Statistical Mechanics of Multilayer Sorption: 2. Systems Containing

ARTICLE pubs.acs.org/JPCC

Statistical Mechanics of Multilayer Sorption: 2. Systems Containing Multiple Solutes Cari S. Dutcher,*,† Xinlei Ge,† Anthony S. Wexler,†,‡ and Simon L. Clegg*,†,§ †

Air Quality Research Center, University of California at Davis, Davis, California 95616, United States Departments of Mechanical and Aerospace Engineering, Civil and Environmental Engineering, and Land, Air and Water Resources, University of California at Davis, Davis, California 95616, United States § School of Environmental Sciences, University of East Anglia, Norwich NR4 7TJ, United Kingdom ‡

ABSTRACT: Using statistical mechanics, the extension of an adsorption isotherm to include an arbitrary number of energetically unique adsorbed monolayers was established previously by Dutcher et al. (J. Phys. Chem. C 2011, 115, 1647416487). The main purpose of that work, although not its only application, was to represent the thermodynamic properties of solutions over very wide ranges of concentration. Here, the model is developed further to obtain expressions for the Gibbs energy, solvent and solute activities, and solute concentrations for mixtures containing arbitrary numbers of solutes. A remarkable result of this study is a novel statistical mechanical derivation of the ZdanovskiiStokesRobinson (ZSR) mixing rule, found in the limit of zero solutesolute interactions. The more general mixing rule derived here includes a modification of one of the energy parameters or a number of adsorption sites to represent the effects of solutesolute interactions on solution properties. The effects of this ternary solutesolute mixing term on the relationship between solvent activity and solute concentration are examined.

1. INTRODUCTION Stokes and Robinson1 first drew attention to the analogy between adsorption and the hydration of ions or molecules in aqueous solutions and successfully used the BrunauerEmmett Teller (BET)2 isotherm to represent water activities of pure aqueous solutions of several electrolytes as a function of concentration. For single solute solutions, equations have been derived for the Gibbs energy,3 water and solute activities,3,4 and thermal properties.5 Ally and Braunstein6 have used statistical mechanics to derive expressions for solvent and solute activities in a twosolute system, based upon the BET isotherm, and for the multiple solute case by extension. Clegg and Simonson7 added ternary (solutesolute) interaction terms to the BET isotherm for an arbitrary number of solutes and demonstrated their use. The application of BET adsorption to aqueous solutions implies that solutes, either molecules or neutral combinations of cations and anions in the case of electrolytes, at high concentration are surrounded by a monolayer of adsorbed water molecules. Beyond the monolayer, the water molecules exist in a multilayer that is energetically equivalent to the bulk solvent. Due in part to the limitation of using only a single adsorption monolayer, BET models are only accurate for solutions with a low adsorption density8 or, equivalently, a low water activity (aw, typically 1) monolayers simply act as an extended solvent field. However, the

Xj, 2 ¼ Xj, 3 1854

1 aw K j aw Kj

! ð34Þ

dx.doi.org/10.1021/jp2084154 |J. Phys. Chem. C 2012, 116, 1850–1864

The Journal of Physical Chemistry C

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Table 2. Special Cases of Equation 30 for Solute Activity (aA) and the Results of Other Studies bulk energy name

no. of solutes

no. of layers

other studies yielding aA (this work) rA NA XA, 1 rA rA NA rA NA XA, 1 rA rA NA rA NA XA, 1 rA rA NA rA rA NA XA, 1 xA rA NA rA rA NA XA, 1 xA rA NA

parameter value

BrunauerEmmettTeller (BET)2

1 (A)

2

1.0

GuggenheimAndersondeBoer

1 (A)

2

KA

Dutcher et al. (n-GAB)12

1 (A)

n

KA

Ally and Braunstein (2 solute BET)6

2 (A, B)

2

1.0 (both solutes)

Ally and Braunstein (multisolute BET),

arbitrary (A, B, ..., Z)

2

1.0 (all solutes)

(GAB)11,9,10

6

postulated

the same expression Abraham3 eq 20 Dutcher et al.12 eq 29 Dutcher et al.12 eq 47 Ally and Braunstein6 eq 25 Ally and Braunstein6 eq 38

Table 3. Molalities (mA, mB) and Solute Activities (aA, aB) in a System Containing Two Solutes, with Three Sorption Layersa quantity mA, mB

expression mA

b

mB ! ¼ 1 ð1 aw KB Þð1 aw KB ð1 CB, 1 Þ CB, 1 a2w KB2 ð1 CB, 2 ÞÞ Mw rB CA, 1 aw KB ðð1 aw KB Þ2 þ ð2 aw KB Þaw KB CB, 2 Þ

! þ ð1 aw KA Þð1 aw KA ð1 CA, 1 Þ CA, 1 a2w KA2 ð1 CA, 2 ÞÞ Mw rA CA, 1 aw KA ðð1 aw KA Þ2 þ ð2 aw KA Þaw KA CA, 2 Þ !rA 1 aw KA aA ¼ xA 1 aw KA ð1 CA, 1 Þ a2w KA2 CA, 1 ð1 CA, 2 Þ

aA

aB

aB ¼

xB

!rB 1 aw KB 1 aw KB ð1 CB, 1 Þ a2w KB2 CB, 1 ð1 CB, 2 Þ

a All equations are without the mixture parameters described in Section 3. b The denominators beneath each of the molalities in the mixture are equal to the molalities in the pure solutions of A and B at water activity aw.

Substituting eqs 3334 into eq 32, the molality of solute j is given by mj ¼

ð1 aw Kj Þð1 aw Kj ð1 Cj, 1 Þ Cj, 1 a2w Kj2 ð1 Cj, 2 ÞÞ Mw rj Cj, 1 aw Kj

Xj, 3 Nw Cj, 2 a2w Kj2

ð35Þ

Again using eqs 33 and 34, the total number of water molecules associated with solute j, written in terms of the number of water molecules in the multilayer of j, is given by Xj, 1 þ Xj, 2 þ Xj, 3 ¼ Xj, 3

ð1 aw Kj Þ2 þ ð2 aw Kj Þaw Kj Cj, 2 Cj, 2 a2w Kj2

!

ð36Þ Using eq 36 to multiply eq 35 by unity, we find mj ¼

Therefore, eq 37 can be rewritten as mj Xj, 1 þ Xj, 2 þ Xj, 3 ¼ moj Nw

ð39Þ

where Xj,13 are the amounts of water adsorbed on solute j in the mixture. For a solution containing two solutes, A and B, using eqs 8 and 38 we find that for the molalities of the solutes in both pure (single solute) aqueous solutions and mixtures at the same water activity mA mB þ o ¼1 ð40Þ moA mB This expression can be rewritten in terms of the amount of water in the mixture (Nw) and that associated with the same amounts of o,B solutes A and B in pure aqueous solutions (No,A w and Nw ) when at the same water activity Nwo, A þ Nwo, B ¼ Nw

ð1 aw Kj Þð1 aw Kj ð1 Cj, 1 Þ Cj, 1 a2w Kj2 ð1 Cj, 2 ÞÞ

ð41Þ

Mw rj Cj, 1 aw Kj ðð1 aw Kj Þ2 þ ð2 aw Kj Þaw Kj Cj, 2 Þ Xj, 1 þ Xj, 2 þ Xj, 3 ð37Þ Nw

As shown in the Appendix, using a similar procedure to that above and the equations for adsorption involving an arbitrary number of layers and solutes, eqs 40 and 41 readily generalize to

From our previous work,12 we know that the molality of solute j in a single solute solution, moj , with three sorption layers is given by

ð42Þ

moj ¼

ð1 aw Kj Þð1 aw Kj ð1 Cj, 1 Þ Cj, 1 a2w Kj2 ð1 Cj, 2 ÞÞ

Z

m

j ∑ o ¼ 1 m j¼A j

and

Mw rj Cj, 1 aw Kj ðð1 aw Kj Þ2 þ ð2 aw Kj Þaw Kj Cj, 2 Þ

Z

Nwo, j ¼ Nw ∑ j¼A

ð38Þ 1855

ð43Þ



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Table 4. Molalities (mj) and Solute Activities (aj) in a System Containing an Arbitrary Number of Solutes (j) and Sorption Layers (n)a quantity mj

expression m

∑ mojj

¼ 1

moj moj

aj aoj

1 aw Kj Mw rj aw Kj

¼

!

1

ð1 aw Kj Þ2

n2

n1

i Q1

∑ ðaiw Kji ð1 Cj, i Þ k ¼ 1 Cj, k Þ

i¼1 m Q

nQ 1

∑ ðmaw m 1 Kj m 1 k ¼ 1 Cj, k Þ þ ððn 1Þ ðn 2Þaw Kj Þaw n 2 Kj n 2 k ¼ 1 Cj, k m¼1

aj ¼ xj 3 aoj 0

1rj

B C 1 aw Kj C aoj ¼ B @ A iQ 1 n1 1 ∑ ðaiw Kji ð1 Cj, i Þ Cj, k Þ i¼1

k¼1

a

All equations are without the mixture parameters described in Section 3. For solutes j, each with nj sorption layers (where nj is less than the maximum n of all solutes present), Cj,i should be set equal to 1.0 for layers i g nj.

Note that the expression for moj for an arbitrary number of layers and solutes (eq 46 in our previous work12) is given in Table 4. Equations 42 and 43 are the widely used ZdanovskiiStokes Robinson (ZSR) relationship. Zdanovskii14 and later Stokes and Robinson13 independently discovered this mixing model for electrolytes and nonelectrolytes, respectively. Stokes and Robinson derived the equation by considering solutes which interact with the solvent by a series of solvation equilibria to form species which mix according to the ideal solution law. The analogy with the present model, especially when expressed in terms of adsorption equilibrium constants,12,15 is clear. The ZSR relationship is only one of a number mixing rules based upon semi-ideal solution theory16 and has been made more flexible by the addition of solutesolute interaction terms17 and by corrections for mixtures containing solutes of different charge types.7 Here, we have derived ZSR, for the first time, using statistical mechanics. The basis of the mixture model derived thus far is that the adsorption energies for all layers are the same for each solute in both the mixture and a binary solution (i.e., Kj = Kjo and Cj,i = Cj,io), and the number of adsorption sites on the solute in the mixture and binary solution are the same (i.e., rj = rjo). Clegg and Simonson7 incorporated solutesolute interactions in the Ally and Braunstein6 BET (i.e., two-layer) adsorption isotherm for mixtures by relaxing the above constraints. The resulting expressions for the adsorption site parameter and energy term, modified by the relative amounts of each solute present, were originally suggested by Sangster et al.18 for the simplified case of molten mixtures of electrolytes with a common anion. The same solutesolute interactions can be introduced into the present model by following the same method detailed above while including a concentration dependence in the definition of rj and Cj,1 found in eqs 2, 12, 14, A.2, and A.11. Here, for simplicity, we assume either Cj,1 is variable and that the number of adsorption sites on a solute is fixed (i.e., rj = rjo) in Sections 3.1.1 and 3.2.1 or that rj is variable and that the energy of adsorption is constrained by rj ln Cj,1 = rjo ln Cj,1o in Sections 3.1.2 and 3.2.2. 3.1.1. Mixture Effects on Cj,1. Assuming a fixed number of adsorption sites (rj = rjo), the Clegg and Simonson expression for each Cj,1 for a mixture containing two solutes A and B can be rewritten as CA, 1 ¼ CoA, 1 expðxB TA, B, 1 =2rA Þ CB, 1 ¼ CoB, 1 expðxA TB, A, 1 =2rB Þ

where T A,B,1 and TB,A,1 are empirical mixture parameters, and the interactions are symmetrical so that T B,A,1 = TA,B,1 . These parameters can be related to the Clegg and Simonson variables through, for example, T A,B,1 = T MNX and TB,A,1 = T NMX , when solute A is comprised of cation M and anion X and solute B is comprised of cation N and anion X. When eq 44 is applied to the two-layer mixing model (i.e., BET), the resulting relationship between solute concentration and water activity is mA ð1 aw KA Þð1 aw KA þ aw KA CoA, 1 expðxB TA, B, 1 =2rA ÞÞ Mw rA CoA, 1 expðxB TA, B, 1 =2rA Þaw KA þ

!

mB ð1 aw KB Þð1 aw KB þ aw KB CoB, 1 expðxA TB, A, 1 =2rB ÞÞ Mw rB CoB, 1 expðxA TB, A, 1 =2rB Þaw KB

!¼1

ð45Þ This can be rewritten as moA

mA mB þ o ¼1 þ ΔA mB þ ΔB

ð46Þ

where ΔA ¼

ð1 aw KA Þ2 ðexpðxB TA, B, 1 =2rA Þ 1Þ Mw rA CoA, 1 aw KA

ð47Þ

and ΔB is analogous to the equation above, but with the subscripts A and B transposed. In general, for a system containing any number of solutes j, Cj, 1 ¼ Coj, 1 expð

Z

ðxj Tj, j , 1 Þ=2rj Þ ∑ j ¼A 0

0

0

ð48Þ

where j0 6¼ j. The mixing model for an arbitrary number of layers and solutes with parameter Tj,j0 ,1 for interactions between solutes j and j0 is Z

m

j ¼1 ∑ o m þ Δj j¼A j

ð44Þ 1856

ð49Þ



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where Δj is given by ð1 aw Kj Þ2 ðexpð Δj ¼ 2

Mw rj aw Kj ðð1 aw Kj Þ

n2

∑

p¼1

ðpaw p 1 Kj p 1

p Q k¼1

Z

∑ ð xj Tj, j , 1 Þ=2rjÞ 1Þ 0

0

j0 ¼ A

Coj, k Þ

þ

ððn 1Þ ðn 2Þaw Kj Þaw n 2 Kj n 2

Z

mA 1 ð1 aw KA Þð1 aw KA þ aw KA ðCoA, 1 Þ1=ð1 þ ΓA Þ Þ @ A Mw rAo ð1 þ ΓA ÞðCoA, 1 Þ1=ð1 þ ΓA Þ aw KA mB 1¼1 þ0 ð1 aw KB Þð1 aw KB þ aw KB ðCoB, 1 Þ1=ð1 þ ΓB Þ Þ @ A Mw rBo ð1 þ ΓB ÞðCoB, 1 Þ1=ð1 þ ΓB Þ aw KB

!

Nj

∑ Nwo, j j¼A

ð51Þ

o, j

N j þ Δ j N w Mw

rB ¼

rBo ð1

ð55Þ where ΔΓj ¼

ð52Þ

þ ΓB Þ

Γj ¼

o

ð53Þ

CB, 1 ¼ ðCoB, 1 ÞrB =rB o

Mw rjo Coj, 1 aw Kj

∑j xj Sj, j =2rjo 0

ð57Þ

0

0

∑

where, similar to eq 50, ΔΓj is given by

ð1 aw Kj Þ2 ðCoj, 1 ðΓj =ð1 þ Γj ÞÞ 1Þ Mw rjo aw Kj ðð1 aw Kj Þ2

n2

∑

p¼1

ðpaw p 1 Kj p 1

p Q k¼1

Coj, k Þ

3.2. Solute Activity. For the case where each solute j in a multicomponent mixture contains three sorption layers, rearranging eqs 12 and 14 and assuming no mixture parameters, we find

ðXj, 1 Xj, 2 Þ þ Cj, 1 aw Kj Xj, 1 Cj, 1 aw Kj

ðXj, 1 Xj, 2 Þ ðXj, 1 Xj, 2 Þ þ Cj, 1 aw Kj Xj, 1

þ

ððn 1Þ ðn 2Þaw Kj Þaw n 2 Kj n 2

!rj ð61Þ

Expressions for Xj,1 and Xj,2, given in eqs 33 and34, combine with eq 61 to yield !rj 1 aw Kj aj ¼ xj ð62Þ 1 aw Kj ð1 Cj, i Þ a2w Kj2 Cj, 1 ð1 Cj, 2 Þ

nQ 1 k¼1

ð59Þ Coj, k Þ

For n possible sorption layers, as shown in the Appendix, the solute activity is given by 0 1r j B aj ¼ xj B @

ð60Þ

Substituting eq 60 into eq 30, we have aj ¼ xj

ð56Þ

where j0 6¼ j. Applying eqs 52, 53, and 57 an arbitrary number of layers and solutes, we find the mixing model including parameter Sj,j0 for interactions between solutes j and j0 is ! Z mj ð1 þ Γj Þ ¼1 ð58Þ ΔΓj þ moj j¼A

where ΓA = x/BSA,B/2roA, ΓB = x /ASB,A/2r oB, and SA,B and SB,A are empirical mixture parameters, and the interactions are symmetrical so that S B,A = S A,B. When eqs 52 and 53 are applied to the two-layer mixing model (i.e., BET), the relationship between solute concentration and water activity becomes

rj Nj ¼

ð1 aw Kj Þ2 ðCoj, 1 ðΓj =ð1 þ Γj ÞÞ 1Þ

In general, for a system containing any number of solutes and sorption layers, function Γj in eqs 52 and 53 is defined by:

CA, 1 ¼ ðCoA, 1 ÞrA =rA

ΔΓj ¼

ð54Þ

which can be rewritten as mA mB ð1 þ ΓA Þ þ ð1 þ ΓB Þ ¼1 ΔΓA þ moA ΔΓB þ moB

3.1.2. Mixture Effects on rj. Assuming TA,B,1 = 0 and allowing rj 6¼ rjo, the Clegg and Simonson expression for each rj and Cj,1 for a mixture containing two solutes A and B can be rewritten as rA ¼ rAo ð1 þ ΓA Þ

k¼1

ð50Þ Coj, k Þ

0

As with eq 43, this expression can be rewritten in terms of the amount of water in the mixture (Nw) and that associated with the same amounts of solutes in pure aqueous solutions (Nwo,A and Nwo,B), when at the same water activity Nw ¼

nQ 1

1 a w Kj n1

iQ 1

i¼1

k¼1

1 ∑ ðaiw Kji ð1 Cj, i Þ

Cj, k Þ

C C A

ð63Þ

Using the expression for the activity of solute j from our previous work for single solute solutions, ajo,12 the activity of solute j in a multicomponent mixture at the same water activity can be written as aj ¼ xj 3 aoj

ð64Þ

Equation 64 assumes no solutesolute mixing interactions. In the sections below, we examine the effects of mixture parameters Tj,j0 ,1 (3.2.1) and Sj,j0 (3.2.2) on solute activities, in the same manner as Sections 3.1.1 and 3.1.2. 1857



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3.2.1. Mixture Effects on Cj,1. As with water activity, mixture parameters Tj,j0 ,1 (with Sj,j0 = 0) can be included in the solute activity expression by substituting eq 48 into the definition of the energy parameter in eq A.11. Rederivation of solute activity, using the same procedure as shown in eqs 1024 and using the modified definition of the energy parameter from eq 48, yields aj ¼

xj

rj Nj Xj, 1 r j Nj

!rj

"

exp

ð∂Cm, 1 =∂Nj ÞXj, 1 Cm, 1

Z

∑

m¼A

!#

ð65Þ

eqs A.8, A.24, A.26, and A.28 from the Appendix can be used to express the number of water molecules in the first layer of solute j, Xj,1, as ð1 Xj, 1 ¼ Nj rj aw Kj Cj, 1

n1

iQ 1

∑ aiw 1 Kji 1 ð1 Cj, i Þ k ¼ 2 Cj, k Þ i¼2

ð1

n1

iQ 1

∑ ðaiw Kji ð1 Cj, i Þ k ¼ 1 Cj, kÞÞ

i¼1

ð66Þ Substituting eq 66 into eq 65 yields !rj 1 a w Kj aj ¼ xj λj " !# Z ð∂Cm, 1 =∂Nj Þrm Nm ð1 aw Km λm Þ exp Cm, 1 λm m¼A

∑

ð67Þ where λj ¼ 1

n1

iY 1

∑ ðaiw Kji ð1 Cj, i Þ k ¼ 1 Cj, k Þ i¼1

ð68Þ

The values of Cm,1 and Cj,1 in eqs 6568 are given by eq 48, and the partial derivative ∂Cm,1/∂Nj in eqs 65 and 67 is at constant Nw and Nj0 , where j 6¼ j0 . 3.2.2. Mixture Effects on rj. Likewise, mixture parameters Sj,j0 (with Tj,j0 ,1 = 0) can be included in the solute activity ! rj " !# Z ð∂Cm, 1 =∂Nj ÞXj, 1 rj Nj Xj, 1 aj ¼ xj exp rj Nj Cm, 1 m¼A Z Y rm Nm Xm, 1 Nm ð∂rm =∂Nj Þ ð69Þ rm Nm m¼A

∑

which, for practical applications, can be rewritten using eq 66 as !rj 1 a K w j aj ¼ xj λj " !# Z rm Nm ð∂Cm, 1 =∂Nj Þð1 aw Km λm Þ exp Cm, 1 λm m¼A Z Y 1 aw Km Nm ð∂rm =∂Nj Þ ð70Þ λm m¼A

3.3. Different Numbers of Sorption Layers (nj) for Each Solute. Even in the presence of solutesolute interaction param-

eters Tj,j0 ,1 and Sj,j0 , the relationship between the water activity and solute concentrations in a mixture has no dependence on the number of sorbed layers on each solute (eqs 42, 49, and 58). The same is true of the relationship between solute activities (eqs 64, 67, and 70). Therefore, in fits of thermodynamic data for pure aqueous solutions of each solute, the numbers of adsorbed layers n can be varied as necessary (i.e, can be different for each solute). Thus, in mjo and ajo (eqs 46 and 48 in our previous work12), the number of adsorbed layers n, which were previously assumed to be the same for all solutes, can be replaced by nj, where nj may be different for each solute j. This is also equivalent to letting all Cj,i = 1.0 for all i g nj. Finally, the limiting case of all Cj,i = 1.0, that is Ej,1 = Ej,2 = ... = EL, reduces to Raoult’s law when rj = 1.0.

4. EXAMPLES In Section 3, the model for solution mixtures developed here was shown to be equivalent to the ZSR mixing rule, as long as the parameters rj and Cj,i are the same in both mixtures and single solute solutions. The ZSR mixing rule has been shown over many years to provide satisfactory predictions of mixed solution thermodynamic properties for a large number of systems. However, this is not true when solutesolute interactions are significant and, depending on the nature of the mixture, an additional term or terms in the equations can improve their predictive accuracy. Here we demonstrate the effects of the additional terms in Tj,j0 ,1 and Sj,j0 , described in Section 3, and compare the results with those obtained by Clegg et al.19 for a similar parameter Bj,j0 which they used to extend the ZSR mixing rule (see their eq 7). The first example is a mixture containing two solutes A and B, which, for simplicity, are both assumed to obey Raoult’s law in binary (single solute) aqueous solutions. In the present model this is equivalent to all the energy parameters in different layers, Coj,i, and the number of sorption sites, roj , being equal to unity, as shown by Dutcher et al.12 Figure 3a shows the influence of mixture parameter TA,B,1 on water activity for a 1:1 mol ratio of A and B. A positive TA,B,1 yields a negative deviation in aw, while negative TA,B,1 corresponds to a positive deviation relative to values calculated without the mixture term. The effects are not symmetrical about the 1:1 line. Solute activities (as), calculated using eq 63, are shown in Figure 3b and are analogous. The influence of the mixture parameter SA,B on water activity is shown in Figure 3c. However, unlike the mixture parameter TA,B,1, the parameter SA,B has a practical lower limit to the values it can take. As shown in eq 56, the term ΔΓj is zero when the energy parameters Coj,i are equal to 1.0, thus the term 1 + Γj in eq 55 cannot be zero or negative which imposes a limit of SA,B > 4 for the 1:1 mixture. The effects of SA,B on calculated solute activities in eq 70 are shown in Figure 3d. Clegg et al.19 introduced a mixture term into the ZSR relationship, which can be written for the present case as mA mB mA mB ðAo þ BA, B 3 aw Þ ¼ 1 o þ o þ mA mB mA þ mB A, B

∑

For eqs 69 and 70, λj is given by eq 68; rm and Cm,1 are given by eqs 52 and 53, and the partial derivatives ∂rm/∂Nj and ∂Cm,1/∂Nj are at constant Nw and Nj0 , where j 6¼ j0 .

ð71Þ

where AoA,B and BA,B are parameters for the interaction of solutes A and B. The term in AoA,B when assuming BA,B = 0 yields unphysical estimates of the water activities of highly concentrated solutions, thus in this comparison we assume AoA,B is zero and only consider the mixture parameter BA,B. Figure 3e shows the influence of BA,B on the relationship between water activity and total solute mole fraction. 1858



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Table 5. Fitted Parameters of the Three-Layer Model for Aqueous Solutions at 298.15 Ka three-layer model range fitted solute (A) malonic acid

glutaric acid

(m: mol kg1)

parameter

value

uncertainty

0.913

0.042

0.099 23.14

0.535 124.7

m: 0.5209.5

rA

aw: 0.990.21

CA,1 CA,2 KA

1.0027

0.0001

m: 0.298.1

rA

0.599

0.012

aw: 0.9950.371

CA,1

0.257

0.424

CA,2

0.997

1.612

KA

1.0027

0.0001

a

The large uncertainties associated with CA,1 and CA,2 above suggest that a simpler, two-layer model, may also have fitted the data for both solutes well.

Figure 3. Calculated water activities (aw) and solute activities (as) of an aqueous mixture containing two solutes (in a 1:1 mol ratio) which individually mix with water according to Raoult’s law, plotted against the total mole fraction of the solutes (xt, calculated as (nA + nB)/(nA + nB + nH2O), where prefix n indicates the number of moles). The solute activities are relative to a reference state of hypothetical pure liquid compounds. (a) Water activity (aw). Solid line, mixing without solute solute interactions; dotted lines, results with values of the mixture parameter TA,B,1 (eqs 49 and 50) as indicated on the plot with label TA,B. (b) Solute activity (as, the activity of A or B). Solid and dotted lines, as in (a) (eqs 67 and 68). (c) Water activity (aw). Solid line, as in (a); dotted and dashed lines, results for positive and negative values of the mixture parameter SA,B as indicated on the plot (eqs 5759). (d) Solute activity (as). Solid, dotted, and dashed lines, as in (c) (eq 70). (e) Water activity (aw). Solid line, as in (a); dotted and dashed lines, results for positive and negative values of the mixture parameter BA,B (indicated on the plot) in the extended ZSR model of Clegg et al.19 using their eq 7. (f) Solute activity (as). Solid, dotted, and dashed lines, as in (e) (eqs 7 and 911 of Clegg et al.19 with conversion to mole fraction based activity).

The value of BA,B has a practical lower limit that is determined by the requirement that the water content of the solution must be nonnegative for all water activities. This is most easily seen in eq 7 of Clegg et al.19 In the present example, where the individual solutions of A and B obey Raoult’s law, it is found that BA,B > Mw/(x/Ax/B) for the general case and BA,B > 4Mw for the 1:1 mixture of solutes A and B shown in Figure 3e. The influence of the mixture parameter BA,B on solute activity was calculated using eq 9 of Clegg et al.,19 including a transformation to a mole fraction basis and to a reference state of the pure

Figure 4. Measured and predicted water activities (aw) of aqueous malonic acidglutaric acid (1:1 mol ratio) solutions at ∼298.15 K, plotted against the total mole fraction of the solutes (xt, calculated as (nMal + nGlu)/(nMal + nGlu + nH2O), where prefix n indicates the number of moles). The total molality of solutes (mt, calculated as mMal + mGlu) is shown on the upper axis. Key: circle, data of Choi and Chan;21 solid line, three-layer model from eq 49 with mixture parameter (TA,B,1 = 2.1154); dashed line, three-layer model from eq 49 with no mixture parameter (TA,B,1 = 0).

liquid solute, and is shown in Figure 3f. Positive values of BA,B induce a positive deviation in the estimated solute activity relative to Raoult’s law. However, the effect of BA,B differs from that of TA,B,1 and SA,B in that BA,B has a larger influence at high concentrations on the solute activity which does not converge to a value of 0.5 in the absence of water. In fact, BA,B can yield unrealistically high values of solute activity (i.e., as > 1). Recognizing that the solute activity must be less than or equal to unity imposes a limit of BA,B < 4Mw ln(2) for the 1:1 mixture considered here. The combined limitations for both water and solute activity mean that 4Mw < BA,B < 4Mw ln(2). In comparison, values of TA,B,1 are not restricted, and 4 < SA,B < ∞ in the adsorption isotherm model. The second example is the application of the model to water activities of aqueous solutions of malonic acid and glutaric acid in a 1:1 mol ratio. Water activities of the two binary solutions have been measured by Peng et al.20 up to supersaturated concentrations (low water activities) and were fitted using the procedure 1859


The Journal of Physical Chemistry C described by Dutcher et al.12 to obtain binary parameters for the three-layer model which are listed in Table 5. Water activities of 1:1 mixtures of these two organic acids were measured by Choi and Chan21 down to aw values of about 0.2 and are compared with predictions of our model in Figure 4. Using only the binary solution parameters noted above, the model tends to overpredict the water activities, especially at high concentrations; adding the mixture parameter TA,B,1 = 2.1154 (determined by fitting the data shown in the figure) results in much better agreement with the measurements. Although data for the solution mixture for a range of ratios of the two acids would be preferable for both fitting and validation, previous experience suggests that the improvement in the representation of data shown in Figure 4 is likely to extend to other relative compositions.

5. CONCLUSIONS We have derived expressions for molality and solvent and solute activities in multisolute solutions by extending the singlesolute, multilayer sorption model of Dutcher et al.12 We showed that many of the same conclusions regarding the thermodynamic predictions seen with the single-solute solutions are true also for mixtures: (1) The solvent activity is equal to the ratio of the total amount of water in the multilayer to the sum of the amount of the water in the multilayer and the outermost monolayer of each solute multiplied by the bulk energy parameter for the respective solute. (2) Conversely, the activity of the solute is related to the ratio of the number of surface sorption sites occupied by the first sorbed monolayer to the total number of available sites. This ratio is multiplied by the dry mole fraction of the given solute. (3) In solutions where all the monolayer sorption energies are equal to that for the multilayer (i.e., unity) and both Kj and the number of sites per molecule (rj) are also equal to 1.0, Raoult’s law is obtained. The widely used ZdanovskiiStokesRobinson (ZSR)13 empirical mixing equation relating solute molalities to solvent activity has been derived using the statistical mechanical formalism for multilayer adsorption. The ZSR mixing rule is equivalent to BET and other adsorption isotherms when the model parameters (i.e., the number of adsorption sites and energy of solvent adsorption for a given solute molecule) are the same in single-solute solutions and mixtures. Thus, there is no specific interaction between the solutes, only between the solutes and the solvent. The fact that measured water activities of many solution mixtures approximately conform to the ZSR rule supports this simplified view of solution behavior. However, as noted by Stokes and Robinson,13 deviations from the rule imply departures of from the semi-ideal behavior that it represents, so we have included solute solute interaction terms in the model to improve predictions for such mixtures. These terms have been shown by Clegg and Simonson7 to be important for the accurate calculation of water activities and salt solubilities in ternary mixtures. In this work, the use of the Tj,j0 ,1 parameter significantly improved the representation of water activities of mixed aqueous malonic and glutaric acids.

ARTICLE

A.1. Statistical Mechanical Derivation for an Arbitrary Number of Layers and Solutes. Consider a system containing

Nw water molecules and Nj molecules of solute j, where j = A, B, ..., Z. The number of distinguishable ways in which the solute molecules can mix is given by ΩA, B, :::Z ¼ ð

Z

Z Y

∑ NjÞ! = j ¼ AðNj!Þ j¼A

ðA:1Þ

For each solute j, there are rj sorption sites, rjNj total sorption sites, and n sorption layers surrounding the solute, consisting of (n 1) monolayers and an outermost multilayer. In the first monolayer, Xj,1 of the rjNj sites are occupied by sorbed water molecules, with only one water molecule allowed per site, leaving (rjNj Xj,1) sites unoccupied. The number of distinguishable ways in which the Xj,1 water molecules can sorb to the available sites on solute j is given by ðrj Nj Þ! ðrj Nj Xj, 1 Þ!Xj, 1 !

Ωj, 1 ¼

ðA:2Þ

For subsequent monolayers, the Xj,i water molecules occupying the ith monolayer are sorbed on top of the Xj,i1 water molecules of the (i 1)th monolayer, with only one water molecule allowed per site per layer, resulting in (Xj,i1 Xj,i) unoccupied sites in the ith monolayer. The number of distinguishable ways in which Xj,i water molecules can sorb to available sites is given by Xj, i1 ! ðXj, i1 Xj, i Þ!Xj, i !

Ωj, i ¼

ðA:3Þ

where 2 e i e n1. Those water molecules not involved in the (n 1) adsorbed monolayers are assigned to the nth (outermost) layer, the multilayer. The Xj,n water molecules occupying this layer are arranged on top of Xj,n1 water molecules of the (n 1)th monolayer, there being no limit to the number of water molecules per site in the multilayer. In this way, each of the Xj,n water molecules in the multilayer has a combinatoric association with the solute j. The number of distinguishable ways that Xj,n water molecules can sorb to the Xj,n1 available sites is given by ðXj, n1 þ Xj, n 1Þ! ðXj, n1 þ Xj, n Þ! ≈ ðXj, n1 1Þ!Xj, n ! Xj, n1 !Xj, n !

Ωj, n ¼

ðA:4Þ

where both Xj,n . 1 and Xj,n1 . 1. The total number of distinguishable arrangements, Ω, is given by Ω ¼ ΩA, B, :::Z

Z Y n Y j¼A i¼1

Ωj, i

ðA:5Þ

Combining eqs A.1A.5, we obtain

’ APPENDIX Here we address the general case of a system containing an arbitrary number solutes and adsorption layers. In Section A.l, we develop the statistical mechanical derivation in the same manner as Section 2. In Section A.2 we derive the practical equations for the relationship between solute concentration and water activity (A.2.1), and solute activity and water activity (A.2.2), for an arbitrary number of adsorption layers in the same manner as Section 3.

Z

∑ Nj Þ! j¼A

ð Ω¼

Z Q j¼A

ðNj !Þ

Z Y j¼A

1 ðrj Nj Þ! ðXj, n þ Xj, n1 Þ! nY Xj, i1 ! ðrj Nj Xj, 1 Þ!Xj, 1 ! Xj, n1 !Xj, n ! i ¼ 2 ðXj, i1 Xj, i Þ!Xj, i !

!

ðA:6Þ 1860


ARTICLE

ðXj, 1 Þ2 ðrj Nj Xj, 1 ÞðXj, 2 Þ

Using Stirling’s approximation, ln(N!) ≈ N ln(N) N, the statistical thermodynamic entropy, S = k ln Ω, can be generalized for n g 3 as 0 1 1 00 Z ! rj C Nk C B B C C BB B Z B rj Nj BN lnBBk ¼ A C C S¼k B j BB C C N r N X @ A A @ @ j j j j, 1 j¼A rj Nj Xj, 1 þ ðXj, 1 Þln Xj, 1 Xj, 2

þ ðXj, n1 Þln þ ðXj, n Þln

Nw ¼

E¼

∑

E¼

ðXj, n1 Þ2 !!

ðA:7Þ

Z

n

ðA:8Þ

Z

n

∑ ∑ Xj, i εj, i j¼A i¼1

ðA:9Þ

Z n1

Z

Xj, n dj ∑ ∑ Xj, i ðεj, i þ djÞ j ∑ j¼A i¼1 ¼A

ðA:10Þ

The total entropy, given by eq A.7, is maximized under the energy constraint from eq A.10 using the Lagrangian method of undetermined multipliers from eq 11 as described in the main text. The equations apply to all sorbed monolayers of all solutes, noting that eq A.8 yields ∂Xj,n/∂Xj,i = 1 in the same way as the two solute system. The results are expressed in terms of energy parameters Cj,i for sorption for each monolayer i of solute j ðXj, n þ Xj, n1 ÞðXj, 1 Xj, 2 Þ εj, 1 ¼ exp Cj, 1 ðA:11Þ kT ðXj, n Þðrj Nj Xj, 1 Þ

ðrj Nj Þ! ðXj, 2 þ Xj, 1 Þ! ðrj Nj Xj, 1 Þ!Xj, 1 ! ðXj, 2 Þ!ðXj, 1 Þ!

ðXj, n þ Xj, n1 ÞðXj, i Xj, iþ1 Þ εj, i ¼ exp kT ðXj, n ÞðXj, i1 Xj, i Þ

j¼A

Nj !Þ

j¼A

Cj, i where 2 e i e n 2

ð

Z Q

Xj, n þ Xj, n1 Xj, n

∑

!!

Following the same assumptions for ε/j,i described for eq 10 in the text, the general form for E given in eq A.9 becomes

! Z

∑ Nj Þ! YZ j¼A ð

Xj, i1 Xj, i þ ðXj, i Þln Xj, i Xj, iþ1 i¼2 ! ðXj, n2 Xj, n1 ÞðXj, n þ Xj, n1 Þ

The maximization of entropy in the system is constrained by the energy of the system. The total change of energy (E) due to the sorption of water molecules from a free liquid state is defined as

ðXj, n1 Þ2 εj, n1 ¼ exp Cj, n1 ðXj, n2 Xj, n1 ÞðXj, n Þ kT

2

j¼A

n2

∑ ∑ Xj, i j¼A i¼1

0

∑

ðrj Nj Þ!ðXj, 1 Þ! Nj !Þ Z Q

ð

!

where k is Boltzmann’s constant. Entropy expressions for n = 1 and n = 2 are shown in Table A1. Each water molecule is statistically affiliated with one, and only one, solute and is either present in one of the (n 1) monolayers or in the outer multilayer of that solute. That is

∑

Z

∑ Z

Z

∑ Nj Þ! ðrj Nj þ Xj, 1 Þ! j¼A ð

1

∑

∑

00 1 1 1 Z !rj C ! !C N k B B C B BB C C C Z B rj Nj C þ X ln ðrj Nj Xj, 1 ÞðXj, 2 þ Xj, 1 Þ þ X ln Xj, 2 þ Xj, 1 C BN lnBBk ¼ A C k j, 1 j, 2 C C B j BB C 2 N r N X X @ @ A A @ A Þ ðX j j j j, 1 j, 2 j¼A j, 1

not applicable !rj !CC Nk C B B B C B BB C rj Nj rj Nj þ Xj, 1 C CC BN lnBBk ¼ A C þ Xj, 1 ln k C B j BB CC N r N X X @ A @ @ A A j j j j, 1 j, 1 j¼A

Cj,1

11 S

1

00 0 Ω n

Table A1. Expressions for the Numbers of Distinguishable Arrangements (Ω), Entropy (S), and the Energy of Adsorption Parameters for the First Monolayer (Cj,1) for the Special Cases n = 1 and n = 2


1861

ðA:12Þ ðA:13Þ



ARTICLE

It should be noted that eq A.12 did not arise for the n = 3 case described in Section 2 since in the case of n > 3 there are additional monolayermonolayermonolayer interactions not present in the n = 3 case. The expression for Cj,n1 for the n = 2 case (i.e., Cj,1) is shown in Table A1. Next, also using eq 11, the entropy for the multilayer of each solute is maximized in the same way as for the n = 3 case. The total numbers of particles (N) of all species are fixed, and the numbers of sorbed water molecules (X) are also fixed for monolayers 1 to (n 1) of each solute. It should be noted that eq A.8 yields ∑j0 ∂Xj0 ,n/∂Xj,n = 1, where j0 6¼ j, for this multi-solute system. The result, similar to eq 16, is a relationship between the energy parameters Kj for sorption in the multilayer each solute Xj, n 1 Kj Xj, n þ Xj, n1

!

1 Xk, n ¼ Kk Xk, n1 þ Xk, n

! ðA:14Þ

where Kj exp(dj/kT) (with an analogous relationship for Kk), and j and k are any pair of solutes A, B, ..., Z. This ratio for a specific solute, j, can be expressed in terms of all the solutes in the system by multiplying eq A.14 by unity (i.e., (1 + ∑k0 Xk0 ,n/Xk,n)/ (1 + ∑k0 Xk0 ,n/Xk,n), where k0 6¼ k) and applying the equalities of eq A.14 for k = A, B, ..., Z Xj, n 1 Kj Xj, n þ Xj, n1

! ¼

Z

∑ Xk, n k¼A

! =

Z

∑ KkðXk, n1 þ Xk, n Þ k¼A ðA:15Þ

Substituting eqs A.11A.15 into eq A.7 yields the most probable distribution, Ω*, of the number of adsorbed water molecules in all layers and for all solutes 0 00 1 11 Z ! r ðN Þ j CC k C B B BB B C CC Z B rj Nj BN lnBBk ¼ A C CC ln Ω ¼ j B BB C CC @@ Nj A rj Nj Xj, 1 AA j¼A @

∑

∑

þ

Z n1

∑ ∑

j¼A i¼1 Z

Xj, i

εj, i kT

n

∑ ∑ Xj, i lnððXj, n1 þ Xj, nÞ=Xj, nÞ j¼A i¼1

ðA:16Þ

derivatives of the energy equalities from eqs A.11A.13 to each ∂Xj,i/∂Nw yields Z

aw ¼

G=kT ¼

∑

j¼A

! Nj ðlnðxj Þ

þ Nw lnð

Z

þ rj lnð1 Xj, 1 =ðrj Nj ÞÞÞ

The expression for the activity of solute j is similarly obtained by partial differentiation of the Gibbs energy expression, eq A.17, with respect to Nj !rj rj Nj Xj, 1 aj ¼ xj ðA:20Þ rj Nj The following two subsections contain brief discussions of the statistical mechanical derivation and expressions for two special cases, n = 1, 2 (Section A.1.1) and n = 2, Kj = 1, and j = A, B (Section A.1.2). A.1.1. Special Case (n = 1, 2). When n g 3, there are three types of adsorption: solutemonolayer, monolayermonolayer, and monolayermultilayer. For the limiting cases of n = 1 (i.e., pure multilayer adsorption) or n = 2 (e.g., BET or GAB adsorption), there are only the following: solutemultilayer (n = 1) or solutemonolayer and monolayermultilayer (n = 2). The statistical mechanical derivation for these two special cases yields the same equations for water and solute activities A.18A.20 as the general case. However, the governing equations for the number of distinguishable arrangements (Ω), the entropy (S), and the definition of the energy parameter (Cj,1), if applicable, differ. The expressions are given in Table A1. A.1.2. Special Case (n = 2, Kj = 1.0, and Two Solutes). The definition of Cj,1 for this case (i.e., two-solute BET) can be shown to be equivalent to that of Ally and Braunstein.6 Using eq A.19 and n = 2, Kj = 1.0, the energy parameter Cj,1 in Table A1 can be rewritten as Cj, 1 ¼

ðA:17Þ

ðXj, 1 Þ2 ðXj, 1 Þð1 aw Þ ¼ ðrj Nj Xj, 1 ÞðXj, 2 Þ ðrj Nj Xj, 1 Þaw

ðA:21Þ

Using eq A.8 and j = A, B, the water activity in eq A.18 can be expressed as aw ¼

Kj ðXj, n1 þ Xj, n ÞÞ ∑ Xj, n= j ∑ ¼A

∑ KjðXj, n1 þ Xj, nÞ

ðA:18Þ

The water activity of the system can also be written in terms of the amounts of water adsorbed in the multilayer and outermost monolayer of any individual solute j using eq A.15 ! ! Xj, n 1 aw ¼ ðA:19Þ Kj Xj, n1 þ Xj, n

Z

j¼A

Z

j¼A

The Gibbs energy is related to the entropy and enthalpy of the system via eq 19. Using eqs A.10, A.15, and A.16, the Gibbs free energy for a system with an arbitrary number of solutes and adsorption layers is Z

∑ Xj, n j¼A

Nw XA, 1 XB, 1 Nw

ðA:22Þ

Substituting eq A.22 into eq A.21 yields Cj, 1 ¼

x/j

where is defined as the dry mole fraction of solute j, so that x/j = Nj/∑Zk=ANk.7 Differentiating eq A.17 with respect to the total water content of the system, Nw, and applying the partial

ðXj, 1 Þ2 ðXj, 1 ÞðXA, 1 þ XB, 1 Þ ¼ ðrj Nj Xj, 1 ÞðXj, 2 Þ ðrj Nj Xj, 1 ÞðNw XA, 1 XB, 1 Þ

ðA:23Þ 1862



ARTICLE

The right-hand side of eq A.23 is the expression for Cj,1 given by Ally and Braunstein6 (their eqs 10 and 11 for j = A and B, respectively). A.2. Practical Equations for Arbitrary Numbers of Layers and Solutes. To apply the model, the amounts of solutes (Nj) and solvent (Nw) and the activities (aj and aw) need to be related to Xj,i and the model parameters rj, Cj,i, and Kj. Expressions for water and solute activities are derived below for the case of an arbitrary number of solutes and sorption layers (n > 3). A.2.1. Water Activity and Solute Concentration. Combining eqs A.11A.13 and A.19, the numbers of water molecules in 00

0n 2

∑

aw h Kjh

h Q

1

the monolayers of solute j can be written in terms of the number of water molecules in the multilayer, Xj,n 0 Xj, i

0n 2

∑

aw h Kjh

h Q

1

1

Cj, k C B B C k¼1 h¼i B B C þ 1 C ¼ Xj, n ð1 aw Kj ÞBð1 aw Kj ÞB C C n 1 Q aw Kj A @ @ A aw n Kjn Cj, k k¼1

ðA:24Þ Substituting eq A.24 into eqs A.11 and A.19, the molality of solute j (defined as mj = Nj/(MwNw)) is given by

1

0

0n 2

1

h Q

∑

11

Cj, k C aw h Kjh Cj, k C BB B C B B CC k¼1 k¼1 1 C h¼1 h¼2 B B B C B B C þ 1 CC ð1 aw Kj ÞBBð1 aw Kj ÞB a K Þ ð1 a K Þ þ ð1 þ C C C B B C CC j, 1 w j w j nQ 1 nQ 1 a w Kj A a w Kj A A @@ @ A @ @ A aw n Kjn Cj, h aw n Kjn Cj, h h¼1

mj ¼

h¼1

Mw rj Cj, 1 aw Kj

Xj, n Nw

ðA:25Þ Note that for the n = 2 case, the expression for Cj,1 from Table A1 should be used instead of eq A.11. Using eq A.24, the total number of water molecules associated with solute j, written in terms of the number of water molecules in the multilayer of j, is given by 0

n

∑

m¼1

Xj, m ¼

0

0

0n 2

1

h Q a w Kj h Cj, k C Bn 1 B B B k ¼ 1 Bð1 a K ÞBð1 a K ÞBh ¼ m C Xj, n B B B C w j B w j B nQ 1 @m ¼ 1 @ @ @ A a w n Kj n Cj, k k¼1

∑

∑

11

h

þ

1

C CC 1 CC C þ 1 C CC aw Kj AA A

ðA:26Þ

Using eq A.26 to multiply eq A.25 by unity, we find

mj ¼

1 aw K j M w rj a w K j

!

0 B B B @

1 ð1 aw Kj Þ2

n2

n1

10

m Q

Xj, m C CB C m¼1 CB B C C@ Nw A A

iQ 1 ðaiw Kji ð1 Cj, i Þ Cj, k Þ k¼1 i¼1

∑

nQ 1

∑ ðmaw m 1 Kj m 1 k ¼ 1 Cj, k Þ þ ððn 1Þ ðn 2Þaw Kj Þaw n 2 Kjn 2 i ¼ 1 Cj, i m¼1

n

∑

1

ðA:27Þ By substituting the expression for the molality of solute j in a single solute solution, moj , with n sorption layers from our previous work,12 eq A.27 can be rewritten as n mj Xj, i Þ=Nw o ¼ ð mj i¼1

∑

ðA:28Þ

where Xj,i are the amounts of water adsorbed on solute j in the mixture. For a solution containing an arbitrary number of solutes, using eqs A.8 and A.28 we find that for the molalities of the solutes in both pure (single solute) aqueous solutions and mixtures at the same water activity Z

m

j ∑ o ¼ 1 m j¼A j

ðA:29Þ

(r j N j ), we find rj Nj ¼

ðXj, 1 Xj, 2 Þ þ Cj aw Kj Xj, 1 Cj, 1 aw Kj

ðA:30Þ

Substituting the total number of j adsorption sites from eq A.30 into the expression for solute activity in eq A.20, we have ! rj ðXj, 1 Xj, 2 Þ ðA:31Þ aj ¼ xj ðXj, 1 Xj, 2 Þ þ Cj, 1 aw Kj Xj, 1 Expressions for X j,1 and X j,2 obtained from A.24 combine with eq A.31 to yield a general expression for solute activity 0 1r j

A.2.2. Solute Activity. Substituting the expression for water activity, eq A.19, into the expression for the energy parameter of the first monolayer, Cj,1 , in eq A.11 and then rearranging eq A.11 for the total number of j adsorption sites

B aj ¼ xj B @

1863

1 a w Kj n1

iQ 1

i¼1

k¼1

1 ∑ ðaiw Kji ð1 Cj, i Þ

Cj, k Þ

C C A

ðA:32Þ



ARTICLE

Equation A.32 also results from substitution of the expression for the number of adsorbed water molecules in the first monolayer of solute j, eq 66, into the statistical mechanical expression for solute activity, eq A.20.

’ AUTHOR INFORMATION Corresponding Authors

*E-mail: [email protected] and [email protected].

’ ACKNOWLEDGMENT The authors gratefully acknowledge support from the Electric Power Research Institute, the National Oceanic and Atmospheric Administration (grant NA07OAR4310191), the Natural Environment Research Council of the U.K. (grant NE/ E002641/1), and the Department of Energy (grant DE-FG0208ER64530). ’ REFERENCES (1) Stokes, R. H.; Robinson, R. A. J. Am. Chem. Soc. 1948, 70, 1870–1878. (2) Brunauer, S.; Emmett, P. H.; Teller, E. J. Am. Chem. Soc. 1938, 60, 309–319. (3) Abraham, M. J. Chim. Phys. Phys.-Chim. Biol. 1981, 78, 57–59. (4) Ally, M. R.; Braunstein, J. Fluid Phase Equilib. 1996, 120, 131–141. (5) Abraham, M.; Abraham, M.-C. Thermodynamic and Transport Properties of Bridging Electrolyte-Water Systems. In Modern Aspects of Electrochemistry; Conway, B. E., White, R. E., Eds.; Springer: US, 2002; Vol. 35, pp 197294. (6) Ally, M. R.; Braunstein, J. J. Chem. Thermodyn. 1998, 30, 49–58. (7) Clegg, S. L.; Simonson, J. M. J. Chem. Thermodyn. 2001, 33, 1457–1472. (8) Haworth, A. Adv. Colloid Interface Sci. 1990, 32, 43–78. (9) Anderson, R. B. J. Am. Chem. Soc. 1946, 68, 686–691. (10) de Boer, J. H. The Dynamical Character of Adsorption; Clarendon Press: Oxford, 1968. (11) Guggenheim, E. A. Applications of Statistical Mechanics; Clarendon Press: Oxford, 1966. (12) Dutcher, C. S.; Ge, X.; Wexler, A. S.; Clegg, S. L. J. Phys. Chem. C 2011, 115, 16474–16487. (13) Stokes, R. H.; Robinson, R. A. J. Phys. Chem. 1966, 70, 2126–2131. (14) Zdanovskii, A. B. Tr. Solyanoi Lab. Akad. Nauk SSSR 1936. (15) Nikitas, P. J. Phys. Chem. 1996, 100, 15247–15254. (16) Hu, Y. F.; Fan, S. S.; Liang, D. Q. J. Phys. Chem. A 2006, 110, 4276–4284. (17) Clegg, S. L.; Seinfeld, J. H. J. Phys. Chem. A 2004, 108, 1008–1017. (18) Sangster, J. M.; Abraham, M. C.; Abraham, M. Can. J. Chem. 1978, 56, 348–351. (19) Clegg, S. L.; Seinfeld, J. H.; Edney, E. O. J. Aerosol Sci 2003, 34, 667–690. (20) Peng, C.; Chan, M. N.; Chan, C. K. Environ. Sci. Technol. 2001, 35, 4495–4501. (21) Choi, M. Y.; Chan, C. K. J. Phys. Chem. A 2002, 106, 4566–4572. (22) Hill, T. L. J. Chem. Phys. 1946, 14, 263–267.

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Statistical Mechanics of Multilayer Sorption: 2. Systems Containing

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