Vapor Mixtures in Microporous Solids - The

Article pubs.acs.org/JPCC

Net Adsorption of Gas/Vapor Mixtures in Microporous Solids Orhan Talu Department of Chemical and Biomedical Engineering, Cleveland State University, Cleveland, Ohio 44115, United States ABSTRACT: Adsorption thermodynamics is based on Gibbs definition, which transforms the nonuniform interfacial region to a uniform three-phase system including a two-dimensional adsorbed phase on a hyper-surface. Gibbs definition is a pure mathematical construct applicable wherever the hyper-surface is located. On the other hand, physical quantification of adsorption and hence its applications require that the hyper-surface be located. Conceptually, the location of hyper-surface differentiates between so-called absolute, excess, and the recently introduced (Gumma and Talu, Langmuir 2010, 26 (22), 17013−17023) net adsorption thermodynamic frameworks. This article details net adsorption thermodynamic framework for mixtures. In addition, a thermodynamic inconsistency is recognized in the calculation of grand potential (or solid chemical potential) with commonly used implementation of excess adsorption in literature. The inconsistency is shown to have a substantial impact on further thermodynamic calculations such as mixture adsorption predictions for even a simple typical example as oxygen−nitrogen-zeolite 5A system at 22 C and moderate pressures. Historically, this inconsistency seems to originate from adopting intuitive concepts for planar surfaces to microporous systems without regard to the differences in the physical nature of these two types of interfaces. Net adsorption framework circumvents the inconsistency as well as providing an unequivocal description of adsorption in micropores.

■

INTRODUCTION Most significant applications of adsorption with microporous solids are separation processes.1 The ability of solids to selectively adsorb components from a fluid is the basis of many industrial processes such as gas dehydration, hydrogen separation and purification, oxygen and/or nitrogen production from air, separation of normal, branched, and cyclic paraffins, paraffin/olefin separations, xylene isomer separations, and so on.2,3 Adsorption is also extensively utilized in air and water pollution control.4 All these applications involve mixtures. Mixture adsorption formulations are complicated, and experimental measurements are tedious, time-consuming, and prone to accuracy problems. Prediction of mixture adsorption equilibrium from pure component information still remains as one of the most challenging problems in adsorption5 evidenced by decades of research on experiment, theory, and more recently on molecular simulations. A new thermodynamic framework for adsorption named net adsorption was recently introduced by Gumma and Talu.6 The original development of net adsorption was primarily motivated by gas storage applications (i.e., adsorptive storage of hydrogen and methane), thus the formulations were developed for only pure component adsorption. In this work, net adsorption is extended to mixtures. As stated by one of the pioneers in adsorption area, Van Ness,7 “It is the function of thermodynamics to relate those properties of a system required for practical or theoretical purposes to the parameters that are most readily measured, and thus to provide maximum return of information for any investment in experiment”. Formal thermodynamic treatment of adsorption dates back to Gibbs.8 Since it is not possible to review voluminous literature on thermodynamics of adsorption here, only significant contributions that shaped the current state © 2013 American Chemical Society

particularly for mixture gas/vapor adsorption in microporous solids are provided here. In this manuscript, the term microporous solid is used in general to denote a solid whose surface area cannot be independently measured without using any probe molecule. Myers and Prausnitz’s9 work in developing adsorbed solution theory (AST) was particularly effective in formulating rigorous solution thermodynamics of mixture adsorbed phases. If the adsorbed phase behaves as an ideal mixture, the formulations predict mixture equilibrium completely from pure component information (as isotherms) in the well-known ideal adsorbed solution theory (IAST). IAST is similar to Rault’s law for vapor−liquid equilibrium (VLE) with a subtle but extremely important difference in the standard state definition. Later, the concepts introduced by Myers and Prausnitz were extended to nonideal systems first using liquid phase activity coefficient models.10−12 The important differences between liquid and adsorbed phase activity coefficients were recognized by Talu and Zwiebel,13,14 who developed a semitheoretical model for adsorbed phase activity coefficients named spreading pressure-dependent (SPD) activity coefficients. Later Valenzuela and Myers15 introduced an empirical model that was shown to follow the experiment and molecular simulation results from zero loading by Talu et al.16 Formulations built around the Gibbs definition of adsorption are the only general thermodynamic treatment of mixture adsorption in literature. It should be noted that there are many mechanistic formulations for mixture microporous adsorption. These formulations are not necessarily thermodynamically consistent; they are often used for mathematical convenience Received: March 1, 2013 Revised: May 30, 2013 Published: June 3, 2013 13059

dx.doi.org/10.1021/jp4021382 | J. Phys. Chem. C 2013, 117, 13059−13071

The Journal of Physical Chemistry C

Article

mathematical (and thermodynamic) consistency is that the location of hyper-surface cannot depend on thermodynamic properties such as temperature and pressure. The word hypersurface is used here to emphasize that Gibbs definition is only a mathematical 2D construct; it does not have any shape, size, topography, and so on. The hyper-surface is commonly referred to as Gibbs-dividing-surface in the literature, and the concepts explained in this paragraph are referred to as Gibbs definition of adsorption.

such as the mixed-Langmuir model, which can provide consistent results only under certain circumstances.17 Although the thermodynamic framework described by adsorbed solution theory is rigorous, an important discrepancy in its common implementation is recognized for the first time in this work. Since the discrepancy stems from misinterpretation of very basic concepts, it is first necessary to review the fundamental definitions; i.e., Gibbs definition of adsorption. Then, the most general relations for pure and mixture adsorption are developed from fundamental equations of thermodynamics, only with the stipulations in Gibbs definition of adsorption, without any assumptions, to highlight the mentioned discrepancy. Essential differences between adsorption on planar surfaces and in micropores are detailed. The Gibbs definition is then applied to microporous solids to show the differences between absolute, excess, and net adsorption. Finally, the significance of mentioned discrepancy is quantified by using a typical set of data from literature supplemented with some new experimental data necessary for the purpose. It is also shown below that the net adsorption framework mentioned above completely circumvents the discrepancy while absolute and excess adsorption requires additional information to perform thermodynamic calculations beyond what is typically provided in literature. The three adsorption frameworks are detailed and contrasted in the following. Other relevant issues such as experimental measurements, thermophysical modeling, impact on kinetics and dynamics, implications on molecular simulations, and practical implications for process design and operation with net adsorption will be addressed in the future in other studies. Gibbs Definition of Adsorbed Phase. Adsorbed phase is not autonomous; it does not exist without adjoining bulk phases as fluid (i.e., gas, vapor, or liquid) and solid. Most adsorbed phase properties in micropores cannot be measured directly; they are inferred from changes occurring in adjoining bulk phases. This inference is best exemplified in pure component adsorption isotherm measurements by (1)a volumetric apparatus where changes in quantity of gas are measured, and (2) a gravimetric apparatus where changes in solid weight are measured. Since micropore adsorption can only be measured as differences, any quantification of thermodynamic properties requires the definition of a reference state. (In this work, the term reference state is used to assign values of thermodynamic properties that can only be measured as differences. By contrast, the term standard state is used in solution thermodynamics formulations of mixtures.) Further complication arises because thermodynamic properties are not uniform in the interfacial region. The density changes dramatically between dense solid and dilute fluid phase. As a result, the potential field varies substantially causing nonuniformity in thermodynamic properties. The interfacial region is bordered by two bulk phases, solid and fluid, with uniform properties far away from the interface where the potential field is uniform. Gibbs8 proposed to replace the “ill-defined,” nonuniform interfacial region with a mathematically equivalent form where properties of two bulk phases are treated uniform, and the differences from actual interfacial region are attributed to an adsorbed phase on a hyper-surface. Mathematically, two bulk phases as three-dimensional (3D) objects are separated by a two-dimensional (2D) hyper-surface. A 2D adsorbed phase does not have a volume. Gibbs does not specify where the hyper-surface should be located. The only restriction for

■

THERMODYNAMIC RELATIONS INDEPENDENT OF HYPER-SURFACE LOCATION First, pure thermodynamic relations for the Gibbs’ definition of adsorption will be developed without specifying the location of dividing hyper-surface. These relations provide the necessary background to demonstrate the implications of absolute, excess, and net adsorption frameworks. Thermodynamic System. Consider a heterogeneous system containing the microporous solid in equilibrium with a gas mixture containing C components. (The word gas is used in this manuscript to depict any f luid). With Gibbs’ definition, there are three distinct phases in the system: two bulk phases as solid and gas, and an adsorbed phase. Any total extensive thermodynamic property (Zt) may be written as Zt = Z s + Z a + Z g

(1)

where the superscripts t, s, a, and g stand for total, solid, adsorbed, and gas phases, respectively. The fundamental property relation for this heterogeneous system is C

d(U t) = Td(St) − Pd(V t ) +

∑ μi d(Nit) + μs d(M) i

(2)

Nti

where is total number of moles of species i with a chemical potential of μi. Chemical potential is the same for component i in all phase at equilibrium. Mass, M, is used as the extensive property for the solid with its chemical potential μs. Mass of pure solid at zero pressure (i.e., when there is no adsorption) is the only extensive property of a microporous solid which can be measured independently as discussed later. Total internal energy (Ut) is a first-order homogeneous function of the extensive properties St, Vt, Nit, and M. Equation 2 can be integrated using Euler’s theory for homogeneous functions. C

U t = TSt − PV t +

∑ μi Nit + μs M i

(3)

Taking the total differential of eq 3 and substituting into eq 2 gives the Gibbs−Duhem relation for adsorption as C

Std(T ) − V td(P) +

∑ Nitd(μi ) + Md(μs ) = 0 i

(4)

Mathematical Definition of Gibbs Adsorption. Only the total extensive property Zt in eq 1 can be measured in experiments with microporous solids. None of the terms on the right-hand-side can be separately measured, which is the main reason for using Gibbs definition in adsorption thermodynamics. Adsorbed phase properties are inferred/back-calculated from eq 1 as

Z a = Zt − Z g − Z s 13060

(5)



Article

causes thermodynamic discrepancy in the calculation of solid potential. Furthermore, this omission also has important implications on placing the hyper-surface as discussed later. Noting phase equilibrium for gas components, the chemical potential can be written as

This fact complicates thermodynamics but does not compromise its rigor. There are three important stipulations of Gibbs definition of adsorption: (1) Adsorbed phase does not have a volume. Va = Vt − Vg − Vs = 0

(6)

dμi = RTd(ln Pi)

(2) Solid phase is a pure component closed system (distinguishing adsorption from absorption), thus it does not contain any gas components. Nis = 0

(for i = 1...C)

(Ideal gas equation of state (EOS) is used in eq 14 and throughout this manuscript for simplicity, all equations are general, and they can be converted using a real-gas EOS if necessary.) Noting that d(P) = ΣCi Pid(LnPi), eq 13 becomes

(7)

(3) Gas phase with uniform properties up to the hypersurface is an open system in equilibrium with adsorbed phase on the hyper-surface. The amount in gas phase can be determined from its density (ρg) if and only if gas phase volume (Vg) is known. Nig

g g

= V ρ yi

C ⎛ v sPi ⎞ dμs = −RT ∑ ⎜nia − ⎟d(ln Pi) ⎝ RT ⎠ i

=

Nit

(8)

−

Nig

−

Nis

=

Nit

g g

− V ρ yi

φ = (μs − μs *) = −RT

− V d (P ) +

∑

(9)

ηi = nia −

+ Mdμ = 0 (10)

C

∑ Nigdμi = 0 i

(11)

Combining eqs 6−11 gives C

−V sd(P) +

∑ Niadμi + Mdμs = 0 i

(12)

Division with M to convert to mass specific variables and rearrangement gives c

dμs = −∑ niad(μi ) + v sd(P) i

⎞

v sPi RT

(17)

The first term is the amount adsorbed. Its value depends on where the hyper-surface is located. The second term (in this context) is the amount of gas that would be in the space occupied by solid. Its magnitude also depends on where the hyper-surface is located. But at a given condition (i.e., T, P, y), the integral of their dif ference, ηi, must be a constant independent of the hyper-surface location, since grand potential is a state property. The second term originating from Poynting factor is a correction to the amount adsorbed due to the volume occupied by the solid. Neglecting it results in an overestimate of the grand potential. Except for the trivial case of Pi = 0, the Poynting factor vanishes if and only if the hyper-surface is placed to make vs zero. For all other cases of hyper-surface placement, specific solid volume, vs, where gas molecules are not present by Gibbs definition, must be somehow measured and included in eq 16 for thermodynamic consistency. Mixture Adsorption in Microporous Solids by Solution Thermodynamics. Applications of adsorption almost always involve mixtures. Here general relations for mixture adsorption are developed without placing the hypersurface. Theoretical developments supplemented with data (either physical or from molecular simulations) aim at

The Gibbs−Duhem relation at isothermal conditions for the gas phase is − V g d (P ) +

s

This is the general Gibbs adsorption isotherm equation. μs* is the chemical potential of pure solid at its reference state. The grand potential (φ) is also defined in the equation as the change in chemical potential of the solid from its reference state. Equation 16 relates solid chemical potential to measurable properties of gas adsorption in microporous solids. It is the only way to calculate the chemical potential of a microporous solid. Equation 16 is completely general and applies regardless of where the hyper-surface is located. The integrand in eq 16 plays an important role and deserves further examination. It can be defined in a new thermodynamic property, ηi, as the contribution of compound i to the grand potential

s

i

⎛

(16)

C

Nitdμi

Pi C

vP ∫i ∑ ⎜⎝nia − RTi ⎟⎠d(ln Pi) i

Gibbs Adsorption Isotherm Equation. As mentioned earlier, only total solid mass at zero pressure can be directly and unequivocally measured. Other properties of the solid vary with properties (e.g., pressure). Chemical potential of the solid (μs) plays a pivotal role in thermodynamics as evident in fundamental property relation in eq 2. It is calculated from measurable quantities using the Gibbs isotherm equation, which is developed as follows: At constant temperature, Gibbs−Duhem relation (eq 4) for the heterogeneous system becomes t

(15)

Solid chemical potential appears as a differential term in eq 15, which is integrated at constant temperature from the reference state defined as pure solid at zero pressure, i.e., without any gas present.

By these three stipulations, the heterogeneous interfacial region is mathematically transformed to two homogeneous bulk phases (i.e., gas and solid) and an interfacial adsorbed phase on the hyper-surface. The amount adsorbed as the most important extensive property of the adsorbed phase is from eqs 5 and 7, Nia

(14)

(13)

(Lower case letters are used in this manuscript for solid mass specific properties, which are intensive properties of the adsorbed phase.) The last term in this equation is the effect of pressure on chemical potential of solid, similar to Poynting factor.18 It has been omitted in almost all of literature on adsorption thermodynamics.19 Omission of the Poynting factor 13061



Article

Pio = f {T , φ}

formulating partial amount adsorbed from a mixture as a function of gas properties. Mathematically, the purpose of mixture adsorption formulations is nia = f {T , P , y}

(for all i = 1...C)

In phase equilibrium calculations, these equations are implicitly solved using Gibbs adsorption isotherm (eq 16) for pure components with

(18)

φ = φio = φjo

(The {} in this manuscript are used to indicate functionalities wherever necessary for clarity.) On the other hand, the primary intensive properties for adsorbed phase are temperature, grand potential, and adsorbed phase composition (x), mathematically: nia = f {T , φ , x}

(for all i = 1...C)

(19)

(20)

Chemical potential in gas phase given in eq 14 can be integrated from a reference state pressure P+i with chemical potential μ+i {T,P+i } at constant temperature. ⎛P ⎞ μi g = μi+ {T , Pi+} + RT ln⎜ +i ⎟ ⎝ Pi ⎠

C

d (φ ) = −∑ ηid(ln Pi) RT i

d(ln Pi) = d(ln Pioxiγi) C

d (φ ) = −∑ ηid(ln Pio) − RT i

(22)

i

d (φ ) = −ηiod(ln Pio) RT

(30)

(31)

ηoi

where is at standard state conditions; at same temperature and grand potential as the mixture. Combining the last two equations to eliminate d(LnPoi ), and rearrangement gives

(23)

⎡ d (φ ) ⎢ ⎢1 − RT ⎢ ⎣

C

∑ i

⎛ ⎜ ∂ ln γi o + ∑ ηi ⎜ φ ηi i ⎝ ∂ RT C

ηi

( )

⎞ ⎤ ⎟ ⎥ ⎟ ⎥=0 ⎠T , x ⎥⎦

(32)

For any arbitrary change d(φ) the term in parentheses must be zero which is rearranged to calculate total amount adsorbed, ηT, as

⎛ Po ⎞ μioa {T , φ} = μiog {T , Pio} = μi+ {T , Pi+} + RT ln⎜ i+ ⎟ ⎝ Pi ⎠

1 = ηT

(24)

Finally, combining eqs 20−24 gives phase equilibrium relation for gas/vapor-adsorbed phase equilibrium (VAE) as

Pyi =

⎡ ∂ ln γi ⎤ ⎥ dφ ⎣ ∂φ ⎦T , x

C

∑ ηi⎢

For all pure components at their standard states

Unlike liquid phase where pressure dependency of activity coefficients are usually neglected (i.e., incompressible liquid assumption), the grand potential dependence for adsorbed phase activity coefficients cannot be neglected since adsorbed phase is highly compressible. Adsorbed phase density varies very substantially from zero (at zero pressure) to large finite, liquid-like values at finite pressures.13 Standard state chemical potential in eq 22 can be written in terms of pure component gas phase properties as

Pioxiγi

(29)

Combining these with some mathematical manipulation gives

where μoa i is the standard state chemical potential of pure component i in adsorbed phase at same temperature and grand potential as the mixture, and γi is the activity coefficient. Note that activity coefficients in adsorbed phase are functions of temperature, grand potential, and composition γi = f {T , φ , x}

(28)

From phase equilibrium relation in eq 25

(21)

With solution thermodynamics, chemical potential in adsorbed phase can be expanded using pure component standard states as μia = μioa {T , φ} + RT ln(xiγi)

(27)

Therefore, for a given set of gas properties as {T,P,y}, the adsorbed phase intensive properties {T,φ,x} can be calculated by simultaneously solving eqs 16, 23, and 25−27. If the adsorbed phase is ideal (γi = 1 for all i), multicomponent adsorption equilibrium is completely predicted from pure component isotherms, which forms the IAST. Total Amount Adsorbed from a Mixture. The phase equilibrium relations above provide grand potential and composition of mixture adsorbed phase at given gas conditions. On the other hand, applications need the amount adsorbed, which is derived next once again without any stipulation about the location of the hyper-surface. Writing the Gibbs Adsorption isotherm, eq 16, in general differential terms with ηi defined in eq 17

Solution thermodynamics is used to develop equilibrium relations between intensive properties of the system as the correspondence {T,P,y} ⇔ {T,φ,x}. Phase equilibrium requires that chemical potential of each species be same in all phases, in this case in gas and adsorbed phases μi g {T , P , y} = μia {T , φ , x}

(26)

C

∑ i

xi − ηio

⎛ ∂ ln γ ∑ xi⎜⎜ φ i i ⎝ ∂ RT C

( )

⎞ ⎟ ⎟ ⎠T , x

(33)

where ηT is defined as C

(25)

ηT =

This equation, first developed by Myers and Prausnitz,9 is similar to modified Rault’s law for vapor−liquid equilibrium18 with the subtle but important difference being that the standard state pressures for adsorbed phase are functions of temperature and grand potential, not just temperature as with VLE. Functionally,

∑ ηi

(34)

i

Adsorbed phase mole fractions are therefore η xi = i ηT 13062

(35)



Article

Equation 33 was first derived by Myers and Prausnitz19 for IAST without the last term, and by Talu and Zwiebel13 for the general nonideal case.

for two layers of close-packed nitrogen molecules on a single graphene sheet.23 Another alternative to surface area is to use the total pore volume of adsorbed phase as the extensive property to replace mass (M) in thermodynamic relations starting with eq 2. In essence, the practical implementation of this approach leads to the so-called “pore-filling” mechanisms adapted in numerous adsorption formulations (e.g., Polanyi theory,24 Dubinin models).25−27 Typically, pore-filling models assume uniform density in direction parallel to the surface. The density is likened to highly compressed fluid phase in the potential field of the solid (or the liquid phase in some cases). This picture is far from reality in the confined spaces of micropores. The spatial organization of molecules in the pores is predominantly dictated by the solid potential, not by the molecule to molecule interactions as in fluid phases. The adsorbed phase density in micropores is/can be very different from liquid density. Simply put, adsorbed phase is not a liquid (even when the adjoining bulk phase is a liquid). As a result, the pore volume estimated from these models (and by other techniques) is based on a fictitious density value. It would be futile to base all further thermodynamic development using an extensive variable based on an unphysical, hypothetical density value. These discussions show that both surface area and pore volume of a microporous solid are not unequivocal and cannot be measured independently without any adsorption occurring. Their values depend on probe molecule used in the measurements. Therefore, these quantities are not a property of the pure solid as commonly cited in the literature. Rather, surface area and pore volume are a property of a solid/probe− gas binary system: a pair property in thermodynamics. As such, they are inherently flawed to be used to quantify the extent of pure solid in thermodynamics. The only quantity that can be independently measured and used to quantify the extent of a pure microporous solid is its mass at vacuum as used in this work starting with eq 2. Obviously, surface area and pore volume bring important physical insight to adsorption phenomena, which have been used for generations, but they cannot be used to represent the extent of solid in the system in thermodynamic equations without major complications and inconsistencies. Given that mass is the only extensive property that can be unequivocally used in micropore adsorption, the derivations in preceding sections show that rigorous thermodynamics requires mass specific solid volume (vs) to calculate the grand potential, or the chemical potential of the solid. Further formulations are significantly simplified by taking vs independent of pressure and temperature: incompressible solid with negligible thermal expansion. For most applications of adsorption involving crystalline (e.g., zeolites, MOFs, etc.) and rigid amorphous (e.g., activated carbon, silica gel, mesoporous materials, activated alumina, etc.) solids, these assumptions are justifiable, therefore vs can be treated as a constant. The effect of pressure and temperature on solid volume is further discussed by Myers.19 If vs changes with pressure (or loading) and temperature, the formulations can be modified with isothermal compressibility and thermal expansion coefficient of the solid.

■

MEASURABLE PROPERTIES OF MICROPOROUS ADSORPTION Up to this point, all equations above are exact thermodynamic relations. They apply regardless of how or where the hypersurface is placed. Before attempting to describe how the hypersurface is conceptually placed, experimentally measured, and theoretically used, it is important to elaborate on the assertion above that vacuum mass is the only independently measurable extensive property of a microporous solid. Adsorbed phase is on a hyper-surface by Gibbs definition. Since it refers to a surface, one would be tempted to use total surface area as the extensive property for adsorbed phase in above formulations starting with eq 2. However, this approach is not possible for microporous solids. A well-defined surface area is clear and simple for a planar solid, or a liquid surface, where the surface curvature is practically infinity, or orders of magnitude larger than molecular size. For these planar interfaces, interfacial area can be independently measured by other techniques not involving any adsorption. By contrast, the concept of interfacial surface area becomes physically meaningless for microporous solids where the pores are of molecular size. It is not possible to measure a unique surface area of a microporous solid unequivocally. Attempts to measure surface area of microporous solids are similar to the “coastline paradox” frequently discussed in fractal geometry as the Richardson effect.20 As was first observed by Lewis Fry Richardson, the coastline length of Great Britain is 2800 km if measured with a 100 km device, but the same coastline measures 3400 km using a 50 km measuring device. In fact, the coastline length unboundedly increases as the measuring device length decreases; the measured value depends on the length of the measuring device. The only measuring device to determine surface area of a microporous solid is gas molecules. Depending on molecular size, each gas used as a probe yields a different surface area value for the same solid. It is well-known that measured surface area for the same solid is larger with smaller molecules.21 At the other extreme, measured surface area approaches zero if the gas molecules are larger than the pores, which, in fact, is the basis for a major application of adsorption as molecular sieving. This argument only considers geometric aspects without the further complications caused by the solid potential field. Even when the solid potential is “switched off,” which is only possible in molecular simulations, surface area of a solid depends on the size of probe molecule as already shown in the literature many times.21,22 In reality, it is not possible to switch-off the solid potential. Any probe gas forms a dense phase in the immediate vicinity of a solid. Often adsorbed phase is irregularly organized and/or forms multilayers due to variations in potential field with location. This result in a miscount of number of molecules assumed to be lining the surface regularly to enable surface area calculation. Even when using analysis techniques with allowance for multilayer formation, the result is often an overcount giving surface area values greater than physically possible. The most commonly known example of this unphysical result is nitrogen BET areas for activated carbon greater than approx. 2630 sq·m/g, which is the theoretical limit

■

LOCATING GIBBS HYPER-SURFACE: ABSOLUTE, EXCESS, AND NET ADSORPTION Location of the Gibbs hyper-surface determines the value of mass specific solid volume, vs. The location of hyper-surface for adsorption in microporous solids can be conceptually defined 13063



Article

Figure 1. Illustration of Gibbs hyper-surface definitions by absolute, excess, and net adsorption. Dark shaded regions are not accessible by fluid molecules.

in one of three ways as was first noted by McBain28 and later elaborated by Coolidge.29,30 Figure 1 illustrates the three possible definitions which correspond to absolute, excess, and the recently introduced net adsorption.6 Note that a hypersurface in 3D maps to a 2D cross section on plain paper; the location of the hyper-surface in the figure is therefore demarcated by showing the extent of bulk solid and gas phases as 2D areas (which would be volumes in 3D). The total physical volume corresponding to Figure 1 illustration is conceptually divided into three parts as inaccessible solid volume (Vis), pore volume (Vp), and free gas volume (Vfg). V t = V is + V p + V fg

effect, simulations are performed inside the single particle in Figure 1a without the adjoining gas phase, and the entire simulation volume corresponds to Vsabs in eq 36 by prescription. On the other hand, unlike molecular simulations, experiments cannot be performed within a single particle no matter how large. Experiments and applications always have some amount of bulk gas in contact with the solid. Hence, the absolute adsorption reference state is almost never used in experimental or application work. Excess Adsorption. In excess adsorption (Figure 1b), the hyper-surface is drawn through the pores to include pore space with gas phase. The remaining volume is commonly referred as “impenetrable” solid volume. Mathematically, the thermodynamic properties are related to physical volumes in Figure 1b with excess adsorption by

(34)

It is clear that the total volume in this equation is identical to the total volume in the fundamental equation, eq 2. Adsorption definitions differ on how the three physical volumes in eq 34 are related to the thermodynamic phase volumes Vs and Vg, which make up the total volume in eq 2. Note Va = 0 by Gibbs definition. Absolute Adsorption. In absolute adsorption (Figure 1a), the hyper-surface is drawn around the particle to include pore space with solid phase volume. Mathematically, the thermodynamic properties are related to physical volumes in Figure 1a with absolute adsorption by g V abs

t

is

=V −V −V

p

φ = −RT

Pi C

⎛

i

ηi = niabs −

(v is + v p)Pi RT is

is

+ v p)Pi ⎞ ⎟d(ln Pi) RT ⎠

V exs = V is

(40) Pi C

⎛

is

⎞

v P ∫0 ∑ ⎜⎝niex − RTi ⎟⎠d(ln Pi) i

ηi = niex −

v isPi RT

(41)

(42)

The value of vis is necessary to calculate grand potential with excess adsorption. Excess adsorption avoids the artificial assumptions about adsorbed phase density, but still requires the use of a probe molecule to locate the hyper-surface. Therefore the solid reference state is also a thermodynamic pair-property. Since the 1930s (starting with McBain28), helium around room temperature was used to locate the hyper-surface with excess adsorption by essentially performing helium isotherm measurements. With the assumption helium does not adsorb, impenetrable solid volume is back-calculated from isotherm data. Over decades now, numerous researchers argued theoretically31,32 and showed by experiments33−43 and by molecular simulations44−50 that helium does adsorb under ordinary conditions albeit to a small extent. In fact, any probe molecule would adsorb on/in a solid because of the potential field. The vast majority of adsorption data in literature is based on excess adsorption, which makes it inherently significant to the field. In addition, thermodynamic relations for mixture predictions are well-established with excess adsorption starting with Myers and Prausnitz.9 These are very important facts for

(36)

(v ∫0 ∑ ⎜⎝niabs −

(39)

φ = −RT

(35)

s V abs = V is + V p

V exg = V t − V is

(37)

(38) p

The values of v and v are necessary to calculate grand potential with absolute adsorption in eq 37. As discussed before, measurement of vis is only possible using probe molecules. Furthermore, measurement of vp involves unrealistic assumptions about adsorbed phase density. These also make the solid reference state a pair property. Although this is conceptually simple and may be intuitive, it is not possible to unequivocally measure the solid volume prescribed by absolute adsorption in experiments. On the other hand, absolute adsorption is the natural result in molecular simulations where calculations are performed with a model system which consists of only pores and the solid. In 13064



Article

contribution of component i to grand potential, ηi, is the net amount adsorbed.

the utility of excess adsorption. Unfortunately, thermodynamic framework for mixtures with excess adsorption, as it was commonly used in literature by this author and many others, is not thermodynamically rigorous because the second term in eq 41 was omitted. The commonly used form of eq 41 in the literature is φ ≠ −RT

ηi = ninet

Conversion between Absolute, Excess, and Net Adsorption. It should be noted at this point that data with absolute, excess, and net adsorption reference states can be interconverted if enough information is available, just like any other thermodynamic data based on different reference states. The conversion is through

Pi C

∫0 ∑ niexd(ln Pi) i

(43)

This essentially equates vis to zero. The excess adsorption formulations commonly used in literature on one hand use a finite value of vis to perform the necessary material balance to calculate niex; on the other hand, assume vis = 0 to calculate grand potential by eq 43. As shown later, the grand potential by this equation deviates substantially from the correct value even for simple applications such as oxygen and nitrogen adsorption in zeolite-A at moderate pressures. Net Adsorption. In net adsorption (Figure 1c), the hypersurface is defined to make the solid volume to be zero. Mathematically, the thermodynamic properties are related to physical volumes in Figure 1b with net adsorption by g V net = Vt

(44)

s V net =0

(45)

φ = −RT

ηi = ninet = niex −

i

v isPi (v is + v p)Pi = niabs − RT RT

(48)

The values of vp and/or vis are necessary for conversions. Unfortunately, the values for these crucial thermodynamic properties are almost never reported especially in recent literature. The established conventions by IUPAC in reporting physisorption data54 should be followed, and respective massspecific volumes must be published with excess and absolute adsorption data to further enable other researchers. Using the net adsorption framework, these extra volume measurements are circumvented. Conversion of excess adsorption data is particularly important since the vast majority of the existing literature data is for excess adsorption. Almost all excess data in the literature is based on nonadsorbing helium assumption, thus vis in eq 48 refers to solid volume per mass of adsorbent impenetrable by helium (or its inverse is the helium density of the solid). If solid volume is not reported with data, the value can be experimentally determined with helium on the same solid sample. This approach is implemented as described below to use a set of literature data with rigorous solution thermodynamics: (1) to assess the impact of neglecting vs in thermodynamic formulations, and (2) to demonstrate mixture net adsorption framework.

Pi C

∫0 ∑ ninetd(ln Pi)

(47)

(46)

The grand potential does not depend on any questionable volume measurement of solid and/or pores using probe molecules. Therefore the solid reference state is a pure component property. The hyper-surface location is invariant with net adsorption. It should be noted for eq 45 that Vsnet = 0 only implies that the reference state volume is taken to be zero; it does imply that the solid has no volume, which would be unphysical. As shown above, excess and absolute amount adsorbed reference states involve respective volume corrections, while a reference volume need not be measured at all for net adsorption since it is taken to be zero. Hence in experiments, net adsorption does not require any probe molecules with adsorbent in situ. Only equipment calibrations are necessary as the container volume in volumetric experiments and bucket mass in gravimetric measurements. These are independent measurements of high accuracy performed only once to calibrate the apparatus without any solid present in the system. In applications, the calculations require only the empty volume of contacting equipment and the pure mass of solid later placed in it. The amount adsorbed is determined only by the mass of solid without any (sometimes questionable) estimation of different porosities. In simulations, net adsorption is calculated from the simulation box volume. The resulting values are directly comparable to net experimental results. This is also just a “calibration” of simulations to compare to experiments. No additional simulation effort is necessary. Net adsorption framework overcomes the long-standing hurdle/complications in comparing molecular simulations to excess experimental data,48,51,52 which will be a topic for a future communication. The net adsorption definition is simple and unequivocal. In fact, the thermodynamic property defined above in eq 17 as the

■

EXPERIMENTAL SECTION Data Set. The demonstration data set is for oxygen− nitrogen/zeolite 5A system at two temperatures (23 and 45 C), and pressures up to 920 kPa published by Talu et.al.17 This excess adsorption data set was collected in two different laboratories with two different volumetric experimental approaches (i.e., open and closed system) with the same solid sample using nonadsorbing helium assumption; the results were in excellent agreement. The data set has also been used in numerous other publications.53−58 The range of conditions, the gases, and the solid are a typical example for a microporous adsorption application, thus the data set is well-suited to demonstrate several practical aspects/issues discussed above. Experiments to Measure vis. Mass specific solid volume, is v , is necessary for thermodynamic calculations since the original data was collected using excess adsorption definition with helium. It is experimentally measured recently for the same sample, which was still available in our laboratories. To determine vis, first the internal volume of an empty column is measured by the usual helium expansion technique from a known/calibrated tank. This measurement gives the total system volume, Vt. This would be the only step necessary for system calibration if net adsorption framework is used from the beginning. To make use of previous excess adsorption results, a known mass of sample (M) is packed in the column, and another set of helium expansions are performed to measure 13065



Article

Figure 2. Measurement of column volumes using helium with and without solid present to convert excess to net adsorption. The difference in slopes is related to the solid volume inaccessible by helium.

Figure 3. Pure component oxygen and nitrogen adsorption isotherm in 5A zeolite by excess and net adsorption definitions.

the gas phase volume Vt − Vis. This step is the same as determining the so-called “void volume” in the usual excess adsorption experiments. The results of both volume measurements are shown in Figure 2 as moles of helium admitted to the column versus final pressure with and without the solid in the column. Slopes of the lines are related to the mentioned volumes. The regression statistics are excellent as should be expected in such experiments. No curvature is observed with the sample in the column, indicating that the amount of helium adsorbed in this experiment is negligible within experimental accuracy. From the data, helium impenetrable solid volume for this sample of

zeolite-5A from Tosoh Corp. is calculated to be vis = 0.3277 (±0.001) cc/g. This is somewhat lower than 0.3719 cc/g reported by Malbrunot et al.37 at room temperature by a similar technique. The difference (which is inconsequential for current purposes) is attributed to the binder in our sample. Pure Component Adsorption. For convenience, isotherm data needs to be represented by mathematical expressions to perform thermodynamic calculations.59 The Virial isotherm equation is used for this purpose here with parameter values taken from the original paper; it provided an excellent fit for excess adsorption data. Net adsorption data can also be represented by the same equation with appropriate shift/ 13066



Article

Figure 4. Grand potential dependence on fluid pressure. Solid curves (filled symbols) show the correct values by eq 16 as net adsorption. The commonly used calculation method with excess adsorption by eq 43 shown as dashed curves (empty symbols) overpredicts the grand potential as shown by arrows.

Figure 5. Grand potential dependence on amount adsorbed. Solid curves (bold symbols) show the correct values by eq 16 as net adsorption. The commonly used calculation method with excess adsorption by eq 43 shown as dashed curves (empty symbols) underpredicts the grand potential as shown by arrows.

density of gas is lower than what it would be without the solid present. Hence the designation “net” adsorption. For light gases such as hydrogen, net adsorption can be negative under certain conditions indicating that the concentration enhancement in pore space is not sufficient to compensate for volume lost to the solid in a container. This is an important issue for adsorptive gas storage applications exploiting the enhancement in concentration. To clearly and unequivocally quantify such systems was, in fact, the original motivation for the development of net adsorption6 as mentioned in the Introduction. Furthermore in this work, net

transformation of amount adsorbed axis by using eq 48. The resulting modified-Virial isotherm equation also represents the recalculated net adsorption data with equal accuracy. Pure component excess and net adsorption isotherms are shown in Figure 3 at 23 C. (Bold arrows in all figures are used to highlight the impact of discussions above.) Excess data is taken from the original article; net data is recalculated using vis determined here. As expected, the net amount adsorbed is always lower than the excess amount adsorbed; the difference is quite substantial especially for oxygen as the lighter species. At the extreme, net adsorption is negative if the overall average 13067



Article

Figure 6. Effect of pressure on binary adsorption predictions by IAST as the total amount adsorbed and selectivity. Solid curves (bold symbols) show the correct results with eq 16 as net adsorption. The commonly used calculation method with excess adsorption by eq 43 shown as dashed curves (empty symbols) severely underpredicts the selectivity at all pressure levels as shown by arrows .

Figure 7. Effect of composition on binary adsorption predictions by IAST as the total amount adsorbed and selectivity. Solid curves (bold symbols) show the correct results with eq 16 as net adsorption. The commonly used calculation method with excess adsorption by eq 43 shown as dashed curves (empty symbols) severely underpredicts the selectivity at all compositions as shown by arrows.

adsorption is shown to be the thermodynamically necessary quantity for grand potential calculations in eq 16. The net adsorption value being negative in some circumstances is inconsequential; numerous thermodynamic properties are negative and/or change sign with properties (e.g., excess Gibbs energy for liquid mixtures, residual gas properties, etc.), especially when they are based on differences from a reference state just like adsorption is by Gibbs definition. The grand potential is calculated by eq 46. The results are shown as a function of pressure in Figure 4. Grand potential is negative, indicating spontaneous adsorption. The figure also

shows grand potential calculated by the erroneous, commonly used form of the Gibbs adsorption isotherm by eq 43 neglecting vis. The difference is quite substantial, especially for the lighter species as oxygen. To quantify the impact, the figure shows one grand potential level of −3.35 mol/kg, which is close to the majority of experiments performed for the binary system. At this level, the correct oxygen pressure is 3142 kPa, while the erroneous eq 43 gives 2658 kPa, almost 16% lower. The impact on the pure component amount adsorbed is shown in Figure 5. The erroneous equation results in an amount 13068



■

Article

CONCLUSIONS Almost all applications of adsorption with microporous solids involve mixtures. Mixture phase equilibrium calculations require the adsorption grand potential (i.e., the chemical potential of solid). The grand potential of a microporous solid cannot be directly measured; it is calculated from changes occurring in the adjoining bulk gas/vapor phase by so-called Gibbs isotherm equation. Here, the general Gibbs isotherm equation is derived from fundamental equations without any particular stipulations about the location of hyper-surface delineating bulk fluid and solid phases. Resulting eq 16 clearly shows that the mass specific volume assigned to the solid by Gibbs definition is also necessary in addition to the amount adsorbed for the grand potential calculation. The two quantities, amount adsorbed and mass specific solid volume, are linked to each other through the location of hypersurface. The conceptual location of hyper-surface differentiates between common definitions of adsorption in the literature as absolute, excess, and net adsorption. For all three, the volume assigned to the solid side of the hyper-surface is necessary for rigorous thermodynamics. This is not a simple measurement issue to calculate the amount adsorbed from raw data in experiments. Rather, this volume appears explicitly in the grand potential, thus its value must be known to calculate grand potential from experimental data. In effect, the amount adsorbed measured with all three definitions is corrected through this volume in the calculation of grand potential. Thermodynamically, the volume assigned to the solid side of hyper-surface is a property of pure solid. This quantity with absolute and excess adsorption cannot be measured for microporous solids independent of adsorption without using any probe molecule. Thus, the resulting value is a pair-property. The thermodynamic framework built with such an approach is limited to the specifics of the approach itself, i.e., one cannot uniquely describe the adsorption of nitrogen in zeolite-A at 50 C. Rather using excess adsorption definition for example, adsorption of nitrogen in zeolite-A at 50 C is measured based on helium reference measured at 22 C (room temperature) at the limit of density approaching zero. Even when the correct reference state is cited with data, the thermodynamic equations must include the mass specific volume as shown for the first time in this work. The impact of ignoring mass specific volume of solid on mixture adsorption formulations is shown to be quite substantial for a typical system pertinent to one common application; nitrogen−oxygen-zeolite 5A system at room temperature and moderate pressures. From a practical standpoint, simplifying assumptions are abundant in engineering practice. Engineering is often a compromise between simplicity and accuracy. Interestingly in this case, the absolute and excess adsorption definitions do not really simplify, rather they actually make adsorption phenomena more complicated. Even if the volume term is mathematically ignored in calculations for simplification, the experiments are more complicated with absolute and excess adsorption definitions. For example, the void volume is measured (typically with helium) every time the solid is changed in a volumetric apparatus. In applications, the solid volume is accounted for by different porosities as interparticle and intraparticle porosities (which are often just estimates). Comparison of molecular simulations to experiment is also complicated with excess adsorption; extra simulations are performed with helium to mimic experiments.

adsorbed of 2.380 mol/kg, about 9% higher than the correct value of 2.188 mol/kg for oxygen. A simple, previously unnoticed omission in applying thermodynamics to microporous adsorption results in substantially lower pressure (−16%) and higher amount adsorbed (+9%) for oxygen to reach the same grand potential level, i.e., the same chemical potential of the microporous solid. The error in this calculation propagates through binary adsorption calculations where pure component standard states are at same chemical potential of the solid. The impact on binary (multicomponent) adsorption is further magnified because the impact is different for the two components. In this case, the impact on oxygen is −16% for pressure and +9% for amount adsorbed. For nitrogen, the impact is −6% for pressure and +4% for amount adsorbed. The overall impact of this erroneous omission is shown next for binary adsorption predictions using IAST with net adsorption. Binary Phase Equilibrium Predictions. Binary adsorption equilibrium has three degrees of freedom. The temperature, pressure, and gas composition are fixed in experiments, while adsorbed phase properties typically as partial amounts adsorbed are measured. Two of the three gas properties are held constant to examine the effect of the third on phase diagrams for adsorption. Most commonly, the effect of composition is examined followed by the effect of pressure. In common practice, phase diagrams for binary adsorption are shown as the total amount adsorbed and selectivity rather than the partial amounts adsorbed measured in experiments. The effect of pressure on the total amount adsorbed and selectivity is shown in Figure 6 by IAST formulations given in eqs 25, 26, 27, and 33 with the activity coefficients, γi, equal to unity. The curves are generated by using two forms of grand potential calculation: (1) the correct form by eq 46 marked as “net,” and (2) the erroneous form by eq 43 marked as “excess.” The bold arrows in the figure indicate how the omission of the −(νisPi/RT) term in calculating grand potential impacts the predicted results. Reduction in the total amount adsorbed is expected since the correction term has a negative sign, but the substantial increase in solid selectivity is not intuitive. The actual selectivity of net adsorbed phase is higher than excess adsorption because net adsorption accounts for the volume occupied by the solid displacing the gas phase. The increase in predicted selectivity by almost 15% mathematically originates from the differential impact of this term on the pure component standard state pressures necessary to reach the same solid chemical potential level. Finally, the effect of gas composition on the phase diagram is shown in Figure 7. Once again the total amount adsorbed is reduced and selectivity is substantially increased by correctly accounting for the solid volume in the grand potential calculation. These are quite substantial differences, particularly for selectivity. It is well-known that a 10% difference in selectivity can influence the choice of adsorption over other separation techniques in practice.53 The magnitude of error can be estimated for other systems under various conditions by the relative magnitude of the two terms in eq 16. In general, the difference becomes significant for adsorption of lighter compounds (where first term is small) and/or at higher pressures (where the second term is large). Additionally, there is no compelling reason for using an erroneous equation in thermodynamic formulations. 13069



Article

(14) Talu, O.; Zwiebel, I. Spreading Pressure Dependent Equation for Adsorbate Phase Activity Coefficients. React. Polym. 1987, 5, 81− 91. (15) Valenzuela, D. P.; Myers, A. L. Adsorption Equilibrium Data Handbook; Prentice-Hall: Englewood Cliffs, NJ, 1989. (16) Talu, O.; Li, J.; Myers, A. L. Activity Coefficients of Adsorbed Mixtures. Adsorption 1995, 1, 103−112. (17) Talu, O.; Li, J.; Kumar, R.; Mathias, P. M.; Moyer, J. D.; Schork, J. M. Measurement and Analysis of Oxygen/Nitrogen/5A-Zeolite Adsorption Equilibria for Air Separation. Gas Sep. Purif. 1996, 10 (3), 149−159. (18) Prauznitz, J. M.; Lichtenthaler, R. N.; de Azevedo, E. D. Molecular Thermodynamics of Fluid-Phase Equilibria; Prentice-Hall: New York, 1998. (19) Myers, A. L. Thermodynamics of Adsorption in Porous Materials. AIChE J. 2002, 48 (1), 145−160. (20) Richardson, Lewis F. The Problem of Contiguity: An Appendix to Statistic of Deadly Quarrels. In General Systems: Yearbook of the Society for the Advancement of General Systems Theory; University of Michigan: Ann Arbor, MI, 1961. (21) Talu, O.; Myers, A. L. Reference Potentials for Adsorption of Helium, Argon, Methane and Krypton in High-Silica Zeolites. Colloids Surf., A 2001, 187, 83−93. (22) Do, D. D.; Do, H. D.; Nicholson, D. Molecular Simulation of Excess Isotherm and Excess Enthalpy Change in Gas-Phase Adsorption. J. Phys. Chem. B 2009, 113, 1030−1040. (23) Bhatia, S. K.; Myers, A. L. Optimum Conditions for Adsorptive Storage. Langmuir 2006, 22, 1688−1700. (24) Polanyi, M. Trans. Faraday Soc. 1932, 28, 316. (25) Dubinin, M. M. Chem. Rev. 1960, 60, 235. (26) Dubinin, M. M.; Astakhov, V. A. Description of Adsorption Equilibria of Vapors on Zeolites Over Wide Ranges of Temperature and Pressure. Adv. Chem. 1971, 102, 69. (27) Rouquerol, F.; Rouquerol, J.; Sing, K. Adsorption by Powders and Porous Solids; Academic Press: London, 1999. (28) McBain, J. W.; Britton, G. T. The Nature of the Sorption by Charcoal of Gases and Vapors Under Great Pressure. J. Am. Chem. Soc. 1930, 52, 2198−2222. (29) Coolidge, A. S. Adsorption at High Pressures I. J. Am. Chem. Soc. 1934, 56, 554−561. (30) Coolidge, A. S.; Fornwalt, H. J. Adsorption at High Pressures. II. J. Am. Chem. Soc. 1934, 56, 561−568. (31) Neimark, A. V.; Ravikovitch, P. I. Calibration of Pore Volume in Adsorption Experiments and Theoretical Models. Langmuir 1997, 13, 5148−5160. (32) Sircar, S. Excess Properties and Thermodynamics of Multicomponent Gas Adsorption. J. Chem. Soc., Faraday Trans. I 1985, 81, 1527−1540. (33) Gumma, S.; Talu, O. Gibbs Dividing Surface and Helium Adsorption. Adsorpt. J. 2003, 9 (1), 17−28. (34) Fernbacher, J. M.; Wenzel, L. A. Adsorption Equilibria at HighPressures in Helium-Nitrogen-Activated Carbon System. I&EC Fundam. 1972, 11, 457−465. (35) Kaneko, K.; Setoyama, N.; Suzuki, T. Ultramicropore Characterization by He Adsorption. In Proceedings of the COPS III Symposium; Roquerol, J., Rodriguez-Resinoso, F., Sing, K. S. W., Unger, K. K., Eds.; Elsevier: Amsterdam, 1994; pp 593−602. (36) Maggs, F. A. P.; Schwabe, P. H.; Williams, J. H. Nature 1960, 186, 956−958. (37) Malbrunot, P.; Vidal, D.; Vermesse, J.; Chahine, R.; Bose, T. Adsorbent Helium Density Measurement and Its Effect on Adsorption Isotherms at High Pressure. Langmuir 1997, 13, 539−544. (38) Sircar, S. Measurement of Gibbsian Surface Excess. AIChE J. 2001, 47 (5), 1169−1176. (39) Sircar, S. In Proceedings of the FOA 7; Kaneko, K., Kanoh, H., Hanzawa, Y., Eds.; IK International: Chiba-City: Japan, 2001; pp 656− 663.

The apparent reason for this awkward situation in adsorption is only historical to the best of our knowledge. Gibbs definition of adsorption to quantify adsorption by invoking a hypersurface was initially applied and tested with flat surface such as at vapor/liquid interfaces. Later, these formulations for flat surfaces were translated to microporous solids without regard to the differences in the physical nature of the systems. These formulations can only be applied if the area of hyper-surface can be independently measured/fixed, which is not possible for microporous solids. Therefore, the commonly used “surfaceexcess” wording for microporous adsorption is somewhat misleading and confusing. The Gibbs definition can still be used to describe adsorption in micropores. The correct physical picture may be better relayed with the wording “volume-excess” Gibbs adsorption which is quantified by the net amount adsorbed. The net adsorption framework detailed in this manuscript simplifies and circumvents all complications in thermodynamic formulations. The rigorous formulations are simpler with mass specific volume being a pure solid property (defined as zero), and experiments are simpler without any need for extra measurements with helium. Furthermore, comparison of molecular simulation with experiment is straightforward, and process modeling by column dynamics is conceptually intuitive and simpler as will be shown in future communications.

■ ■ ■

AUTHOR INFORMATION

Notes

The authors declare no competing financial interest.

ACKNOWLEDGMENTS The author gratefully acknowledges stimulating discussions with Profs. Alan L. Myers and Sasidhar Gumma. REFERENCES

(1) Yang, R. T. Gas Separation by Adsorption Processes; Imperial College Press: London, 1997. (2) Tien, C. Adsorption Calculations and Modeling; ButterworthHeinemann: Woburn, MA, 1984. (3) Crittenden, B. D.; Thomas, W. J. Adsorption Technology and Design; Butterworth-Heinemann: Woburn, MA, 1998. (4) Noll, K. E.; Gounaris, V.; Hou, W. S. Adsorption Technology for Air and Water Pollution Control; Lewis: Chelsea, MI, 1992. (5) Talu, O. Needs, status, techniques and problems with binary gas adsorption experiments. Adv. Colloid Interface Sci. 1998, 76−77, 227− 269. (6) Gumma, S.; Talu, O. Net Adsorption: A Thermodynamic Framework for Supercritical Gas Adsorption and Storage in Porous Solids. Langmuir 2010, 26 (22), 17013−17023. (7) Van Ness, H. C. I&EC Fundam. 1969, 8 (3), 464−473. (8) Gibbs, J. W. The Collected Works of J. W. Gibbs; Longmans and Green: New York, 1928. (9) Myers, A. L.; Prausnitz, J. M. Thermodynamics of Mixed-Gas Adsorption. AIChE J. 1965, 11, 121+. (10) Costa, E.; Sotelo, J. L.; Callega, G.; Marron, C. Adsorption of Binary and Ternary Hydrocarbon Gas Mixtures on Activated Carbon Experimental Determination and Theoretical Prediction of the Ternary Equilibrium Data. AIChE J. 1981, 27, 5−12. (11) Suwanayuen, S.; Danner, R. P. Vacancy Solution Theory of Adsorption from Gas Mixtures. AIChE J. 1980, 26, 76−83. (12) Cochran, T. W.; Kabel, R. L.; Danner, R. P. Vacancy Solution Model of Adsorption Using Flory−Huggins Activity Coefficient Equations. AIChE J. 1985, 31, 268−277. (13) Talu, O.; Zwiebel, I. Multicomponent Adsorption Equilibria of Nonideal Mixtures. AIChE J. 1986, 22 (8), 1263−1276. 13070



Article

(40) Springer, C.; Major, C. J.; Kammermeyer, K. Low Pressure Adsorption of Helium on Microporous Solids. J. Chem. Eng. Data 1969, 14, 78−82. (41) Staudt, R.; Bohn, S.; Dreisbach, F.; Keller, J. U. In Proceedings of the COPS IV Symposium; McEnaney, B., Mays, T. J., Roquerol, J., Rodriguez-Resinoso, F., Sing, K. S. W., Unger, K. K., Eds.; Royal Society of Chemistry: Cambridge, UK, 1997; pp 261−266. (42) Suzuki, I.; Kakimato, K.; Oki, S. Volumetric Determination of Adsorption of Helium over Some Zeolites with a Temperature Compensated, Differential Tensimeter Having Symmetrical Design. Rev. Sci. Instrum. 1987, 58, 1226−1230. (43) Talu, O.; Kabel, R. L. Isosteric Heat of Adsorption and the Vacancy Solution Model. AIChE J. 1987, 33 (3), 510−514. (44) Myers, A. L.; Calles, J.; Calleja, G. Comparison of Molecular Simulation of Adsorption with Experiment. Adsorption 1997, 3, 107− 115. (45) Neimark, A. V.; Ravikovitch, P. I. Calibration of Pore Volume in Adsorption Experiments and Theoretical Models. Langmuir 1997, 13, 5148−5160. (46) Liu, J.; Culp, J. T.; Natesakhawat, S.; Bockrath, B. C.; Zande, B.; Sankar, S. G.; Garberoglio, G.; Johnson, J. K. Experimental and Theoretical Studies of Gas Adsorption in Cu3(BTC)2: An Effective Activation Procedure. J. Phys. Chem. C 2007, 111, 9305−9313. (47) Talu, O.; Myers, A. L. Prediction of Adsorption for Polar and Non-polar Gases on Silicalite by Molecular Simulation. Fundamentals of Adsorption 5; Levan, D. M., Ed.; Kluwer: Boston, 1996; pp 945−952. (48) Talu, O.; Myers, A. L. Molecular Simulation of Adsorption: Gibbs Dividing Surface and Comparison with Experiment. AIChE J. 2001, 47, 1160−1168. (49) Talu, O.; Myers, A. L. Force Constants for Adsorption of Helium and Argon in High-Silica Zeolites. Fundamentals of Adsorption 6; Meunier, F., Ed.; Elsevier: New York, 1998; pp 861−866. (50) Siperstein, F.; Myers, A. L.; Talu, O. Long Range Corrections for Computer Simulations of Adsorption. Mol. Phys. 2002, 100 (13), 2025−2030. (51) Heuchel, M.; Snurr, R. Q.; Buss, E. Adsorption of CH4−CF4 Mixtures in Silicalite: Simulation, Experiment and Theory. Langmuir 1997, 13, 6795−6804. (52) Smit, B. Simulating the Adsorption Isotherm of Methane, Ethane and Propane in the Zeolite Silicalite. J. Phys. Chem. 1995, 99, 5597−5603. (53) Mathias, P. M.; Kumar, R.; Moyer, J. D.; Schork, J. M.; Srinivasan, S. R.; Auvil, S. R.; Talu, O. Correlation of Multicomponent Gas Adsorption by the Dual-Site Langmuir Model. Application to Nitrogen/Oxygen Adsorption on 5A-Zeolite. Ind. Eng. Chem. Res. 1996, 35, 2477−2483. (54) Rouquerol, J.; et al. Recommendations for the Characterization of Porous Solids. Pure Appl. Chem. 1994, 66 (8), 1739−1758. (55) He, Y. F.; Yun, J. H.; Seaton, N. A. Adsorption Equilibrium of Binary Methane/Ethane Mixtures in BPL Activated Carbon: Isotherms and Calorimetric Heats of Adsorption. Langmuir 2004, 20 (16), 6668−6678. (56) Heymans, N.; Vaesen, S.; De Weireld, G. A Complete Procedure for Acidic Acid Gas Separation by Adsorption on MIL-53 (Al). Microporous Mesoporous Mater. 2012, 154 (SI), 93−99. (57) Talu, O. Measurement and Analysis of Mixture Adsorption Equilibrium in Porous Solids. Chem. Ing. Tech. 2011, 83 (1−2), 67−82. (58) Ridha, F. N.; Webley, P. A. Anomalous Henry’s Law Behavior of Nitrogen and Carbon Dioxide Adsorption in Alkali-Exchanged Chabazite Zeolites. Sep. Purif. Tech. 2009, 67 (3), 336−343. (59) Siperstein, F. R.; Myers, A. L. Mixed-Gas Adsorption. AIChE J. 2001, 47 (5), 1141−1159.

13071


Vapor Mixtures in Microporous Solids - The

Recommend Documents