^^)) are output in block 114. [0056] Note that the algorithm can be to model adsorption isotherms for multiple temperatures instead of a fixed system temperature as shown in FIG.1. In such a case, temperature (T) would be included in block 102, and new values for the adsorption equilibrium constant (^^^^^^^ ^ ^^^^) and the condensation-evaporation equilibrium constant (^^^^^^^^,^^^^) would be calculated using Eqs. (18)- (19) in block 108. [0057] gBET Isotherm for Multicomponent Adsorption Equilibria [0058] Referring now to FIG. 2, the gBET isotherm formulation can be further generalized for multicomponent multilayer adsorption equilibria as shown. In this example, vacant sites 202 (dotted circles) are shown on the base layer 204 of the adsorbent 206. Adsorbate molecules 1 (gray 206), 2 (red 208) and 3 (blue 210) are shown on layers I (212), II (214) and II (216). [0059] Taking into account the competitive multicomponent adsorption on the adsorbent surface, the occupied adsorbent surface area in the first layer 212 due to adsorbate component ^^^^, Θ1^^^^, is: ^^^^ Θ1^^^^ = ^^^^ 1^^^^^^^^^^^^ 1^^^^^^^^^^^^ = ^^^^ ^^^^^^^^ ^^^^ ^^^^ ^^^^ ^^^^^^^^ 1^^^^ ^^^^ ^^^^ (20) [0060] Here ^^^^1^^^^ is the moles is the gas phase mole fraction. The total occupied adsorbent surface area due to the multicomponent adsorption in the first layer 212 is: ^^^^ ^^^^ ^^^^1^^^^^^^^^^^^ ^^^^^^^^ ) Here, ^^^^ denotes the number [0061] Beyond the first layer 212, the subsequent adsorption layers 214, 216 due to condensation- evaporation of the mixed-gas can be expressed as follows: ^^^^^^^^^^^^(^^^^) + Θ1 ↔ Θ2 (22) Here ^^^^^^^^^^^^ denotes the gas mixture with specific gas phase composition, ^^^^, dew point pressure, ^^^^^^^^^^^^^^^^, and dew point liquid phase composition, ^^^^∗. The occupied surface coverage on the second layer 214 and ^^^^^^^^ℎ layer 216…m can be written as follows: ^^^^ ^^^^ Θ 1^^^^^^^^^^^^ 2 = ^^^^^^^^^^^^Θ1 = Θ0(^^^^^^^^^^^^)� ^^^^^^^^^^^^ ^^^^ ^^^^ ^^^^ ^^^^ 1^ (23) ^^^^=1 ^^^ ^^^^ ^^^^ Θ^^^^ = ^^^^^^^^^^^^Θ^^^^−1 = Θ0(^^^^^^^^^^^^)^^^^−1� 1^^^^^^^^^^^^ ^^^^^^^^^^^^ ^^^^ (24) ^^^^ ^^^^ ^^^^ Where Θ and Θ^^^^ are layers, respectively. ^^^^^^^^ is the condensation-evaporation equilibrium constant, and it should correspond to the reciprocal of the dew point pressure, ^^^^^^^^^^^^^^^^ , of the mixed-gas at the system temperature. Making use of Eqs. (21), (23), and (24), the sum of vacant and occupied surface areas results in the total adsorbent surface area, written as follows: ∞ ^^^^^^^^ =� Θ^^^^ = Θ0 + Θ1 + Θ2 + ⋯ (25) The resulting simplified 1 − ^^^^^^^^^^^^ + ∑ ^ ^^^ ^ ^^ =^ ^^^^1^^^^^^^^^^^^ ^^^^ 1 ^^^^^^^^ ^^^^^^^^^^^^ ^^^^ [0062] The total adsorbed vacant surface area is: ∞ ^^^^^^^^ ^^^^ + 2Θ2 + 3Θ3 + ⋯ (27) Upon simplification, ^^^^ ^^^^ ^^^^ Θ0 ^^^^1^^^^^^^^^^^^ ^^^^^^^^^^^^ For multicomponent ^^^^ ^^^^ ^^^^^^^^ =�^^^^^^^^^^^^^^^^ + ^^^^∗�^^^^∗ ^^^^ ^^^^^^^^ (29) where ^^^^^^^^ is the amount adsorbed ^^^^∗ is the total amounts adsorbed due to condensation on subsequent layers; ^^^^^^ ∗ ^^ is the dew point liquid phase composition. Combining Eqs. (26), (28), and (29) results in the gBET isotherm for multicomponent adsorption equilibria. ∑^^^^ ^^^^ ^^^ ∗ ^^^^ ∗ ^^^^=1 ^^^^^^^^^^^^^^^^ + ^^^^ ∑^^^^=1 ^^^^^^^^ ^^^^^^^^ ∑^ ^^^ ^ ^= ^^ 1^^^^ ^^^^^ 1 ^^^^ ^^^^^^ ^ ^^ ^^^^^^^^^^^^^^^ = 1^^^^ ^^^^ ^^^^ loadings of component ^^^^ and ^^^^ on the monolayer follows: ^^^^^^^^^^^^^^^^ ^^^^^^^^ ^^^^^^^^^^^^^^^^^ ^^^^^^^^ = = ^^^^ ^^^ ^^^^0 ^^^^ ^^^^1^^^^^^^^^^^^ ^^^^^^^^^^^^ ^^^^ The total amount and condensation on subsequent layers can be calculated from the following expression. ^^^^^^^^ ^^^^ = ^^^^^^^^ + ^^^^∗^^^^∗ ^^^^ (33) [0063] Note that the gBET system involving the adsorbate component ^^^^ and the adsorbent vacant sites represented with the reference molecule ^^^^. Similarly, the binary gas adsorption system is considered as a ternary system of two adsorbates and the adsorbent vacant sites, and so on. The adsorbent surface heterogeneity and the nonideality of the monolayer where the adsorbates are in direct contact with the adsorbent surface are addressed with the area-based aNRTL model. The activity coefficients are functions of the monolayer adsorbed phase composition and the molecular areas of the species. gBET makes no assumptions on the nonideality of the condensed liquid above the monolayer since it is up to the thermodynamic model appropriately chosen for the dew point calculations. In order to calculate the amount adsorbed of ^^^^ adsorbate components, one Eq. (30), ^^^^ number of Eq. (31), and one Eq. (32) coupled with the aNRTL activity coefficient model need to be solved simultaneously. [0064] Now referring to FIG.3, a calculation algorithm of the gBET isotherm for multicomponent adsorption equilibria at a fixed system temperature is shown in accordance with one embodiment of the present disclosure. Input data is provided in block 302, which includes mixed-gas adsorption data (i.e., pressure (P), gas phase mole fraction for each adsorbate component (^^^^^^^^) and adsorption isotherm for each adsorbate component (^^^^^^^^ ^^^^)), pure component gBET parameters (see e.g., FIG. 1)(i.e., saturation adsorption loading for monolayer adsorption of each adsorbate component (^^^^^ 0 ^^^), adsorption equilibrium constant (^^^^^^^ ^ ^^^^), condensation-evaporation equilibrium constant (^^^^^^^^,^^^^) and binary interaction parameter ( ^^^^^^^^^^^^ )), adsorbate areas (i.e., adsorbate area for each adsorbate component (^^^^^^^^) and adsorbate area for a reference molecule (^^^^^^^^)), vapor-liquid equilibria (i.e., dew point pressure for each adsorbate component (^^^^^^^^^^^^^^^^ ^^^^ ) and a dew point liquid phase composition (^^^^^ ∗ ^^^). Initial values or guesses for the gBET binary parameters are provided in block 304. The gBET binary parameters include a correction factor (^^^^^^^^) and a binary interaction parameter (^^^^^^^^^^^^) for the species ^^^^ – species ^^^^ pair. Initial values or guesses for an amount adsorbed of each adsorbate component due to monolayer adsorption (^^^^^^^^), an amount adsorbed of the reference molecule due to monolayer adsorption (^^^^^^^^) and a total of amounts adsorbed of each adsorbate component due to condensation on subsequent layers (^^^^^ ∗ ^^^) at a given gas phase mole fraction for each adsorbate component (^^^^^^^^) and pressure (P) are provided in block 306. In addition, an adsorbed phase mole fraction for each adsorbate component (^^^^^^^^) and the reference molecule (^^^^^^^^) in the monolayer only are calculated using Eqs. (39)-(40) in block 306. Finally, a gas phase mole fraction for each adsorbate component (^^^^^^^^) and the reference molecule (^^^^^^^^) are calculated from area-based aNRTL activity coefficient model using Eq. (35) in block 306. Eqs. (30)-(32) are solved for new values for ^^^^^^^^, ^^^^^^^^ and ^^^^^ ∗ ^^^ in decision block 308. In addition, a total amount of adsorbate component ^^^^ due to monolayer adsorption and condensation on subsequent layers (^^^^^ ^ ^^ ^^ ^^) is calculated using Eq. (33) in decision block 308. If the guessed values for ^^^^^^^^, ^^^^^^^^ and ^^^^^ ∗ ^^^ from block 306 do not match the calculated values from Eqs. (30)-(32) in 308, as determined in decision block 308 (False), the process loops back to block 306 where the new values for ^^^^^^^^, ^^^^^^^^ and ^^^^^ ∗ ^^^ are used. If, however, the guessed values for ^^^^^^^^ , ^^^^^^^^ and ^^^^^ ∗ ^^^ from block 306 the calculated values from Eqs. (30)-(32) in decision block 308, as determined in decision block 308 (True), the process proceeds to decision block 310. If Eq. (41) is not minimized, as determined in decision block 310, the process loops back to block 304 where the new values for the correction factor (^^^^) and the binary interaction parameter (^^^^^^^^^^^^) are used. If, however, Eq. (41) is minimized, as determined in decision block 310, the binary parameters (i.e., binary interaction parameter (^^^^^^^^^^^^) and correction factor (^^^^)) and mixed-gas adsorption equilibria (i.e., pressure (P) and adsorption isotherm for each adsorbate component (^^^^^^^^ ^^^^) at gas phase mole fraction for each adsorbate component (^^^^^^^^)) are output in block 312. [0065] Note that the algorithm can be modified to model adsorption isotherms for multiple temperatures instead of a fixed system temperature as shown in FIG.3. In such a case, temperature (T) would be included in block 102, and new values for the adsorption equilibrium constant (^^^^^^^ ^ ^^^^), the condensation-evaporation equilibrium constant (^^^^^^^^,^^^^) are calculated using Eqs. (18)-(19) and the binary interaction parameter (^^^^^^^^^^^^) using Eq. (37). [0066] Area-based Adsorption Nonrandom Two-Liquid Model [0067] Developed on the basis of the Nonrandom Two-Liquid theory [35], the aNRTL model accounts the dominant adsorbate-adsorbent interactions of the monolayer adsorbed phase through the difference in adsorbate-adsorbent interaction energetic strength. The corresponding excess Gibbs energy expression and the activity coefficient equation are given below. [14,18,32] ^^^^^^^^ ^^^^+1 ∑^^^^+1 ^^^^ ^^^^ = ^^^^ ^^^^=1 ^^^^ ^^^^ ^^^^^^^^^^^^ ^^^^^^^^^^^^
^^^^ ^^^^ = ^^^^ ^^^^ ∑^^^^ ^^^^=1 ^^^^^^^^ + ^^^^^^^^ (39) where ^^^^^^^^ and ^^^^^^^^ are the adsorbed are the interaction energies; ^^^^ is the nonrandomness factor set to 0.3 per the NRTL convention [35]; ^^^^^^^^^^^^ is the binary interaction parameter for the species ^^^^ – species ^^^^ pair with the temperature dependence containing an entropic contribution term, ∆^^^^^^^^^^^^, and an enthalpic contribution term, ∆ℎ^^^^^^^^. The aNRTL model uses a symmetric reference state, i.e., ^^^^^^^^ → 1 at ^^^^^^^^ → 1 and ^^^^^^^^ → 1 at ^^^^^^^^ → 1, to address the nonideality of the monolayer. The detailed derivation and notable features of area-based aNRTL model is available in the literature. [14] [0068] Results and Discussion [0069] To validate the gBET isotherm for multicomponent multilayer adsorption equilibria, three different binary adsorption mixtures on three adsorbents at various ranges of compositions and pressures have been investigated. The binary mixtures include O2 (1) + N2 (2) adsorption on TiO2 anatase at five different compositions and 78.2 K, [36] benzene (1) + cyclohexane (2) adsorption on activated charcoal at four different compositions and 303.15 K, [37] and H2O (1) + CO2 (2) adsorption on activated alumina F-200 at varying relative humidity of H2O and constant partial pressure of CO2 at 303.15 K [26]. Also investigated are the single component adsorption of O2 and N2 on TiO2 anatase at 78.2 K, [36] benzene and cyclohexane on activated charcoal at 303.15 K, [37] and CO2 and H2O on activated alumina F-200 at 303.15 K [26]. [0070] To determine the BET and gBET model parameters from the single and binary adsorption equilibria data, a maximum likelihood principle-based objective function is minimized, as given in Eq. (41). [38] To compare the performance of model against the experimental data, the average relative deviation (ARD%) is calculated using Eq. (42). ^^^^ ^^ ^^^^^^^^^^^^^^^^ ^^^^ ^^^^^^^^^^^^^^^^ 2 ^^^^^^^ ^ ^^ ^^^^ − ^^^^^^^^ ^^^^ ^^^^ ^^ ^^^^^^^^^^^^^^^^ ^^^^ ^^^^^^^^^^^^^^^^ 100 ^^^^^^ ^^^^^^^^^^^^ % = �� ^^^^ ^^^^ − ^^^^^^^^ ^^^^ ^^^^ ^^^^^^^^ ^^^^^^^^^^^^^^^^ � (42) ^^^^ where ^^^^ is the data point index; ^^^^ is the are model-calculated result and the data measurement, i.e., adsorption loadings for single and binary component systems as a function of pressure and composition, respectively. ^^^^^^^^^^^^^^^^^^^^ is the experimental data standard deviation with a default value of 0.05 mmol/g. [0071] Single Component Adsorption Isotherm [0072] Referring now to FIGS. 4A-4C, the BET and gBET representations for the single component multilayer adsorption of the six adsorbates on the three adsorbents are shown in accordance with one embodiment of the present disclosure. The regressed parameters for BET and gBET using the experimental data are reported in Table 1 and Table 2, respectively in FIG.5. The BET isotherm has three adjustable parameters (^^^^0 ^^^^, ^^^^^^^^ ^^^^ , and ^^^^^^^^,^^^^) while the gBET isotherm has four model parameters (^^^^0 ^^^^ , ^^^^^^^^ ^^^^ , ^^^^^^^^,^^^^ , and ^^^^^^^^^^^^ ). Note that ^^^^0 ^^^^ can be calculated from the adsorbent surface area and ^^^^,^^^^ parameter is treated as the reciprocal of the product of the saturation pressure and a correction factor ^^^^, i.e., ^^^^ 1 ^^^^,^^^^ = ^^^^^^^^^^^^ ^^^^^^ ^ ^^ ^^^^^^^^^^^×^^^^. In this disclosure, ^^^^^^^^ is calculated from VLE models while ^^^^ is regressed. [0073] FIG.4A shows the adsorption of both O2 and N2 on TiO2 anatase at 78.2 K exhibits a rapid increase in adsorption loading as the pressure approaches the corresponding saturation pressures, typical of multilayer adsorption isotherm behavior. Both BET and gBET give similar representations and capture the experimental data above ~ 0.01 bar. At pressures below ~ 0.01 bar, the BET isotherm significantly deviates from the experimental data while the gBET isotherm accurately represents the data, although both the BET and gBET isotherms obey Henry’s law in the low-pressure region with slope of unity in the log-log scale. Table 1 and Table 2 in FIG. 5 show the ARD%’s from gBET are less than 9% and 6% for O2 and N2, respectively, while the ARD%’s from BET can be as high as 23% and 13% for O2 and N2, respectively. The BET and gBET model parameters are nearly identical for the saturation loading, ^^^^0 ^^^^, and the condensation- evaporation constant, ^^^^^^^^,^^^^, while the BET and gBET model parameters are significantly different for the adsorption equilibrium constant, ^^^^^^^^ ^^^^ ^^^^ . This difference is due to the fact that the ^^^^^^^^ in gBET together with the adsorbate-adsorbent interaction parameter, ^^^^^^^^^^^^, take into account the adsorbent surface heterogeneity and the monolayer adsorbed phase nonideality. Table 1 and Table 2 in FIG. 5 show the ^^^^^^^ ^ ^^^^^^^^^^^^’s for O2 and N2 as calculated from the VLE model; the factor ^^^^ is found to be close to ~ 1.1 to 1.2, for both BET and gBET. [0074] The single component adsorption isotherms of benzene and cyclohexane on activated charcoal at 303.15 K with the BET and gBET representations are shown in FIG. 4B. The corresponding model parameters are given in Table 1 and Table 2 in FIG.5. FIG.4B shows BET qualitatively represents the benzene and cyclohexane isotherm data with the ARD%’s larger than 6% while gBET quantitively matches the experimental data for the complete range of pressure with the ARD%’s less than 1.4%. Table 1 and Table 2 in FIG. 5 show the ^^^^^^^ ^ ^^^^ parameter values from BET are significantly higher than those for the ^^^^^^ ^ ^^ ^^^ parameters from gBET. On the other hand, the ^^^^0 ^^^^ parameter and the ^^^^^^^^,^^^^ parameter values from both BET and gBET are similar. Furthermore, the ^^^^ is found to be ~ 3 for BET and ~ 4 to 5 for gBET. [0075] The adsorption isotherms of water and carbon dioxide on activated alumina F-200 at 303.15 K together with the BET and gBET model results are shown in FIG. 4C. The model parameters are reported in Table 1 and Table 2 in FIG.5. FIG. 4C shows the Type II multilayer adsorption isotherm data for H2O where the adsorption loading rapidly increases with the increase in H2O partial pressure. In contrast, the adsorption data for CO2 suggest monolayer adsorption where the isotherm exhibits concave down shape. Both BET and gBET result in satisfactory fitting for H2O adsorption isotherm data with the ARD%’s around 5%. In fact, the ^^^^^^^^^^^^ parameter value for gBET is zero and the ^^^^0 and ^^^^ value ^^^^ ^^^^ s for both BET and gBET are The ^^^^^^^^ parameter value from gBET is the ^^ ^^^^ ^^^^^^ parameter from BET due to the inclusion of adsorbate size in the gBET isotherm. The corresponding ^^^^ value is ~ 1.2 for both BET and gBET. For CO2 adsorption isotherm, gBET provides better representation of the data than BET. The ARD% is 2.4% for gBET and 5.0% for BET. The ^^^^^^^^^^^^ value for gBET is –2.49 indicative of nonideal monolayer adsorbed phase. The ^^^^0 value BET is almost half of the ^^^^0 ^^^^ value from while the ^^^^^^^^ ^^^^ value from BET is ten times of the ^^^^^^^^ ^^^^ value from gBET. Due to the absence of multilayer adsorption behavior for CO2, the factor ^^^^ is set to unity for both BET and gBET. [0076] Dew Point Calculations [0077] Referring now to FIGS.6A-6C, the vapor-liquid equilibria of the three pairs of adsorbates are shown in accordance with one embodiment of the present disclosure. To calculate multicomponent multilayer adsorption loadings of adsorbate components, the gBET isotherm for multicomponent adsorption, Eqs. (30) – (32), requires information about the dew point pressure, ^^^^^^^^^^^^^^^^, and the dew point liquid phase composition, ^^^^∗, of the mixed-gas at the system temperature. The dew point calculations for the three binary mixtures have been carried out using Aspen Properties V1439. The choice of thermodynamic models depends on the type of components in the multicomponent mixtures and the thermodynamic conditions. These thermodynamic models are also used to calculate the ^^^^^^^ ^ ^^^^^^^^^^^^ ’s required for the single component adsorption isotherm calculations previously described. [0078] The O2 (1) + N2 (2) binary mixture is modeled using the well-established Peng-Robinson equation of state (PR EOS) 39, 40. The PR EOS binary interaction parameter, ^^^^^^^^^^^^, was regressed against the literature vapor-liquid equilibrium (VLE) experimental data at 77 – 88 K 41-43. Excellent agreement between the experimental data and the model results was obtained with ^^^^^^^^^^^^ = −0.0153 ± 0.0017. The dew point calculations at 78.2 K are shown in FIG. 6A which illustrates the monotonic decrease in dew point pressure with the increase in the vapor phase and liquid phase O2 mole fraction. [0079] The benzene (1) + cyclohexane (2) binary mixture is modeled using the ideal gas equation of state for the vapor phase and the nonrandom two-liquid (NRTL) activity coefficient 35, 39 model for the liquid phase solution nonideality. The asymmetric NRTL binary interaction parameters, ^^^^ ≠ ^^^^ , were regressed using the li 44, 45 ^^^^^^^^ ^^^^^^^^ terature VLE data at 303.15 K with ^^^^12 = 0.580 ± and ^^^^21 = −0.025 ± 0.026. The model representation with experimental data is presented in FIG. 6B. The binary mixture of benzene (1) + cyclohexane (2) shows a maximum pressure azeotrope at the vapor and liquid phase mole fractions of benzene around 0.5. It suggests that the selectivity shall be reversed as the vapor phase mole fraction of benzene crosses the azeotropic composition. [0080] The H2O (1) + CO2 (2) binary system is modeled using Perturbed-Chain Statistical Associating Fluid Theory (PC-SAFT) of Gross and Sadowski 39, 46. Yan and Chen 47 accurately regressed and validated the PC-SAFT model parameters for the binary mixture that remain valid at high CO2 pressures. FIG.6C shows the VLE results based on the PC-SAFT model parameters from Yan and Chen 47. The dew point pressure sharply decreases with an increase in the vapor phase mole fraction of H2O at 303.15 K. [0081] Mixed-gas Adsorption Equilibria [0082] The mixed-gas adsorption equilibria for the three binary adsorption systems with varying gas phase compositions are calculated with both IAST and the gBET isotherm. The IAST results were calculated from the BET isotherm with the BET parameters reported in Table 1 in FIG.5 for single component adsorption. The gBET results were calculated with the gBET parameters reported in Table 2 in FIG.5 for single component adsorption. As illustrated above for both BET and gBET, the factor ^^^^ relating ^^^^^^^^,^^^^ and ^^^^^^^^^^^^^^^^ ^^^^ has been treated as an adjustable parameter in fitting the single component adsorption isotherm data. Therefore, the factor ^^^^ relating ^^^^^^^^ and ^^^^^^^^^^^^^^^^ for the mixed-gas adsorption gBET isotherm is also treated as an adjustable parameter and regressed from the binary adsorption equilibria data along with the binary aNRTL interaction parameter ^^^^12. Interesting, it was found that the factor ^^^^ determined from the binary adsorption equilibria data is same or close to that determined from single component adsorption isotherm data for the same adsorbent. The regressed parameters for the mixed-gas adsorption systems and the corresponding ARD%’s from both IAST and gBET are presented in Table 3 in FIG.7. In addition, the predictive capability of the gBET isotherm has been investigated with ^^^^12 set to zero while keeping the same value for ^^^^. The results are also given in Table 3 in FIG.7. [0083] Referring to FIGS. 8A-8E, the IAST and gBET results for the adsorption of O2 (1) + N2 (2) binary mixture on TiO2 anatase at 78.2 K at five different gas phase compositions are shown in accordance with one embodiment of the present disclosure. FIG.8A to FIG.8E show that IAST consistently overpredicts the O2 loading and underpredicts the N2 loading as ^^^^1 increases from 0.149 to 0.853. To the contrary, gBET satisfactorily predicts all the adsorption loadings of both adsorbate components for the complete range of experimental data including the crossover of the O2 loading and the N2 loading at ^^^^1 = 0.298, 0.502, and 0.702 with ^^^^ = 1.17. Table 2 in FIG. 5 and Table 3 in FIG.7 show the factor ^^^^ remains the same for the same adsorbent whether for single or multicomponent adsorption. Table 3 in FIG. 7 further indicates that the ARD% from IAST is 61.3% while the ARD% from gBET is merely 13.9 % with ^^^^12 set to 0 and reduced to 8.0% when ^^^^12 is adjusted. [0084] Now referring to FIGS. 9A-9D, the model results for the adsorption of benzene (1) + cyclohexane (2) on activated charcoal at 303.15 K up to 0.16 bar and at four different gas phase compositions are shown in accordance with one embodiment of the present disclosure. Note that the literature 37 reported the total adsorption loading in mass-basis and the gas phase composition in mole-basis. No individual component loadings data were reported. FIG. 9A to FIG. 9D show the IAST predictions match well the total adsorption loading data up to 0.08 bar and then deviate on the high side from the data above 0.08 bar. In contrast, gBET accurately predicts the total adsorption loading data for the complete range of pressure with ^^^^ = 3.79. It is interesting that IAST predicts an adsorption selectivity switching from cyclohexane to benzene at ^^^^1 ≈ 0.3 while gBET predicts the adsorption selectivity switch at around the azeotrope point, i.e., ^^^^1 ≈ 0.5. Table 3 in FIG.7 shows that the ARD% are 3.9% for IAST, 1.9% for gBET with ^^^^12 set to 0, and 1.5% for gBET with ^^^^12 adjusted. [0085] The adsorption of CO2 from the binary mixture of H2O (1) + CO2 (2) is of high interest due to the immense interest in adsorption for CO2 capture applications. FIG.8 shows the experimental data 26 and the modeling results for the binary adsorption of H2O (1) + CO2 (2) on activated alumina F-200 at 303.15 K, constant CO2 partial pressure of 1.012 bar, and variable H2O relative humidity in accordance with one embodiment of the present disclosure. The experimental data shows a rapid increase in the H2O loading consistent with multilayer adsorption while the CO2 loading remains relatively constant and independent of the relative humidity. FIG. 10 shows the IAST results are available for only up to 37% relative humidity. Beyond this range, the IAST calculations fail because the constant spreading pressure constraint for the adsorbates cannot be satisfied due to the mismatch in the high spreading pressure for H2O and the low spreading pressure for CO2 for the system. Previous studies have reported similar failures and the related challenges in applying IAST for multilayer adsorption.26 Furthermore, inconsistent with the experimental data, IAST predicts rapid increase in the H2O loading and rapid drop in the CO2 loading with increasing relative humidity. On the other hand, gBET quantitatively estimate both the H2O loading and the CO2 loading across the entire range of relative humidity at ^^^^12 = 6.30 and ^^^^ = 1.13 with the ARD% of 11.9%. The ARD% for gBET is 44.0% if ^^^^12 is set to 0. Note that ^^^^ was manually adjusted to 1.13 to optimize the fit and ^^^^12 was regressed from the CO2 loading data alone. The high ^^^^12 value of 6.30 suggests strong nonideality in the monolayer adsorbed phase. The ARD% for IAST is not reported in Table 3 in FIG. 7 due to the failure in computing the binary adsorption above 37% relative humidity. [0086] Now referring to FIG. 11, a block diagram of a system 1100 in accordance with one embodiment of the present disclosure is shown. The system 1100 for identifying an adsorption equilibrium constant and a condensation-evaporation equilibrium constant for an adsorbate component in a gas adsorption system 1102 includes a memory 1104, an input/output interface 1106, and one or more processors 1108 communicably coupled to the memory 1104 and the input/output interface 1106. The one or more processors 1108: calculate an adsorption equilibrium constant (^^^^^^^^ ^^^^^^^^^^^^ −∆^^^^^^^^^^^^^^^^,^^^^ 1 1 ^^^^ ) for the adsorbate component (i) using ^^^^^^^^ ^^^^ = ^^^^^^^^ ^^^^ exp� ^^^^ � ^^^^ − ^^^^^^^^^^^^^^^^�� (see Eq. (18)), calculate a condensation-evaporation component (i) using ^^^^ ^^^^^^^^^^^^ −∆^^^^^^^^^^^^^^^^^^^^,^^^^ 1 1 ^^^^,^^^^ = ^^^^ ^^^^ ^^^^^^^^^^^^ ^^^^,^^^^ exp� ^^^^ � ^^^^ − ^^^^^^^^^^^^^^^^�� (see Eq. (19)), wherein: ^^^^^^^^ is the adsorption constant at a reference temperature (^^^^^^^^^^^^^^^^), ^^^^ ^^^^^^^^^^^^ ^^^^,^^^^ is the condensation-evaporation equilibrium constant the reference temperature , ^^^^ is a gas constant, ^^^^ is a system temperature, ∆^^^^^^^^^^^^^^^^,^^^^ is an enthalpy of adsorption, ∆^^^^^^^^^^^^^^^^^^^^,^^^^ is an enthalpy of condensation, and provide the adsorption equilibrium constant and the condensation-evaporation equilibrium constant to the input/output interface. The gas adsorption system is configured using the adsorption equilibrium constant and the condensation-evaporation equilibrium constant. Note that the one or more processors 1108 can be part of one or more controller, computers, servers or other devices suitable for performing the method. The memory 1104 can be any type of data storage. The input/output interface 1106 can be any component capable of interfacing with the one or more processors. The components can be local, remote or a combination thereof. Moreover, the components can be part of a distributed computing architecture, design system or control system. In another aspect, the gas absorption system 1102 comprises a CO2 capture system, a hydrocarbon processing system, an air separation system, a direct air capture system, a trace elements and heavy metals removal system, or an aqueous organic acids separation system. In another aspect, the gas absorption system 1102 comprises a liquid adsorption system mathematically formulated as a pseudo gas adsorption system. In another aspect, the liquid adsorption system comprises a trace elements removal system, a heavy metals removal system, an aqueous organic separation system, or an ethanol-water separation system. In another aspect, the input/output interface 1106 comprises an interface directly or indirectly to the gas adsorption system 1102. [0087] In one aspect, a product is produced using the gas adsorption system 1102. In another aspect, the gas adsorption system 1102 is further configured using a saturation adsorption loading for monolayer adsorption of the adsorbate component (^^^^0 ^^^^), a correction factor (^^^^^^^^) and a binary interaction parameter (^^^^^^^^^^^^) for the species ^^^^ – species ^^^^ pair. In another aspect, the adsorption equilibrium constant condensation-evaporation equilibrium constant are calculated using a regression analysis. aspect, the one or more processors 1108 calculate the adsorption equilibrium constant and the condensation-evaporation equilibrium constant by: providing input data comprising a pressure (P), an adsorption isotherm (^^^^^ ^ ^^ ^^ ^^), an adsorbate area for the adsorbate component (^^^^^^^^), an adsorbate area for a reference molecule (^^^^^^^^), and a saturation pressure (^^^^^^^^^^^^^^^^ ^^^^ ); calculating a saturation adsorption loading for monolayer adsorption of the adsorbate component ^^^^^^^^) from an experimental adsorbate surface area or ^^^^ ^^^0 ( 0 ^0 ^ ^^^ = 0 ^^^^^^^^ where ^^^^ is a total adsorbent surface providing initial values for the adsorption constant (^^^^^^^^), a correction factor (^^^^^^) ^^^^ ^^ and a binary interaction parameter (^^^^^^^^^^^^) for the species ^^^^ – species ^^^^ pair; providing initial values for an amount adsorbed of the component due to monolayer adsorption (^^^^^^^^), an amount adsorbed of the reference molecule due to monolayer adsorption (^^^^^^^^) and a total of amounts adsorbed of the adsorbate component due to condensation on ∗ layers (^^^^^^^^); calculating an adsorbed phase mole fraction for the adsorbate component in the monolayer only (^^^^^^^^) using ^^^^ ^^^^ ^^^^ = ^^^^ ∑^^^^ ^^^^=1 ^^^^^^^^ +^^^^^^^^; calculating an adsorbed phase mole fraction for the reference molecule in the ^^^^ (^^^^^^^^) using ^^^^^^^^ = ^^^^ ∑^^^^ ^^^^^^^^ +^^^^ ; calculating a gas phase mole fraction for the adsorbate component ( ^^^^ ) using ln ^^^^^ 2 ^^^^^^^^ 2 ^^^^^^^^^^^^^^^�Ω^^^^^^^^−1� , where ∆ℎ^^^^^^^^ �∑^^^^ ^^^^=1 ^^^^ in: ^^^^^^^^^^^^ = ∆^^^^^^^^^^^^ + ^^^^ ^^^^ , Ω^^^^^^^^ ^^^^^^^^^^^^Ω^^^^� exp�−^^^^^^^^ � and ^^^^ is a about 0.3 per phase mole fraction for the reference molecule ( ^^^^^^^^ ) using ln ^^^^^^^^ = ^^^^2^^^^2^^^^^^^^^^^^�Ω^^^^ −1� ^^^^ ^^^^+1 ^^^^ ^^^^ ^^^^ ; calculating a new value for ^^^^^^^^, ^^^^^^^^ and ^^^^∗ ^^^^^^^^ = �^^^^^^^^+^^^^ ∗ ^^^^�^^^^^^^^ ^^^^ ^^^^ ^^^^ ^^^^^^ ^ ^^ ^^^^^^^ ^^^^^^^^ ^^^^^^^^ = ^^^^0 ^^^^ = �1−^^^^^^^^,^^^^^^^^�� ^^^^ 1^^^^ , ^^^^ (1−^^^^ ^^^^ ^^^^ 1^^^^ ^^^^ ^^^^ ^^^^,^^^^ )+^^^^^^^^ ^^^^� calculating a due to monolayer adsorption and condensation on subsequent layers (^^^^^^^^ ^^^^ ∗ ^^^^) using ^^^^^^^^ = ^^^^^^^^ + ^^^^^^^^; and repeating steps (a)-(h) whenever the initial values for ^^^^^^^^, ^^^^^^^^ and ^^^^^ ∗ ^^^ do not match the calculated values for ^^^^^^^^, ^^^^^^^^ and ^^^^^ ∗ ^^^; repeating steps (a)-(i) whenever = ∑ ^ ^^^^^^^^^^^^^^^^ ^^^^ ^^^^^^^^^^^^^^^^ 2 ^^^^ ^^^^ ^^^ −^^^^ ^^^^ ^^^^ ^^^^ ^^^^ ^^^^=1� ^^^^^^^^^^^^^^^^^^^^ � is not minimized. (see FIG.1). In another aspect, the adsorption constant and the condensation-evaporation equilibrium constant are calculated for multiple temperatures. In another aspect, the gas absorption system 1102 comprises: an adsorbent comprising titanium dioxide anatase; and the adsorbate component comprises oxygen or nitrogen. In another aspect, the gas absorption system 1102 comprises: an adsorbent comprising activated charcoal; and the adsorbate component comprises benzene or cyclohexane. In another aspect, the gas absorption system 1102 comprises: an adsorbent comprising activated alumina; and the adsorbate component comprises water or carbon dioxide. [0088] Again referring to FIG.11, a block diagram of a system 1150 in accordance with another embodiment of the present disclosure is shown. The system 1150 for identifying a total amount of two or more adsorbate components absorbed for a mixed-gas multilayer adsorption system 1102 includes a memory 1104, an input/output interface 1106, and one or more processors 1108 communicably coupled to the memory 1104 and the input/output interface 1106. The one or more processors 1108: calculate the total amount of the two or more adsorbate components (i) absorbed (^^^^^^^^ ^^^^) using ^^^^^^^^ ^^^^ = ^^^^^^^^ + ^^^^∗^^^^∗ ^^^^ (see Eq. (33)), wherein: ^^^^^^^^ is an amount adsorbed of each adsorbate component (^^^^) due to monolayer adsorption, ^^^^∗ is a total of amounts adsorbed due to condensation on subsequent layers, ^^^^^ ∗ ^^^ is a dew point liquid phase composition, and provide the total amount of the two or more adsorbate components (i) absorbed (^^^^^ ^ ^^ ^^ ^^) to the input/output interface 1106. The mixed-gas multilayer adsorption system 1102 is configured using the total amount of two or more adsorbate components absorbed. Note that the one or more processors 1108 can be part of one or more controller, computers, servers or other devices suitable for performing the method. The memory 1104 can be any type of data storage. The input/output interface 1106 can be any component capable of interfacing with the one or more processors. The components can be local, remote or a combination thereof. Moreover, the components can be part of a distributed computing architecture, design system or control system. In another aspect, the mixed-gas multilayer adsorption system 1102 comprises a CO2 capture system, a hydrocarbon processing system, an air separation system, a direct air capture system, a trace elements and heavy metals removal system, or an aqueous organic acids separation system. In another aspect, the mixed-gas multilayer adsorption system 1102 comprises a liquid adsorption system mathematically formulated as a pseudo gas adsorption system. In another aspect, the liquid adsorption system comprises a trace elements removal system, a heavy metals removal system, an aqueous organic separation system, or an ethanol-water separation system. In another aspect, the input/output interface 1106 comprises an interface directly or indirectly to the mixed-gas multilayer adsorption system 1102. [0089] In one aspect, a product is produced using the mixed-gas multilayer adsorption system 1102. In another aspect, the mixed-gas multilayer adsorption system 1102 is further configured using a binary interaction parameter (^^^^^^^^^^^^), a correction factor (^^^^) and a pressure (P). In another aspect, the total amount of each of two or more adsorbate components (i) absorbed for the mixed-gas multilayer adsorption system (^^^^^^^^ ^^^^) are calculated using a regression analysis. In another aspect, the one or more processors 1108 calculate the total amount of each of the two or more adsorbate components (i) absorbed for the mixed-gas multilayer adsorption system (^^^^^^^^ ^^^^ ) by: providing input data comprising a pressure (P), a gas phase mole fraction for each adsorbate component (^^^^^^^^), the total amount of the two or more adsorbate components (i) absorbed (^^^^^^^^ ^^^^), a saturation adsorption loading for monolayer adsorption of each adsorbate , an adsorption equilibrium constant for each adsorbate component (^^^^^^ ^ ^^ ^^^), a condensation-evaporation equilibrium constant for each adsorbate component (^^^^^^^^,^^^^), a binary interaction parameter (^^^^^^^^^^^^) for the species ^^^^ – species ^^^^ pair, an adsorbate area for each adsorbate component (^^^^^^^^), an adsorbate area for a reference molecule (^^^^^^^^), a dew point pressure for each adsorbate component (^^^^^^^^^^^^^^^^ ^^^^ ), and a dew point liquid phase composition (^^^^^ ∗ ^^^); providing initial values for the binary interaction parameter (^^^^^^^^^^^^) and a correction factor (^^^^); providing initial values for an amount adsorbed of each adsorbate component due to monolayer adsorption (^^^^^^^^ ), an amount adsorbed of the reference molecule due to monolayer adsorption (^^^^^^^^) and a total of amounts adsorbed of each adsorbate component due to condensation on layers (^^^^^ ∗ ^^^) at a given gas phase mole fraction for each adsorbate component (^^^^ ^^^^) and (P); calculating an adsorbed phase mole fraction for each adsorbate component in the monolayer only (^^^^ ^^^^^^^^ ^^^^ ) using ^^^^^^^^ = ∑^^^^ ^^^^=1 ^^^^^^^^ +^^^^^^^^; calculating an adsorbed phase mole fraction for the reference molecule in the (^^^^^^^^) using ^^^^^^^^ = ^^^^ ^^^^ ∑^^^^ ^^^^=1 ^^^^^^^^ +^^^^^^^^; calculating a gas phase mole fraction for each adsorbate component (^^^^^^^^) using ln ^^^^^^^^ = ^^^^2^^^2 ^^^^ ∑ ^^^^+1 ^^^^ ^^^^^^^^^^^^^^^^^�Ω^^^^^^^^−1� , w ∆ℎ^^^^^^^^ ^^^^ ^^^^=1�∑^^^^ ^^^^=1 ^^^^ 2 herein: ^^^^^^^^^^^^ = ∆^^^^^^^^^^^^ + , Ω^^^^^^^^ = exp�−^^^^^^^^^^^^^^^^� and ^^^^ is a ^^^^^^^^^^^^Ω^^^^^^^^� ^^^^ for the reference molecule (^^^^^^^^) using ln ^^^^^^^^ = ^^^^^^^^ ∑ ^^^^+1 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Ω^^^^^^^^−1 ^^^^=1 �∑^^^^ ^^^^ 2 ; calculating a new value for ^^^^^^^^, ^^^^^^^^ and ^^^^^ ∗ ^^^ ^^^^^^^^^^^^Ω^^^^^^^^� using ^^^^^^^^ = �^^^^^^^^+^^^^ ∗ ^^^^ ^^^^ � ^^^^^^^^ = ^^^^^^^^ ^^^^^^ ^ ^^ ^^^^^^^ ^^^^^^^^ ^^^^^^^^ ^^^^0 ^^^^ = ^^^^ 1^^^^ ^^^^ , calculating (i) absorbed (^^^^^^^^ ^^^^ ^^^^) using ^^^^^^^^ = ^^^^ + ^^^^∗; repeating steps (a)-(h) whenever the initial values for ^^^^ , ^^^^ and ^^^^∗ ^^^^ ^^^^ ^^^^ ^^^^ calculated values for ^^^^ , ^^^^^^^^ and ^^^^∗ ^^^^ ; and (a)-(i) whenever ^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^ ^^^^ 2 ^^^^ ^^^^ −^^ ^^^^ ^^^^^^^^^^^^ ^^^^ ^^^^ ^^^^ ^^^^^^ ^^^^ is not (see e.g., FIG. 3). In another aspect, the total amount of more adsorbate components (i) absorbed for the mixed-gas multilayer adsorption system (^^^^^^^^ ^^^^) are calculated for multiple temperatures. In another aspect, the mixed-gas multilayer adsorption system 1102 comprises: an adsorbent comprising titanium dioxide anatase; and the adsorbate component comprises oxygen and nitrogen. In another aspect, the mixed-gas multilayer adsorption system 1102 comprises: an adsorbent comprising activated charcoal; and the adsorbate component comprises benzene and cyclohexane. In another aspect, the mixed-gas multilayer adsorption system 1102 comprises: an adsorbent comprising activated alumina; and the adsorbate component comprises water and carbon dioxide. [0090] Referring now to FIG.12, a flowchart of a computerized method 1200 for identifying an adsorption equilibrium constant and a condensation-evaporation equilibrium constant for an adsorbate component in a gas adsorption system is shown in accordance with one embodiment of the present disclosure. The computerized method 1200 for identifying an adsorption equilibrium constant and a condensation-evaporation equilibrium constant for an adsorbate component in a gas adsorption system includes providing one or more processors, a memory communicably coupled to the one or more processors and an input/output interface communicably coupled to the one or more processors in block 1202. The one or more processors calculate an adsorption equilibrium constant (^^^^^^^^ ^^^^ ^^^^ ^^^^^^^^^^^^ −∆^^^^^^^^^^^^^^^^,^^^^ 1 1 ^^^^ ) for the adsorbate component (i) using ^^^^^^^^ = ^^^^^^^^ exp� ^^^^ � ^^^^ − ^^^^^^^^^^^^^^^^�� (see Eq. (18)) in block 1204. The one or more processors equilibrium constant (^^^^ ) for the adsorbate compo ^^^^^^^^^^^^ −∆^^^^^^^^^^^^^^^^^^^^,^^^^ 1 1 ^^^^,^^^^ nent (i) using ^^^^^^^^,^^^^ = ^^^^^^^^,^^^^ exp� ^^^^ � ^^^^ − ^^^^^^^^^^^^^^^^�� (see Eq. ^^^^^^^^^^^^ (19)), wherein: ^^^^ is the adsorption constant ^^^^^^^^,^^^^ is condensation- equilibrium constant the reference temperature (^^^^^^^^^^^^^^^^ ), ^^^^ is a gas constant, ^^^^ is a system temperature, ∆^^^^^^^^^^^^^^^^,^^^^ is an enthalpy of adsorption, and ∆^^^^^^^^^^^^^^^^^^^^,^^^^ is an enthalpy of condensation in block 1206. The adsorption equilibrium constant and the condensation-evaporation equilibrium constant are provided to the input/output interface in block 1208, and the gas adsorption system is configured using the adsorption equilibrium constant and the condensation-evaporation equilibrium constant in block 1210. Note that the one or more processors can be part of one or more controller, computers, servers or other devices suitable for performing the method. The memory can be any type of data storage. The input/output interface can be any component capable of interfacing with the one or more processors. The components can be local, remote or a combination thereof. Moreover, the components can be part of a distributed computing architecture, design system or control system. In another aspect, the gas absorption system comprises a CO2 capture system, a hydrocarbon processing system, an air separation system, a direct air capture system, a trace elements and heavy metals removal system, or an aqueous organic acids separation system. In another aspect, the gas absorption system comprises a liquid adsorption system mathematically formulated as a pseudo gas adsorption system. In another aspect, the liquid adsorption system comprises a trace elements removal system, a heavy metals removal system, an aqueous organic separation system, or an ethanol-water separation system. In another aspect, the input/output interface comprises an interface directly or indirectly to the gas adsorption system. [0091] In one aspect, the method produces a product using the gas adsorption system. In another aspect, the gas adsorption system further uses a saturation adsorption loading for monolayer adsorption of the adsorbate component (^^^^0 ^^^^ ), a correction factor (^^^^^^^^ ) and a binary interaction parameter (^^^^^^^^^^^^) for the species ^^^^ – species ^^^^ pair. In another the adsorption equilibrium constant and the condensation-evaporation equilibrium constant are calculated using a regression analysis. In another aspect, calculating the adsorption equilibrium constant and the condensation- evaporation equilibrium constant comprises: providing input data comprising a pressure (P), an adsorption isotherm (^^^^^ ^ ^^ ^^ ^^), an adsorbate area for the adsorbate component (^^^^^^^^), an adsorbate area for a reference molecule (^^^^^^^^), and a saturation pressure (^^^^^^^^^^^^^^^^ ^^^^ ); calculating a saturation adsorption loading for monolayer adsorption of the adsorbate (^^^^0 ^^^^) from an experimental adsorbate ce area or ^^^^^^^^ = ^^ 0 surfa 0 ^^ ^^^^^^^^ where ^^^^0 is a total adsorbent surface area; providing initial values for the adsorption cons ^^^^ tant (^^^^^^^^ ), a correction factor (^^^^^^^^) and a binary interaction parameter (^^^^^^^^^^^^) for the species ^^^^ – providing initial values for an amount adsorbed of the adsorbate component due to monolayer adsorption (^^^^^^^^ ), an amount adsorbed of the reference molecule due to monolayer adsorption (^^^^^^^^) and a total of amounts adsorbed of the adsorbate component due to condensation on subsequent layers (^^^^^ ∗ ^^^); calculating an adsorbed phase mole fraction for the adsorbate component in the monolayer only (^^^^ ^^^^^^^^ ^^^^) using ^^^^^^^^ = ∑^^^^ ^^^^=1 ^^^^^^^^ +^^^^^^^^; calculating an adsorbed phase mole fraction for the reference molecule in the ) using ^^^^^^^^ ^^^^^^^^ ∑^^^^ ^^^^=1 ^^^^^^^^ +^^^^^^^^; calculating a gas phase mole fraction for the adsorbate component (^^^^^^^^) using ln ^^^^^^^^ = , wherein: ^^^^ ∆ℎ^^^^^^^^ ^^^^^^^^ = ∆^^^^^^^^^^^^ + , Ω^^^^^^^^ = exp�−^^^^^^^^^^^^^^^^� and ^^^^ is nonrandomness factor set to about 0.3 per the NRTL convention; calculating a gas phase mole fraction for the reference molecule (^^^^ ) using ln ^^^^ ∑ ^^^^+1 ^^^^^ 2 ^^^^^^^^ 2 ^^^^^^^^^^^^^^^�Ω^^^^^^^^−1� ^^^^ ^^^^ = ^^^^^^^^ ^^^^=1�∑^^^^ ^^^^=1 ^^^^ 2 ; calculating a new ^^^^^^^^^^^^Ω^^^^^^^^� value for ^^^^^^^^, ^^^^^^^^ and ^^^^^ ∗ ^^^ using ^^^^^^^^ +^^^^ ∗ ^^^^�^^^^^^^^ ^^^^ ^^^^ ^^^^ ^^^^^^ ^ ^^ ^^^^^^^ ^^^^^^^^ = ^^^^0 ^^^^ = �1−^^^^ ^^^^�� ^^^^ 1^^^^ (1−^^^^ ^^^^ ^^^^ , ^^^^,^^^^ ^^^^ ^^^^ ^^^^,^^^^ )+^^^^^^^^ ^^^^� calculating a due to monolayer adsorption and condensation on subsequent layers (^^^^^^^^ ^^^^) using ^^^^^^^^ ^^^^ = ^^^^^^^^ + ^^^^∗ ^^^^; repeating steps (a)-(h) whenever the initial values for ^^^^ , ^^^^^^^^ and ^^^^∗ do not match values for ^^^^^^^^, ^^^^ and ^^^^^ ∗ ^^; and ^ ^^^^^^^^^^^^ 2 steps (a)-(i) = ∑ �^ ^^^^ ^^^^ ^^^^ ^^^^^^^^^^^^^^^^ ^^^^ ^^^^^^^ ^^^^ −^^^^^^^^ ^^^^ ^^^^^^^^^^^^^^^^^^^^ � is not minimiz ^^^^=1 ed. e.g., FIG.1). In another constant and the condensation-evaporation equilibrium constant are calculated for multiple temperatures. In another aspect, the gas absorption system comprises: an adsorbent comprising titanium dioxide anatase; and the adsorbate component comprises oxygen or nitrogen. In another aspect, the gas absorption system comprises: an adsorbent comprising activated charcoal; and the adsorbate component comprises benzene or cyclohexane. In another aspect, the gas absorption system comprises: an adsorbent comprising activated alumina; and the adsorbate component comprises water or carbon dioxide. [0092] The foregoing computerized method 1200 can be implemented as a non-transitory computer readable medium containing program instructions that cause the one or more processors to perform the foregoing computerized method 1200. [0093] Now referring to FIG. 13, a flowchart of a computerized method 1300 for identifying a total amount of two or more adsorbate components absorbed for a mixed-gas multilayer adsorption system is shown in accordance with one embodiment of the present disclosure. The computerized method 1300 includes providing one or more processors, a memory communicably coupled to the one or more processors and an input/output interface communicably coupled to the one or more processors in block 1302. The one or more processors calculate the total amount of the two or more adsorbate components (i) absorbed (^^^^^^^^ ^^^^) using ^^^^^^^^ ^^^^ = ^^^^^^^^ + ^^^^∗^^^^∗ ^^^^ (see Eq. (33)), wherein: ^^^^^^^^ is an amount adsorbed of each adsorbate component (^^^^) due to monolayer adsorption, ^^^^∗ is a total of amounts adsorbed due to condensation on subsequent layers, and ^^^^^^ ∗ ^^ is a dew point liquid phase composition in block 1304. The one or more processors provide the total amount of the two or more adsorbate components (i) absorbed (^^^^^ ^ ^^ ^^ ^^) to the input/output interface in block 1306, and the mixed-gas multilayer adsorption system is configured using the total amount of the two or more adsorbate components (i) absorbed (^^^^^ ^ ^^ ^^ ^^) in block 1308. Note that the one or more processors can be part of one or more controller, computers, servers or other devices suitable for performing the method. The memory can be any type of data storage. The input/output interface can be any component capable of interfacing with the one or more processors. The components can be local, remote or a combination thereof. Moreover, the components can be part of a distributed computing architecture, design system or control system. In another aspect, the mixed-gas multilayer adsorption system comprises a CO2 capture system, a hydrocarbon processing system, an air separation system, a direct air capture system, a trace elements and heavy metals removal system, or an aqueous organic acids separation system. In another aspect, the mixed-gas multilayer adsorption system comprises a liquid adsorption system mathematically formulated as a pseudo gas adsorption system. In another aspect, the liquid adsorption system comprises a trace elements removal system, a heavy metals removal system, an aqueous organic separation system, or an ethanol-water separation system. In another aspect, the input/output interface comprises an interface directly or indirectly to the mixed-gas multilayer adsorption system. [0094] In one aspect, a product is produced using the mixed-gas multilayer adsorption system. In another aspect, the mixed-gas multilayer adsorption system is further configured using a binary interaction parameter (^^^^^^^^^^^^), a correction factor (^^^^) and a pressure (P). In another aspect, the total amount of each of more adsorbate components (i) absorbed for the mixed-gas multilayer adsorption system (^^^^^^^^ ^^^^) are calculated using a regression analysis. In another aspect, the one or more processors calculate the total amount of each of the two or more adsorbate components (i) absorbed for the mixed-gas multilayer adsorption system (^^^^^^^^ ^^^^) by: providing input data comprising a pressure (P), a gas phase mole fraction for each adsorbate component (^^^^^^^^), the total amount of the two or more adsorbate components (i) absorbed (^^^^^^^^ ^^^^ ), a saturation adsorption loading for monolayer adsorption of each adsorbate component (^^^^^ 0 ^^^), an adsorption equilibrium constant for each adsorbate component (^^^^^^ ^ ^^ ^^^ ), a condensation-evaporation equilibrium constant for each adsorbate component (^^^^^^^^,^^^^), a interaction parameter (^^^^^^^^^^^^) for the species ^^^^ – species ^^^^ pair, an adsorbate area for each component (^^^^^^^^), an adsorbate area for a reference molecule (^^^^^^^^), a dew point pressure for each adsorbate component (^^^^^^^^^^^^^^^^ ^^^^ ), and a dew point liquid phase composition (^^^^^ ∗ ^^^); providing initial values for the binary interaction parameter (^^^^^^^^^^^^) and a correction factor (^^^^); providing initial values for an amount adsorbed of each adsorbate component due to monolayer adsorption (^^^^^^^^ ), an amount adsorbed of the reference molecule due to monolayer adsorption (^^^^^^^^) and a total of amounts adsorbed of each adsorbate component due to condensation on subsequent layers (^^^^^ ∗ ^^^) at a given gas phase mole fraction for each adsorbate component (^^^^^^^^) and pressure (P); calculating an adsorbed phase mole fraction for each adsorbate component in the monolayer only (^^^^ ^^^^^^^^ ^^^^) using ^^^^^^^^ = ∑^^^^ ^^^^=1 ^^^^^^^^ +^^^^^^^^; calculating an adsorbed phase mole fraction for the ^^^^ reference molecule in the ^^^^ ^^^^) using ^^^^^^^^ = ∑^^^^ ^^^^=1 ^^^^^^^^ +^^^^^^^^; calculating a gas phase mole fraction for each adsorbate component (^^^^ ) using ^^^^+1 ^^^^2^^^^2^^^^^^^^^^^^�Ω^^^^ −1� ^^^^ ^^^^ ^^^^ ^^^^=1 ^^^^ ^^^^ , wherein: ^^^^^^^^^^^^ ^^^^ � ∆ℎ ∆^^^^ + ^^^^^^^^ ^^^^ , Ω^^^^^^^^ = exp�−^^^^^^^^^^^^^^^^� and ^^^^ is a 0.3; calculating molecule (^^^^ ^^^^2^^^^2^^^^^^^^^^^^�Ω^^ −1� reference ^^^^) using ln ^^^^^^^^ = ^^^^^^^^ ∑ ^^^^+1 ^^^^ ^^^^ ^^^^^^ ^^^^=1 ^^^^ ^^^^ �2 ; calculating a new value for ^^^^^^^^, ^^^^^^^^ and ^^^^^ ∗ ^^^ using ^^^^ ∗ ^^^ ^^^^ =�^^^^^^^^+^^^^^^^^ � ^^^^^^^^ ^^^^^^ = ^ ^^^^ ^^^^ = ^^^^^^^^ ^^^^ ^^^^ ^^ 0 ^^^^ , calculating (i) absorbed (^^^^^ ^ ^^ ^^ ^^) using ^^^^^ ^ ^^ ^^ ^^ = ^^^^ + ^^^^∗; repeating steps (a)-(h) whenever the init ∗ ^^^^ ^^^^ ial values for ^^^^^^^^, ^^^^^^^^ and ^^^^^^^^ do not match the calculated values for ^^^^^^^^ , ^^^^^^^^ and ^^^^∗ ^^^^ ; and repeating step (a)-(i) whenever ^^^^^^^^^^^^ = ^^^^^^^^^^^^ 2 ^^^^ ^^^^ ^^^^ −^^^ ^^^^ ^^^^^^^^^^^^^^^^ ∑ ^^^^ ^^^^=1� ^^^^ ^^^^ ^^^^^ ^^^^ ^^^^^^^^^^^^^^^^^^^^ � is not (see e.g., FIG. 3). In another aspect, the total amount of adsorbate components (i) absorbed for the mixed-gas multilayer are calculated for multiple temperatures. In another aspect, the mixed- multilayer adsorption system comprises: an adsorbent comprising titanium dioxide anatase; and the adsorbate component comprises oxygen and nitrogen. In another aspect, the mixed-gas multilayer adsorption system comprises: an adsorbent comprising activated charcoal; and the adsorbate component comprises benzene and cyclohexane. In another aspect, the mixed-gas multilayer adsorption system comprises: an adsorbent comprising activated alumina; and the adsorbate component comprises water and carbon dioxide. [0095] The foregoing computerized method 1300 can be implemented as a non-transitory computer readable medium containing program instructions that cause the one or more processors to perform the foregoing computerized method 1300. [0096] The thermodynamic Brunauer-Emmett-Teller isotherm for single component multilayer adsorption isotherm has been extended for multicomponent multilayer adsorption equilibria. Taking into account the adsorbent surface heterogeneity, the competitive adsorption on the monolayer, the condensation-evaporation on the subsequent layers, the adsorbed monolayer phase nonideality, and the condensed phase nonideality in subsequent layers, the generalized Brunauer- Emmett-Teller isotherm provides the first rigorous and comprehensive thermodynamic framework for multicomponent multilayer adsorption equilibria. For single component multilayer adsorption, the gBET isotherm model parameters include the three adsorbate-specific Langmuir isotherm parameters for the monolayer adsorption: the saturation loading ^^^^0 ^^^^, the adsorption equilibrium constant ^^^^^^^^ ^^^^ , and the adsorbate-adsorbent binary interaction parameter ^^^^^^^^^^^^ . In addition, the adsorbate-specific condensation-evaporation equilibrium constant ^^^^ from adsorbate saturation pressure and a correction factor ^^^^, is required for the condensation on the subsequent layers. For multicomponent multilayer adsorption, the mixed-gas condensation- evaporation equilibrium constant ^^^^^^^^ is calculated from the dew point pressure of the mixed-gas and ^^^^. One adsorbate-adsorbate binary interaction parameter ^^^^^^^^^^^^ per ^^^^ − ^^^^ pair may be introduced to further account for the adsorbed monolayer phase nonideality. A simple yet powerful adsorption thermodynamic model for simulation and design of adsorption processes, gBET outperforms IAST and accurately correlates and predicts adsorption loadings of multicomponent multilayer adsorption systems over the complete range of pressure and gas phase composition. The resulting gBET isotherm further enables tracking of adsorbent vacant sites, loading and composition of monolayer adsorbed phase, and loading and composition of adsorbed phase beyond monolayer. Future studies should explore applications of gBET for various industrial multicomponent multilayer adsorption systems. [0097] One skilled in the art of single gas and mixed-gas adsorption will recognize that the present disclosure provides sufficiently accurate parameters for rigorous adsorption design for systems of industrial interest. [0098] Process Simulation [0099] A series of recent advances will now be described in the generalization of the classical Langmuir isotherm of single component adsorption by deriving an activity coefficient model to account for the adsorbed phase adsorbate−adsorbent interactions, substituting adsorbed phase adsorbate and vacant site concentrations with activities, and extending to multicomponent competitive adsorption equilibrium, both monolayer and multilayer. Requiring a minimum set of physically meaningful model parameters, the generalized Langmuir isotherm for monolayer adsorption and the generalized Brunauer−Emmett−Teller isotherm for multilayer adsorption address various thermodynamic modeling challenges including adsorbent surface heterogeneity, isosteric enthalpies of adsorption, BET surface areas, adsorbed phase nonideality, adsorption azeotrope formation, and multilayer adsorption. Also discussed below is the importance of quality adsorption data that cover sufficient temperature, pressure, and composition ranges for reliable determination of the model parameters to support adsorption process simulation, design, and optimization. [00100] Applications of Generalized Langmuirian Isotherms [00101] Isosteric Enthalpies of Adsorption. Isosteric enthalpies of adsorption play an integral part in the mass and energy balances of adsorption units and, together with adsorption isotherms, determine process performance and product purity. Isosteric enthalpies of adsorption for homogeneous adsorbents do not change with adsorption loading, while isosteric enthalpies of adsorption for heterogeneous adsorbents decrease with increase in adsorption loading. [29] For single component adsorption, gL expresses the loading dependence through the binary interaction parameter τiϕ for absorbed phase activity coefficients. The gL expression for the isosteric enthalpy of adsorption of adsorbate component i is shown in Eq. (43). ^^^^ (^^^^ ,^^^^) = Δ^^^^ ^^^^ ln ^^^^^ ^^^^ ln^^^^^^^^ ^^,^^^^ + ^^^^^^^^2 ^^^ ^^^^^^^^,^^^^ ^^^^ ^^^^^^^^^^ � ^^^^^^^^ − ^^^^^^^^ � (43) ^^^^^^^^ Here ^^^^ ln^^^^^^^^ ^^^^ ln ^^^^^^^^ ^^^^^^^^ and ^^^^^^^^ equals zero, the adsorbent surface is homogeneous while nonzero τiϕ adsorbent surface heterogeneity. FIGS.14A-14C are adapted from [32]. FIG.14A shows how the predicted isosteric enthalpy of adsorption varies with τiϕ. With negative τiϕ, FIG.14A shows ^^^^^^^^^^^^,^^^^ is a strong concave-up function of loading at low With positive τiϕ, FIG.14B shows ^^^^ ^^^^^^^^,^^^^ is a strong concave down function of loading at high loadings. With gL parameters regressed from data for adsorption isotherm and isosteric enthalpy of adsorption simultaneously, [14,32] FIG. 14C shows the gL estimates for isosteric enthalpy of adsorption align well with the experimental data for CO2 at 305.95 K on NaX, [49] and CH4 and C2H6 at 297.0 K on BPL activated carbon. [50] [00103] BET Surface Areas. BET surface areas of specific adsorbents are widely determined from the two-parameter BET equation for various physical chemistry and materials science applications. Consistent with concerns on adsorption isotherm reproducibility in the literature, [51] reported BET surface areas often suffer reproducibility issues. The source of discrepancies in reported BET surface areas can be attributed to inability of the BET isotherm to represent the complete adsorption isotherm and reliance solely on a linear fit to data over the P/Psat range from 0.05 to 0.3 for adsorbent surface area calculations. A recent study prepared jointly by more than one hundred adsorption experts reported that multiple linear fit solutions to BET are possible and can dramatically alter the calculated adsorbent surface area. One such example is the adsorbent surface areas for NU-1104 MOF reported by 61 different laboratories, which vary from the lowest estimate of 1757 m2/g to the highest estimate of 9341 m2/g, or a factor of ∼5. [22] The same study reported similar observations for many other adsorbents and concluded that the errors are mainly due to the limitations of the conventional BET theory, the limited relative pressure range, and the limited number of data points used in performing the linear fit. [22] The tBET and gBET isotherms overcome the limitations of the conventional BET theory, reliably represent experimental isotherm data over the complete relative pressure range or prior to the onset of capillary condensation for both homogeneous adsorbents and heterogeneous adsorbents, [21] and they may serve as a refined tool to address the reproducibility issues with adsorbent surface area calculations. [00104] Multicomponent Monolayer Adsorption Equilibrium. Establishing accurate and thermodynamically consistent representations for multicomponent adsorption equilibrium from single-component isotherms and binary adsorption equilibrium data is essential for simulation and design of adsorption units and processes. One relatively ideal binary adsorption system, H2S (1) − CO2 (2) at 0.156 bar, and two azeotrope-forming binary adsorption systems, C3H8 (1) − H2S (2) at 0.081 bar and C3H8 (1) − CO2 (2) at 0.405 bar on zeolite H-mordenite at 303.15 K are highlighted to demonstrate the widely different adsorption equilibrium results from the IAST, cL-eL, Sips- LRC, DPL, and gL isotherms. [14] Single-component isotherm data for CO2 at 283.15, 303.15, and 323.15 K are shown in FIG. 15 along with model results from cL, Sips, DPL, and gL. [18] FIG.15 is adapted from [18] and shows that cL fails to represent CO2 isotherm data, Sips fails to follow Henry’s Law behavior at low pressures, and both DPL and gL provide reasonable representations of the isotherm data and fidelity to Henry’s Law at low pressures. The regressed isotherm parameters for CO2, H2S, and C3H8 are reported in Tables 4 through Table 6 in FIG.16 and Table 7 in FIG.18. [00105] FIGS. 17A-17C are adapted from [14] where symbols are data and lines are model results. FIG.17A shows that for the H2S (1) − CO2 (2) binary adsorption at 0.156 bar, IAST and eL qualitatively represent the adsorption equilibrium data, [52] although both slightly underestimate ^^^^1 ′ with ^^^^1 < 0.2 and overestimate ^^^^1 ′ with ^^^^1 > 0.2. Both the DPL and Sips-LRC predictions deviate significantly from experimental data while Sips-LRC also incorrectly predicts an adsorption azeotrope. In contrast, binary adsorption equilibrium data for the complete range of gas phase composition can be accurately matched by gL with τ12 = −5.01. FIG.17B and FIG.17C further show that, for the C3H8 (1) − H2S (2) binary at 0.081 bar and the C3H8 (1) − CO2 (2) binary at 0.405 bar, IAST and eL both predict ideal adsorption, inconsistent with the experimental data. Sips-LRC and DPL provide qualitative representations of the data and correctly predict azeotrope formation. The gL isotherm accurately correlates the C3H8 (1) − H2S (2) and the C3H8 (1) − CO2 (2) binary adsorption data including azeotropes with τ12 adjusted to 10.96 and 14.24, respectively. These results highlight the widely divergent adsorption equilibrium predictions obtained using commonly practiced isotherm models. The average relative deviations of the model predictions vs the experimental data are summarized in Table 8 in FIG.18. Obviously, these results should put into question the reliability of current process simulation and design for adsorption units. [00106] To illustrate this still further, selectivities for the three binary adsorption systems on zeolite H-mordenite at 303.15 K are investigated using IAST and gL, as shown in FIG.19, which is adapted from [14] where symbols are data and lines are model results. Adsorption selectivity is one key process variable that determines the performance of adsorption units. The selectivity of adsorbate component (1) over adsorbate component (2), S12, is defined by Eq. (44). ^ ′ ′ ^^^^ =�^^^1� ^^^^2 12 ^^^^1 /� ^^^^2� (44) [00107] For the binary adsorption of S12 remains above unity. However, for both the binary adsorption of C3H8 (1) − H2S (2) at 0.081 bar and that of C3H8 (1) − CO2 (2) at 0.405 bar, the data show selectivity reversal due to the formation of adsorption azeotropes, i.e., S12 moves from above to below unity as ^^^^1 increases from zero to unity. The IAST predictions suggest that selectivities should be independent of the gas phase composition. In contrast, gL accurately represents the selectivities including their composition dependence for the three binaries and the selectivity reversal due to azeotrope formation. [00108] Adsorption Azeotropes. Adsorption azeotropes form when the equilibrium gas phase and adsorbed phase compositions, ^^^^^^^^ and ^^^^^ ′ ^^^, respectively, become equal, causing the equilibrium line to cross the 45° line on the ^^^^^ ′ ^^^ - ^^^^^^^^ plot. Although adsorption azeotropes can be predicted or correlated with various isotherm models, the exact thermodynamic condition for adsorption azeotropes remained elusive. [34] [00109] The activity-based intrinsic adsorption equilibrium constants of adsorbate species in a binary adsorption system of adsorbate components (1) and (2) at constant temperature can be expressed as follows: ^^^^0 1 = ^^^^1 ^^^^^^^^^^^^1 = ^^^^1^^^^1 ^^^^^^^^^^^^^^^^^^^^^^^^1 (45) [00110] At the azeotropic these relationships into Eqs. (45) condition for adsorption azeotropes. ^^^^^^^^^^^^^^^^^^^^ ^^^^ ^^^^ 1− ^^^^^^^^ = 1 ^^^^0 1 = 2 ^^^^0 2 (47) [00111] In other words, at the and the intrinsic adsorption equilibrium constant for the two adsorbate components (1) and (2) in the binary adsorption system should be identical and equal to ^^^^^^^^^^^^^^^^^^^^ 1− ^^^^^^^^ . Valid regardless of temperature, pressure, and loading, this thermodynamic azeotropes is analogous to the thermodynamic condition for azeotrope formation in vapor−liquid equilibrium shown below. ^^^^ ^^^^sat = ^^^^ ^^ sat 1 1 2 ^^2 (48) Here ^^^^1 sat and ^^^^2 sat are the saturation pressures of the two components (1) and (2) in vapor−liquid equilibrium. [00112] To illustrate the thermodynamic condition for adsorption azeotrope formation, adsorption of the C2H4 (1) − CO2 (2) binary on molecular sieve 5A at 313.15 K is investigated. [34,53] FIGS.20A-20C are adapted from [14] where symbols are data and lines are model results. FIG. 20A shows that, with τ12 = −2.60, the binary mixture does not form an azeotrope at the relatively low pressure of 0.003 bar since the ^^^^1⁄ ^^^^0 1 curve does not cross the ^^^^2⁄ ^^^^0 2 o curve. However, FIG. 20B shows that, with the pressure increased to 0.02 bar, the ^^^^1⁄ ^^^^0 1 curve and the ^^^^2⁄ ^^^^0 2 curve cross at ^^^^1 = ^^^^′ 1 = 0.68 and an azeotrope forms. FIG. 20C further shows the corresponding θ − x′ y phase diagram and, the azeotropes at 0.02 and 0.06 bar are maximum surface coverage azeotropes. The single-component isotherm parameters are reported in Table 9 in FIG. 18. A comprehensive study on the thermodynamic condition for azeotrope formation is available elsewhere. [34] [00113] Generalized Langmuir Isotherm and Real Adsorbed Solution Theory. Although being completely different thermodynamic frameworks, both gL and RAST have been designed to model nonideal mixed-gas adsorption equilibrium. For a binary gas adsorption system of adsorbate component (1) and adsorbate component (2) at a system temperature, gL treats the adsorbed phase as a ternary system consisting of occupied sites with adsorbate component (1), occupied sites with adsorbate component (2), and phantom molecule ϕ for vacant sites. The reference states for the three components are chosen to be saturated occupied sites with adsorbate component (1), saturated occupied sites with adsorbate component (2), and fully vacant sites, respectively. The adsorbed phase nonideality is then characterized with three aNRTL binary interaction parameters: τ1ϕ, τ2ϕ, and τ12. Here τ1ϕ and τ2ϕ are to be determined from single-component adsorption isotherms, and τ12 is the only parameter to be identified from binary gas adsorption equilibrium data. In contrast, for the same binary gas adsorption system of adsorbate component (1) and adsorbate component (2) at a system temperature, RAST treats the adsorbed phase as a binary system consist of occupied sites with adsorbate component (1) and occupied sites with adsorbate component (2). The reference states are the single-component adsorption systems of adsorbate component (1) and adsorbate component (2) at the same system “spreading pressure” as that of the binary system. As shown in Eq. (49), “spreading pressure” can be viewed as a measure of vacant sites surface fraction θϕ as shown below. �1 − ^^^^^^^^� ^^^^^^^^�^^^^^^ 0 ^^�^^^^ = ^^^^0 ^^^^ ^^^^^^^^(^^^^) 0 ^^^^0 ^ ^^^^^^^^ = ∫ 0 ^^^^^^^^ = −^^^^^^^^ ln�^^^^^^^^ ^^^^ ^^^^^^^ ^^^^ � (49) For RAST solutions addition, as the system “spreading pressure” approaches zero unity, the is in the Henry’s Law region, and the adsorbed phase is defined to be ideal. If the SPD-aNRTL activity coefficient model is used in RAST for the adsorbed phase nonideality, ^^^^1 ′ 2 ∞ (or ^^^^1 ′ 2 ∞) for composition and temperature dependence and β for spreading pressure dependence two adjustable parameters to be identified from binary gas adsorption equilibrium data. [33] [00114] The binary adsorption system of N2 (1) − O2 (2) on LiLSX at 303.15 K is used to highlight both the consistency and the difference in gL and RAST thermodynamic frameworks for mixed-gas adsorption equilibrium. [33] FIGS.21A-21C are adapted from [33]. FIG.21A shows that the single-component isotherms of N2 and O2 at 303.15 K [33,43,[54] are accurately represented by the gL isotherm. The same single-component gL isotherms of N2 and O2 were then used in both the gL and RAST thermodynamic frameworks for mixed-gas adsorption equilibrium. With single-component isotherms represented using gL, RAST with the SPD-aNRTL activity coefficient model is used to calculate the binary adsorption loadings at four pressures, i.e., 0.1013, 1.013, 6.078, and 10.0 bar, by first identifying ^^^^1 ′ 2 ∞ = 1.55 from the binary adsorption experimental data at 6.078 bar [33,34,54] with β fixed at 0.353 g/mmol (^^^^ ≡ 1 ^^^^0 ^^^^ ). Subsequently, the aNRTL binary interaction parameter, −1.83, of the gL isotherm for the binary adsorption system is regressed from the RAST-SPDa-NRTL results at 6.078 bar. FIG.21B shows that the gL correlation at 6.078 bar and the gL predictions at 0.1013, 1.013, and 10.0 bar for the binary adsorption system accurately match the RAST-SPD-aNRTL results for both adsorbate components. [00115] FIG. 21C further shows that the SPD-aNRTL binary interaction parameter ^^^^1 ′ 2 approaches zero at system pressures of 0.1013 and 1.013 bar, indicative of an ideal adsorbed phase at low spreading pressures, i.e., ψ < 1 mmol/g. Note that, at a given system pressure, there is a range of spreading pressure reflecting the variation in the mixed-gas composition and the loading. Increasing the system pressure to 6.078 and 10.0 bar, the spreading pressure changes to ∼1 to 4 mmol/g and ^^^^1 ′ 2 changes to ∼−0.4 to −1.3, indicative of an increasingly nonideal adsorbed phase with increasing spreading pressure and increasing system pressure. Eventually ^^^^1 ′ 2 reaches a plateau and converges to a constant value with ^^^^1 ′ 2 ∞ = 1.55 at infinite spreading pressure. [00116] The correlations and predictions from both RAST-SPDaNRTL and gL for the binary adsorption system of N2 (1) − O2 (2) on LiLSX at 303.15 K at different pressures show that both thermodynamic frameworks are consistent with each other and may yield the same adsorption equilibrium results. The difference between them is that gL does not require determination of spreading pressure for every mixed-gas adsorption data point since gL already takes vacant sites into account as part of the thermodynamic treatment. Note that, due to the different definitions for reference states, τ12 in gL would not equal to ^^^^1 ′ 2 ∞ in RAST-SPD-aNRTL. [00117] Multicomponent Multilayer Adsorption Equilibrium. Multicomponent multilayer adsorption equilibrium may happen with mixed-gas adsorption systems containing condensable adsorbate species. FIGS. 22A-22F are adapted from [55] where symbols are data and lines are model results. FIG.22A shows the single component Type II adsorption isotherms of O2 and N2 on anatase at 78.2 K51 with BET and gBET isotherm representations. [55] Note that the same adsorption data is shown in FIG. 4A with the BET and tBET isotherms. [55] Although both the BET and gBET isotherms provide consistent representation at high pressures, the BET isotherm deviates from the isotherm data at pressures below 10−2 bar yielding average relative deviation of 23.4% for O2 and 12.9% for N2. In contrast, the gBET isotherm accurately represents the isotherm data at all pressures, and results in average relative deviation of 8.5% and 5.5% for O2 and N2, respectively. On the basis of the BET isotherms, IAST predictions for adsorption of the O2 (1) − N2 (2) binary on anatase at 78.2 K51 at gas phase composition of y1= 14.9%, 29.8%, 50.2%, 70.2%, and 85.3% are shown in FIG.22B through FIG.22F, respectively. With the single-component O2 adsorption loading generally larger than the N2 loading, IAST consistently overpredicts the O2 adsorption loading and underpredicts the N2 loading for all mixed-gas phase compositions. The overall average relative deviation for the IAST predictions is 61.3%. In contrast, with the dew point liquid phase composition predicted by the Peng−Robinson equation of state, gBET reliably predicts the experimental adsorption loading data with an overall average relative deviation of 13.9%. The Peng−Robinson equation of state predictions for the dew point liquid phase composition are shown in FIG.23, which is adapted from [55] where symbols are data and lines are model results. The gBET representation for binary multilayer adsorption data is further improved when τ12 for monolayer adsorption is adjusted to correlate the data. With τ12 = 1.77, the average relative deviation drops to 8.0%. The corresponding single-component BET and gBET isotherm parameters are given in Table 1 and Table 2. [00118] Adsorption Equilibrium Data. Successful development and validation of rigorous and predictive thermodynamic models for multicomponent adsorption equilibrium, monolayer or multilayer, ultimately depend on the availability of quality experimental adsorption equilibrium data. Ideally, data for single-component adsorption isotherms for the complete range of pressure at multiple temperatures are needed for proper identification of gL and gBET model parameters. For Type I adsorption isotherms, the single-component adsorption isotherms must have data points at low pressures, i.e., the Henry’s Law region, medium pressures, i.e., the concave down region, and high pressures to properly determine the adsorption equilibrium constant, the aNRTL binary parameter for adsorbate−adsorbent interactions, and the saturation loading. For Type II and Type III isotherms with multilayer adsorption, additional isotherm data for the concave up region showing the condensation-evaporation behavior are required to identify the condensation- evaporation equilibrium constant. Data for adsorption isotherms at multiple temperatures, although relatively rare, are exceedingly useful to help validate the isotherm data at individual temperatures, and from the Clausius−Clapeyron equation, to estimate the isosteric enthalpy of adsorption as a function of loading. Alternatively, data on isosteric enthalpy of adsorption as a function of loading would facilitate reliable estimation of adsorption isotherms over a range of temperature. Likewise, data for binary gas adsorption equilibrium over reasonable ranges of pressure and gas composition are essential for validation of gL/gBET predictions for ideal mixed-gas adsorption equilibrium, and for proper determination of the aNRTL binary interaction parameters for nonideal mixed-gas adsorption equilibrium. While treating the experimental data, one must properly consider data uncertainty, consistency of data from different sources, and thermodynamic consistency of data for different properties. With availability of quality data for single and binary gas adsorption equilibrium, reliable predictions for multicomponent adsorption equilibrium, monolayer or multilayer, are possible. [00119] Accurate, rigorous, and predictive thermodynamic models serve as key scientific foundations for simulation, design, and optimization of industrial chemical processes. For decades, advances in adsorptive separations have been hampered by a lack of rigorous adsorption thermodynamic models for multicomponent adsorption equilibrium. Derived from the concentration-based classical Langmuir isotherm of single component adsorption isotherm, a series of activity-based Langmuirian isotherms have been formulated to constitute a comprehensive family of rigorous adsorption thermodynamic models for multicomponent adsorption equilibrium, monolayer and multilayer. The resulting generalized Langmuir isotherm for multicomponent monolayer adsorption equilibrium and the generalized Brunauer−Emmett−Teller isotherm for multicomponent multilayer adsorption equilibrium address and elucidate various thermodynamic modeling challenges such as adsorbent surface heterogeneity and adsorption isotherm, isosteric enthalpies of adsorption vs loadings, BET surface areas for heterogeneous adsorbents, adsorbed phase nonideality and multicomponent competitive adsorption, thermodynamic condition for adsorption azeotrope formation, and monolayer vs condensed layer adsorption in multilayer adsorption. The family of generalized Langmuirian isotherms represent a promising rigorous alternative to the models currently in practice in adsorption thermodynamics and their implementation in process simulators should enable accurate simulation and rapid development of adsorption processes. [00120] Additional background information regarding the process simulations can be found in [56]. [00121] It is understood that particular embodiments described herein are shown by way of illustration and not as limitations of the invention. The principal features of this invention can be employed in various embodiments without departing from the scope of the invention. Those skilled in the art will recognize or be able to ascertain using no more than routine experimentation, numerous equivalents to the specific procedures described herein. Such equivalents are considered to be within the scope of this invention and are covered by the claims. [00122] All publications and patent applications mentioned in the specification are indicative of the level of skill of those skilled in the art to which this invention pertains. All publications and patent applications are herein incorporated by reference to the same extent as if each individual publication or patent application was specifically and individually indicated to be incorporated by reference. [00123] The use of the word “a” or “an” when used in conjunction with the term “comprising” in the claims and/or the specification may mean “one,” but it is also consistent with the meaning of “one or more,” “at least one,” and “one or more than one.” The use of the term “or” in the claims is used to mean “and/or” unless explicitly indicated to refer to alternatives only or the alternatives are mutually exclusive, although the disclosure supports a definition that refers to only alternatives and “and/or.” Throughout this application, the term “about” is used to indicate that a value includes the inherent variation of error for the device, the method being employed to determine the value, or the variation that exists among the study subjects. [00124] As used in this specification and claim(s), the words “comprising” (and any form of comprising, such as “comprise” and “comprises”), “having” (and any form of having, such as “have” and “has”), “including” (and any form of including, such as “includes” and “include”) or “containing” (and any form of containing, such as “contains” and “contain”) are inclusive or open- ended and do not exclude additional, unrecited features, elements, components, groups, integers, and/or steps, but do not exclude the presence of other unstated features, elements, components, groups, integers and/or steps. In embodiments of any of the compositions and methods provided herein, “comprising” may be replaced with “consisting essentially of” or “consisting of”. As used herein, the term “consisting” is used to indicate the presence of the recited integer (e.g., a feature, an element, a characteristic, a property, a method/process step or a limitation) or group of integers (e.g., feature(s), element(s), characteristic(s), property(ies), method/process steps or limitation(s)) only. As used herein, the phrase “consisting essentially of” requires the specified features, elements, components, groups, integers, and/or steps, but do not exclude the presence of other unstated features, elements, components, groups, integers and/or steps as well as those that do not materially affect the basic and novel characteristic(s) and/or function of the claimed invention. [00125] The term “or combinations thereof” as used herein refers to all permutations and combinations of the listed items preceding the term. For example, “A, B, C, or combinations thereof” is intended to include at least one of: A, B, C, AB, AC, BC, or ABC, and if order is important in a particular context, also BA, CA, CB, CBA, BCA, ACB, BAC, or CAB. Continuing with this example, expressly included are combinations that contain repeats of one or more item or term, such as BB, AAA, AB, BBC, AAABCCCC, CBBAAA, CABABB, and so forth. The skilled artisan will understand that typically there is no limit on the number of items or terms in any combination, unless otherwise apparent from the context. [00126] As used herein, words of approximation such as, without limitation, “about”, “substantial” or “substantially” refers to a condition that when so modified is understood to not necessarily be absolute or perfect but would be considered close enough to those of ordinary skill in the art to warrant designating the condition as being present. The extent to which the description may vary will depend on how great a change can be instituted and still have one of ordinary skill in the art recognize the modified feature as still having the required characteristics and capabilities of the unmodified feature. In general, but subject to the preceding discussion, a numerical value herein that is modified by a word of approximation such as “about” may vary from the stated value by at least ±1, 2, 3, 4, 5, 6, 7, 10, 12 or 15%. [00127] All of the compositions and/or methods disclosed and claimed herein can be made and executed without undue experimentation in light of the present disclosure. While the compositions and methods of this invention have been described in terms of preferred embodiments, it will be apparent to those of skill in the art that variations may be applied to the compositions and/or methods and in the steps or in the sequence of steps of the method described herein without departing from the concept, spirit and scope of the invention. All such similar substitutes and modifications apparent to those skilled in the art are deemed to be within the spirit, scope and concept of the invention as defined by the appended claims. [00128] To aid the Patent Office, and any readers of any patent issued on this application in interpreting the claims appended hereto, applicants wish to note that they do not intend any of the appended claims to invoke paragraph 6 of 35 U.S.C. § 112, U.S.C. § 112 paragraph (f), or equivalent, as it exists on the date of filing hereof unless the words “means for” or “step for” are explicitly used in the particular claim. 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Claims

CLAIMS What is claimed is: 1. A computerized method for identifying an adsorption equilibrium constant and a condensation-evaporation equilibrium constant for an adsorbate component in a gas adsorption system comprising: providing one or more processors, a memory communicably coupled to the one or more processors and an input/output interface communicably coupled to the one or more processors; calculating using the one or more processors, an adsorption equilibrium constant (^^^^^^ ^ ^^ ^^^) for the adsorbate component (i) using ^^^^^^^^ = ^^^^^^^^ ^^^^^^^^^^^^ exp −∆^^^^^^^^^^^^^^^^,^^^^ 1 1 ^^^^ ^^^^ � ^^^^ � ^^^^ − ^^^^^^^^^^^^^^^^��; calculating using evaporation equilibrium constant (^^^^^^^^,^^^^) for the adsorbate component (i) using ^^^^ ^^^^^^^^^^^^ −∆^^^^^^^^^^^^^^^^^^^^,^^ 1 1 ^^^,^^^^ = ^^^ ^^ ^ ^^^^^,^^^^ exp� ^^^^ � ^^^^ − ^^^^^^^^^^^^^^^^�� ; wherein: ^^^^^^^^ ^^^^^^^^^^^^ is (^ ^^^^^^^^^^^^ ^^^ ) ^^^^ ^^^^^^^^^^^^ ^^^^,^^^^ is the condensation-evaporation equilibrium constant the reference (^^^^^^^^^^^^^^^^ ), ^^^^ is a gas constant, ^^^^ is a system temperature, ∆^^^^^^^^^^^^^^^^,^^^^ is an enthalpy of adsorption, and ∆^^^^^^^^^^^^^^^^^^^^,^^^^ is an enthalpy of condensation; providing the adsorption equilibrium constant and the condensation-evaporation equilibrium constant to the input/output interface; and configuring the gas adsorption system using the adsorption equilibrium constant and the condensation-evaporation equilibrium constant.
2. The method of claim 1, further comprising producing a product using the gas adsorption system.
3. The method of claim 1, further comprising configuring the gas adsorption system using a saturation adsorption loading for monolayer adsorption of the adsorbate component (^^^^^ 0 ^^^ ), a correction factor (^^^^^^^^) and a binary interaction parameter (^^^^^^^^^^^^) for the species ^^^^ – species ^^^^ pair.
4. The method of claim 1, wherein the input/output interface comprises an interface to the gas adsorption system.
5. The method of claim 1, wherein the gas absorption system comprises a CO2 capture system, a hydrocarbon processing system, an air separation system, a direct air capture system, a trace elements and heavy metals removal system, or an aqueous organic acids separation system.
6. The method of claim 1, wherein the gas absorption system comprises a liquid adsorption system mathematically formulated as a pseudo gas adsorption system.
7. The method of claim 6, wherein the liquid adsorption system comprises a trace elements removal system, a heavy metals removal system, an aqueous organic separation system, or an ethanol-water separation system.
8. The method of claim 1, wherein the adsorption equilibrium constant and the condensation- evaporation equilibrium constant are calculated using a regression analysis.
9. The method of claim 1, wherein calculating the adsorption equilibrium constant and the condensation-evaporation equilibrium constant comprise: providing input data comprising a pressure (P), an adsorption isotherm (^^^^^ ^ ^^ ^^ ^^), an adsorbate area for the adsorbate component (^^^^^^^^), an adsorbate area for a reference (^^^^^^^^), and a saturation pressure (^^^^^^^^^^^^^^^^ ^^^^ ); calculating adsorption loading for monolayer adsorption of the adsorbate (^^^^ ^^^0 component 0 0 ^ ^^^^ ) from an experimental adsorbate surface area or ^^^^^^^^ = 0 ^^^^^^^^ where ^^^^ is a total adsorbent area; (a) providing initial values for the adsorption equilibrium constant (^^^^^^^ ^ ^^^^), a correction factor (^^^^^^^^) and a binary interaction parameter (^^^^^^^^^^^^) for the species ^^^^ – species ^^^^ pair; (b) providing initial values for an amount adsorbed of the adsorbate component due to monolayer adsorption (^^^^^^^^ ), an amount adsorbed of the reference molecule due to monolayer adsorption (^^^^^^^^) and a total of amounts adsorbed of the adsorbate component due to condensation on subsequent layers (^^^^^ ∗ ^^^); (c) calculating an adsorbed phase mole fraction for the adsorbate component in the monolayer only (^^^^ ) using ^^ ^^^^^^^^ ^^^^ ^^^^^^ = ∑^^^^ ^^^^=1 ^^^^^^^^ +^^^^^^^^; (d) calculating mole fraction for the reference molecule in the monolayer only (^^^^^^^^) using ^^^^^^^^ = ∑^^^^ ^^^^=1 ^^^^^^^^ +^^^^^^^^; (e) calculating a for the adsorbate component (^^^^^^^^) using ^^^^+1 ^^^^2^^^^2^^^^^^^^^^^^�Ω^^^^^^^^ − 1� ln ^^^^ = ^^^^ � ^^^^ ^^^^ ^^^^ ^^^^ wherein: a nonrandomness factor set about 0.3 per the (f) calculating a gas phase mole fraction for the reference molecule (^^^^^^^^) using ln ^^^^ ^^^^2^^^^2^^^^^^^^^^^^�Ω^^^^^ −1� ^^^^ = ^^^^^^^^ ∑ ^^^^+1 ^^^^ ^^^^ ^^^ ^^^^=1 ^^^^ ^^^^ 2 ; (g) calculating a new ^^^^^^^^ �^^^^ +^^^^ ∗ ^^^^ ^^^^ � ^^^^^^^^ ^^^^ ^^^^ ^^^^ ^^^^ = ^^^^ ^^^^^^^^ = ^^^^ ^^^^^^^^ ^^^^ ^^^^ (h) adsorbed due to monolayer adsorption and condensation on subsequent layers (^^^^^^^^ ^^^^ ∗ ^^^^) using ^^^^^^^^ = ^^^^^^^^ + ^^^^^^^^; (i) repeating steps (a)-(h) whenever the initial values for ^^^^^^^^, ^^^^^^^^ and ^^^^^ ∗ ^^^ do not match the calculated values for ^^^^^^^^, ^^^^^^^^ and ^^^^^ ∗ ^^^; and ^^^^^^^^^^^ (j) ^^ ^^^^ ^^^^^ ^^^^ ^^^ −^^^^ ^^^^^^^^^^^^ = ∑ ^^^^=1� ^^^ ^^^^ ^^^^ ^^^^ ^^^^^^^^^^^^^^^^^^^^ � is not minimized.
10. and the condensation- evaporation equilibrium constant are calculated for multiple temperatures.
11. The method of claim 1, wherein the gas absorption system comprises: an adsorbent comprising titanium dioxide anatase; and the adsorbate component comprises oxygen or nitrogen.
12. The method of claim 1, wherein the gas absorption system comprises: an adsorbent comprising activated charcoal; and the adsorbate component comprises benzene or cyclohexane.
13. The method of claim 1, wherein the gas absorption system comprises: an adsorbent comprising activated alumina; and the adsorbate component comprises water or carbon dioxide.
14. A computerized method for identifying a total amount of two or more adsorbate components absorbed for a mixed-gas multilayer adsorption system comprising: providing one or more processors, a memory communicably coupled to the one or more processors and an input/output interface communicably coupled to the one or more processors; calculating using the one or more processors, the total amount of the two or more adsorbate components (i) absorbed (^^^^^^^^) using ^^^^^^^^ = ^^^^ + ∗ ∗ ^^^^ ^^^^ ^^^^ ^^^^ ^^^^^^^^ ; wherein: ^^^^^^^^ is an amount adsorbed of each adsorbate component (^^^^) due to monolayer adsorption, ^^^^∗ is a total of amounts adsorbed due to condensation on subsequent layers, and ^^^^^^ ∗ ^^ is a dew point liquid phase composition; providing the total amount of the two or more adsorbate components (i) absorbed (^^^^^^^^ ^^^^) to the input/output interface; and configuring the mixed-gas multilayer adsorption system using the total amount of the two or more adsorbate components (i) absorbed (^^^^^^^^ ^^^^).
15. The method of claim 14, further comprising producing a product using the mixed-gas multilayer adsorption system.
16. The method of claim 14, wherein the mixed-gas multilayer absorption system comprises a CO2 capture system, a hydrocarbons processing system, an air separation system, a direct air capture system, a trace elements and heavy metals removal system, or an aqueous organic acids separation system.
17. The method of claim 14, wherein the mixed-gas multilayer adsorption system comprises a liquid adsorption system mathematically formulated as a pseudo gas adsorption system.
18. The method of claim 17, wherein the liquid adsorption system comprises a trace elements removal system, a heavy metals removal system, an aqueous organic separation system, or an ethanol-water separation system.
19. The method of claim 14, wherein the input/output interface comprises an interface to the mixed-gas multilayer adsorption system.
20. The method of claim 14, further comprising configuring the mixed-gas multilayer adsorption system using a binary interaction parameter (^^^^^^^^^^^^), a correction factor (^^^^) and a pressure (P).
21. The method of claim 14, wherein the total amount of each of the two or more adsorbate components (i) absorbed for the mixed-gas multilayer adsorption system (^^^^^ ^ ^^ ^^ ^^) are calculated using a regression analysis.
22. The method of claim 14, wherein calculating the total amount of each of the two or more adsorbate components (i) absorbed for the mixed-gas multilayer adsorption system (^^^^^ ^ ^^ ^^ ^^) comprises: providing input data comprising a pressure (P), a gas phase mole fraction for each adsorbate component (^^^^^^^^), the total amount of the two or more adsorbate components (i) absorbed (^^^^^ ^ ^^ ^^ ^^), a saturation adsorption loading for monolayer adsorption of each adsorbate component (^^^^^ 0 ^^^), an adsorption equilibrium constant for each adsorbate component ( ^^^^^^^^ ^^^^ ), a condensation- evaporation equilibrium constant for each adsorbate component (^^^^^^^^,^^^^ ), a binary interaction parameter (^^^^^^^^^^^^) for the species ^^^^ – species ^^^^ pair, an adsorbate area for each adsorbate component (^^^^^^^^), an adsorbate area for a reference molecule (^^^^^^^^), a dew point pressure for each adsorbate component (^^^^^^^^^^^^^^^^ ^^^^ ), and a dew point liquid phase composition (^^^^^ ∗ ^^^); (a) providing initial values for the binary interaction parameter (^^^^^^^^^^^^) and a correction factor (^^^^); (b) providing initial values for an amount adsorbed of each adsorbate component due to monolayer adsorption (^^^^^^^^ ), an amount adsorbed of the reference molecule due to monolayer adsorption (^^^^^^^^) and a total of amounts adsorbed of each adsorbate component due to condensation on subsequent layers (^^^^^ ∗ ^^^) at a given gas phase mole fraction for each adsorbate component (^^^^^^^^) and pressure (P); (c) calculating an adsorbed phase mole fraction for each adsorbate component in the monolayer only (^^^^ ^^^^^^^^ ^^^^) using ^^^^^^^^ = ∑^^^^ ^^^^=1 ^^^^^^^^ +^^^^^^^^; (d) calculating mole fraction for the reference molecule in the ^^^^ monolayer only (^^^^ ) using ^ ^^^^ ^^^^ ^^^^^^^ = ∑^^^^ ^^^^^^^^ +^^^^^^^^; (e) a for each adsorbate component (^^^^^^^^) using ^^^^+1 ^^^^2^^^^2^^^^^^^^^^^�Ω − 1� ^^^^ ^^^^ ^^^^ ^^^^ ^ ^^^^^^^^ wherein: a nonrandomness factor set to about 0.3; (f) calculating a gas phase mole fraction for the reference molecule (^^^^^^^^) using ln ^^^^ = ^^^^ ∑ ^^^^+1 ^^^^^ 2 ^^^^^^^^ 2 ^^^^^^^^^^^^^^^�Ω^^^^^^^^−1� ^^^^ ^^^^ ^^^^=1�∑^^^^ 2 ; ^^^^=1 ^^^^^^^^^^^^^^^^Ω^^^^^^^^� (g) calculating a new ^^^^ �^^^^ ^^^^ ^^^^ ^^^^ ^^^^ ^^^^ ^ ^^^^ ^ = ^^^^^^^^ = ^^^ ^^^ ^^^^0 ^^^^ �1−^^^^ ^^^^ ^^^^,^^^^^^^^�� 1^^^^ (1−^^^^ ^^^^)+^^^^ ^^^^ , ^^^^� ^^^^ ^^^^,^^^^ ^^^^ (h) components (i) absorbed ^^ (^^^^^^ ^^^^) using ^^^^^^^^ ^^^^ = ^^^^^^^^ + ^^^^ ; (i) repeating steps (a)-(h) whenever the initial values for ^^^^^^^^, ^^^^^^^^ and ^^^^^ ∗ ^^^ do not match the calculated values for ^^^^^^^^, ^^^^^^^^ and ^^^^^ ∗ ^^^; and ^^^^^ 2 ^^^^ ^^^^ ^^^^^^^^^^^ −^ ^^^^ ^^^^^^^^^^^^^^^^ (j) ^^^^^^^^^^^^ = ∑ ^^^^ ^^^^ ^^^^ ^^^^^^^ ^^^^ ^^^^=1� ^^^^^^^^^^^^^^^^^^^^ � is not minimized.
23. The method of claim 14, wherein the total amount of each of the two or more adsorbate components (i) absorbed for the mixed-gas multilayer adsorption system (^^^^^^^^ ^^^^) are calculated for multiple temperatures.
24. The method of claim 14, wherein the mixed-gas multilayer adsorption system comprises: an adsorbent comprising titanium dioxide anatase; and the adsorbate component comprises oxygen and nitrogen.
25. The method of claim 14, wherein the mixed-gas multilayer adsorption system comprises: an adsorbent comprising activated charcoal; and the adsorbate component comprises benzene and cyclohexane.
26. The method of claim 14, wherein the mixed-gas multilayer adsorption system comprises: an adsorbent comprising activated alumina; and the adsorbate component comprises water and carbon dioxide.
27. A non-transitory computer readable medium containing program instructions that cause the one or more processors to perform the method as recited in claim 1.
28. A non-transitory computer readable medium containing program instructions that cause the one or more processors to perform the method as recited in claim 14.
29. A system for identifying an adsorption equilibrium constant and a condensation- evaporation equilibrium constant for an adsorbate component in a gas adsorption system comprising: a memory; an input/output interface; and one or more processors communicably coupled to the memory and the input/output interface, wherein the one or more processors: calculate an adsorption equilibrium constant (^^^^^^^^ ^^^^ ) for the adsorbate component (i) using ^^^^^^^^ = ^^^^^^^^ ^^^^^^^^^^^^ −∆^^^^^^^^^^^^^^^^,^^^^ 1 1 ^^^^ ^^^^ exp� ^^^^ � ^^^^ − ^^^^^^^^^^^^^^^^��, calculate a constant (^^^^^^^^,^^^^) for the adsorbate component (i) using ^^^^ ^^^^^^^^^^^ −∆^^^^ 1 1 ^^,^^^^ = ^^^^ ^ ^^^^^^^^^^^^^^^^,^^^^ ^^ ^^^^,^^^^ exp� ^^^^ � ^^^^ − ^^^^^^^^^^^^^^^^��, wherein: ^^^^ temperature (^^^^^^^^^^^^^^^^) ^^^^ ^^^^^^^^^^^^ ^^^^,^^^^ is the condensation-evaporation equilibrium constant the reference temperature (^^^^^^^^^^^^^^^^), ^^^^ is a gas constant, ^^^^ is a system temperature, ∆^^^^^^^^^^^^^^^^,^^^^ is an enthalpy of adsorption, ∆^^^^^^^^^^^^^^^^^^^^,^^^^ is an enthalpy of condensation, and provide the adsorption equilibrium constant and the condensation-evaporation equilibrium constant to the input/output interface; and wherein the gas adsorption system is configured using the adsorption equilibrium constant and the condensation-evaporation equilibrium constant.
30. The system of claim 29, wherein a product is produced using the gas adsorption system.
31. The system of claim 29, wherein the gas adsorption system is further configured using a saturation adsorption loading for monolayer adsorption of the adsorbate component (^^^^^ 0 ^^^ ), a correction factor (^^^^^^^^) and a binary interaction parameter (^^^^^^^^^^^^) for the species ^^^^ – species ^^^^ pair.
32. The system of claim 29, wherein the comprises an interface to the gas adsorption system.
33. The system of claim 29, wherein the gas absorption system comprises a CO2 capture system, a hydrocarbon processing system, an air separation system, a direct air capture system, a trace elements and heavy metals removal system, or an aqueous organic acids separation system.
34. The system of claim 29, wherein the gas absorption system comprises a liquid adsorption system mathematically formulated as a pseudo gas adsorption system.
35. The system of claim 34, wherein the liquid adsorption system comprises a trace elements removal system, a heavy metals removal system, an aqueous organic separation system, or an ethanol-water separation system.
36. The system of claim 29, wherein the adsorption equilibrium constant and the condensation- evaporation equilibrium constant are calculated using a regression analysis.
37. The system of claim 29, wherein the one or more processors calculate the adsorption equilibrium constant and the condensation-evaporation equilibrium constant by: providing input data comprising a pressure (P), an adsorption isotherm (^^^^^^^^ ^^^^), an adsorbate area for the adsorbate component (^^^^^^^^), an adsorbate area for a reference molecule (^^^^^^^^), and a saturation pressure (^^^^^^^^^^^^^^^^ ^^^^ ); calculating adsorption loading for monolayer adsorption of the adsorbate ^^^0 ^^^^ ) from an experimental adsorbate surface area or ^^^^0 ^^^^0 component (^ ^^^^ = ^^^^^^^^ where ^^^^0 is a total adsorbent surface area; (a) providing initial values for the adsorption equilibrium ^^^ ^ ^^^^), a correction factor (^^^^^^^^) and a binary interaction parameter (^^^^^^^^^^^^) for the species ^^^^ – species ^^^^ pair; (b) providing initial values for an of the adsorbate component due to monolayer adsorption (^^^^ ), an amount the reference molecule due to adsorption (^^^^^^^^) and a total of amounts adsorbed of the adsorbate component due to condensation on subsequent layers (^^^^^ ∗ ^^^); (c) calculating an adsorbed phase mole fraction for the adsorbate component in the monolayer only (^^^^ ^^^^^^^^ ^^^^) using ^^^^^^^^ = ∑^^^^ ^^^^=1 ^^^^^^^^ +^^^^^^^^; (d) calculating an adsorbed phase mole fraction for the reference molecule in the ^^^^ monolayer only (^^^^ ) u ^^^^ ^^^^ sing ^^^^^^^^ = ∑^^^^ ^^^^=1 ^^^^^^^^ +^^^^^^^^; (k) calculating a for the adsorbate component (^^^^^^^^) using ^^^^+1 ^^^^2^^^^2^^^^^^^^^^^^�Ω^^^^^^ − 1� ^^^^ = ^^^^ ^^^^ ^^^^ ^^ ^^^^ wherein: a nonrandomness factor set about 0.3 per the (l) calculating a gas phase mole fraction for the reference molecule (^^^^^^^^) using ln ^^^^ = ^^ ∑ ^^^^+1 ^^^^^ 2 ^^^^^^^^ 2 ^^^ ^^^^^^^^^^^^ � Ω^^^^^^^^−1 � ^^^^ ^^^^^^ ^^^^=1 2 ; (m) calculating a new ^^^^ �^^^^ +^^^^ ∗ ^^^^�^^^^ ^^^^ ^^^^ ^^^^^^ ^^^^ ^^^^ = ^^ ^^^^ ^^^^^^^^ = ^^^^ ^^^^^^^^^^^^^^^^ ^^^^^^^^ ^^^^^^ ^ ^^^^^^^^ ^^^^^^^^ = ^^^^0 ^^^^ = ^^^^ ^ 1^^^^ ^^^^ ^^^�1−^^^^^^^^,^^^^^^^^�+^^^^^^ ^ ^^ ^^^^^^^, and 1^^^^ ^ ^^^^ (n) component ^^^^ adsorbed due to monolayer adsorption and using ^^^^^^^^ ^^^^ = ^^^^^^^^ + ^^^^ (o) repeating steps (a)-(h) whenever the initial values for ^^^^^^^^, ^^^^^^^^ and ^^^^^ ∗ ^^^ do not match the calculated values for ^^^^^^^^, ^^^^^^^^ and ^^^^^ ∗ ^^^; and ^^^^^^^^^^^^^^^^ ^^^ ^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^ = ^^^^ ^^^^^ ^^^^ −^^^^ ^ (p) ∑ ^^^ ^^^^ ^^^^ ^^^^ ^^^^=1� ^^^^^^^^^^^^^^^^^^^^ � is not minimized.
38. The system of claim 29, wherein the adsorption equilibrium constant and the condensation- evaporation equilibrium constant are calculated for multiple temperatures.
39. The system of claim 29, wherein the gas absorption system comprises: an adsorbent comprising titanium dioxide anatase; and the adsorbate component comprises oxygen or nitrogen.
40. The system of claim 29, wherein the gas absorption system comprises: an adsorbent comprising activated charcoal; and the adsorbate component comprises benzene or cyclohexane.
41. The system of claim 29, wherein the gas absorption system comprises: an adsorbent comprising activated alumina; and the adsorbate component comprises water or carbon dioxide.
42. A system for identifying a total amount of two or more adsorbate components absorbed for a mixed-gas multilayer adsorption system comprising: a memory; an input/output interface; and one or more processors communicably coupled to the memory and the input/output interface, wherein the one or more processors: calculate the total amount of the two or more adsorbate components (i) absorbed (^^^^^ ^ ^^ ^^ ^^) using ^^^^^^^^ ∗ ∗ ^^^^ = ^^^^^^^^ + ^^^^ ^^^^^^^^ , wherein: ^^^^^^^^ is an amount adsorbed of each adsorbate component (^^^^) due to monolayer adsorption, ^^^^∗ is a total of amounts adsorbed due to condensation on subsequent layers, ^^^^^^ ∗ ^^ is a dew point liquid phase composition, and provide the total amount of the two or more adsorbate components (i) absorbed (^^^^^^^^ ^^^^) to the input/output interface; and wherein the gas adsorption system is configured using the total amount of two or more adsorbate components absorbed.
43. The system of claim 42, wherein a product is produced using the mixed-gas multilayer adsorption system.
44. The system of claim 42, wherein the mixed-gas multilayer absorption system comprises a CO2 capture system, a hydrocarbons processing system, an air separation system, a direct air capture system, a trace elements and heavy metals removal system, or an aqueous organic acids separation system.
45. The system of claim 42, wherein the mixed-gas multilayer adsorption system comprises a liquid adsorption system mathematically formulated as a pseudo gas adsorption system.
46. The system of claim 45, wherein the liquid adsorption system comprises a trace elements removal system, a heavy metals removal system, an aqueous organic separation system, or an ethanol-water separation system.
47. The system of claim 42, wherein the input/output interface comprises an interface to the mixed-gas multilayer adsorption system.
48. The system of claim 42, further comprising configuring the mixed-gas multilayer adsorption system using a binary interaction parameter (^^^^^^^^^^^^), a correction factor (^^^^) and a pressure (P).
49. The system of claim 42, wherein the total amount of each of the two or more adsorbate components (i) absorbed for the mixed-gas multilayer adsorption system (^^^^^^^^ ^^^^) are calculated using a regression analysis.
50. The system of claim 42, wherein calculating the total amount of each of the two or more adsorbate components (i) absorbed for the mixed-gas multilayer adsorption system (^^^^^ ^ ^^ ^^ ^^) comprises: providing input data comprising a pressure (P), a gas phase mole fraction for each adsorbate component (^^^^^^^^), the total amount of the two or more adsorbate components (i) absorbed (^^^^^ ^ ^^ ^^ ^^), a saturation adsorption loading for monolayer adsorption of each adsorbate component (^^^^^ 0 ^^^), an adsorption equilibrium constant for each adsorbate component ( ^^^^^^^^ ^^^^ ), a condensation- evaporation equilibrium constant for each adsorbate component (^^^^^^^^,^^^^ ), a binary interaction parameter (^^^^^^^^^^^^) for the species ^^^^ – species ^^^^ pair, an adsorbate area for each adsorbate component (^^^^^^^^), an adsorbate area for a reference molecule (^^^^^^^^), a dew point pressure for each adsorbate component (^^^^^^^^^^^^^^^^ ^^^^ ), and a dew point liquid phase composition (^^^^^ ∗ ^^^); (a) providing initial values for the binary interaction parameter (^^^^^^^^^^^^) and a correction factor (^^^^); (b) providing initial values for an amount adsorbed of each adsorbate component due to monolayer adsorption (^^^^^^^^ ), an amount adsorbed of the reference molecule due to monolayer adsorption ( ^^^^^^^^ ) and a total of amounts adsorbed of each adsorbate component due to on subsequent layers (^^^^^ ∗ ^^^) at a given gas phase mole fraction for each adsorbate component (^^^^^^^^) and pressure (P); (c) calculating an adsorbed phase mole fraction for each adsorbate component in the monolayer only (^^^^ ) using ^^^ ^^^^^^^^ ^^^^ ^^^^^ = ^^^^^^^^ +^^^^^^^^; (d) calculating an adsorbed phase mole fraction for the reference molecule in the ^^^^ monolayer only (^^^^ ) using ^ ^^^^ ^^^^ ^^^^^^^ = ∑^^^^ ^^^^=1 ^^^^^^^^ +^^^^^^^^; (e) calculating a gas phase adsorbate component (^^^^^^^^) using ^^^^^^^^�Ω^^^^^^^^ − 1� ln ^^^^ = ^^ ^^^^ ^^^^ ^^^^ ^^^^^^� wherein: a nonrandomness factor set to about (f) a gas (^^^^^^^^) using ln ^^^^ ^^^^2^^^^2^^^^^^^^^^^^�Ω^^^^−1� ^^ ^^^^ ∑ ^^^^+1 ^^^^ ^^^^ ^^^^ ^^ = ^^^^ ^^^^=1�∑^^^^ ^^^^=1 ^^^^ 2 ; ^^^^^^^^^^^^Ω^^^^^^^^� (g) calculating a new ^^^^^^^^ �^^^^+^^^^ ∗�^^^^ ^ ^^^^ ^^^^ ^^^^^^^^ = ^^^^ ^^^^ ^^^^ ^^^^^^^ ^^^^ 0 ^^^^ = ^^^^^^^^^^^^ ^^^^ 1^^^^ ^^^^ (h) calculating components (i) absorbed (^^^^^^^^ ^^^^) using ^^^^^^^^ ^^^^ = ^^^^ ∗ ^^^^ + ^^^^^^^^; (i) repeating steps (a)-(h) whenever the initial values for ^^^^^^^^, ^^^^^^^^ and ^^^^^ ∗ ^^^ do not match the calculated values for ^^^^^^^^, ^^^^^^^^ and ^^^^^ ∗ ^^^; and ^^^^^^^^^ 2 ^^^ ^^^^ ^^^^^^^ ^^^^ ^^^^^^^^^^^^^^^^ (j) repeating step - = ∑ ^^^^ ^^^^^ ^^^^ −^^^^^^^^ ^^^^ ^^^^=1� ^^^^^^^^^^^^^^^^^^^^ � is not minimized.
51. The system of claim 42, wherein the total amount of each of the two or more adsorbate components (i) absorbed for the mixed-gas multilayer adsorption system (^^^^^^^^ ^^^^) are calculated for multiple temperatures.
52. The system of claim 42, wherein the mixed-gas multilayer adsorption system comprises: an adsorbent comprising titanium dioxide anatase; and the adsorbate component comprises oxygen and nitrogen.
53. The system of claim 42, wherein the mixed-gas multilayer adsorption system comprises: an adsorbent comprising activated charcoal; and the adsorbate component comprises benzene and cyclohexane.
54. The system of claim 42, wherein the mixed-gas multilayer adsorption system comprises: an adsorbent comprising activated alumina; and the adsorbate component comprises water and carbon dioxide.
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