Statistical Mechanics of Multilayer Sorption: Extension of the Brunauer

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Statistical Mechanics of Multilayer Sorption: Extension of the BrunauerEmmettTeller (BET) and GuggenheimAndersonde Boer (GAB) Adsorption Isotherms 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 School of Environmental Sciences, University of East Anglia, Norwich NR4 7TJ, United Kingdom § 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 ‡

ABSTRACT: Adsorption isotherms have important practical applications, including in solution chemistry where they have been used to model solvent and solute activities in liquid mixtures at extreme concentrations, such as pure melts. The conventional BrunauerEmmettTeller (BET) and Guggenheim Andersonde Boer (GAB) models are less successful for dilute solutions (high water activities), for which most data are available. In this work we extend these models to include arbitrary numbers of additional adsorbed monolayers and, using statistical mechanics, derive expressions for the Gibbs energies of solutions and for solute and solvent activities. We demonstrate consistency with results that can be obtained by other methods, and show how a particular simple case is equivalent to Raoult’s law. Test applications of the equations to water activity data for aqueous solutions over very wide ranges of concentration show that the addition of a single extra adsorbed layer to the fitted adsorption model greatly increases its accuracy, especially for low concentrations.

1. INTRODUCTION Multilayer adsorption isotherm models describe a wide range of sorption phenomena, from adsorption of natural gas by charcoal1 to sorption of metals by algae in wastewater treatment.2 First developed by Langmuir,3 multilayer adsorption is the adsorption of molecules in a lattice formation to sites on a substrate. The building block of multilayer adsorption is monolayer adsorption: the adsorption of molecules on identical, independent adsorption sites, where each site can only be occupied by a single molecule. Langmuir developed multilayer adsorption by applying the monolayer adsorption to a finite or infinite stack of monolayers, where the adsorbed molecules of the previous layer act as adsorption sites for the next layer. Langmuir’s classic picture of adsorption is the backbone of multilayer adsorption isotherm modeling which is used in a wide range of scientific fields, including food science,48 environmental chemistry,913 catalytic chemistry,1416 polymer and protein science,17,18 and solution chemistry.19,20 The most widely used adsorption model is the Brunauer EmmettTeller (BET) isotherm.21 Starting with Langmuir’s treatment of multilayer adsorption, the BET model can be derived by assigning an energy of adsorption, EA,1, to the molecules that adsorb directly to the surface of A (i.e., the first monolayer). For a liquid sorbate, the energy of adsorption of the molecules adsorbed in all other monolayers, EA,i, is assumed to be equal to the heat of liquefaction (condensation) EL of the pure sorbate. That is, molecules sorbed in layers 2 to ∞ are the same, energetically, as if they had simply condensed to the bulk liquid. r 2011 American Chemical Society

Figure 1a shows schematically the BET isotherm, where an infinite number of monolayers with energies EA,1 (layer 1) and EL (layers 2 to ∞) adsorb onto some fraction of the rA surface sites. The energy parameters in the model, CA,i, are proportional to the exponent of EA,i EL; thus, an energy parameter of 1.0 in layers 2 to ∞ represents EA,2 = EA,3 = ... = EA,∞ = EL. With only two parameters, rA and CA,1, the BET model accurately predicts sorption behavior for many systems, especially in systems with a low adsorbate to sorbent ratio. For example, in solution chemistry and food science, where water is treated as the adsorbate and a solute is the sorbent, the BET model successfully reproduces concentrations in solutions at water activities (aw, equivalent to the equilibrium relative humidity) of up to about 0.4. Stokes and Robinson20 first noted the parallel between multilayer sorption and the formation of hydration shells around ions in aqueous solutions due to intermolecular forces.20,22 Immediately surrounding the solute entity (ion or molecule) is a layer of chemisorbed water molecules. As distance from the solute increases, the arrangement of the water molecules goes from ordered (chemisorbed) to semiordered, to disordered (the bulk solvent). Figure 2a demonstrates how the BET model (top) applies to a solution (bottom), and shows three solute species surrounded by a Received: April 26, 2011 Revised: June 28, 2011 Published: August 01, 2011 16474

dx.doi.org/10.1021/jp203879d | J. Phys. Chem. C 2011, 115, 16474–16487

The Journal of Physical Chemistry C

Figure 1. Schematic of four adsorption isotherm models: (a) Brunauer EmmettTeller (BET),21 (b) GuggenheimAndersonde Boer (GAB),2325 (c) that of Timmermann,26 and (d) eq 45 of this work. Each model assumes a surface with rA adsorption sites on solute or substrate A, and each layer i contains XA,i water molecules. Each monolayer is sorbed onto the layer below it. The energy parameters KA and/or CA,i that apply in each isotherm are shown on the right of each panel for all layers. They are related to the exponent of the difference between the energy of adsorption to the layer and the bulk energy of liquefaction by eqs 7, 16, and 23. An energy parameter CA,i, or product KACA,i, of 1.0 corresponds to an energy of adsorption to the layer that is no different from that of the unbound or bulk sorbate. In this work, CA,1 is equivalent to “c” in the BET,21 GAB,2325 and Timmermann26 models, KA is equivalent to “k” in GAB and “f” in the Timmermann model, and n is equivalent to “h + 1” in the Timmermann model.

chemisorbed monolayer of water molecules. Beyond the monolayer, the water molecules exist in a multilayer that is energetically equivalent to the bulk. There is no intermediate region, analogous to outer shell(s) of weakly bound solvent, in the BET model. Modifications of the BET model by adding extra adsorbed layers yield an extension to the range of relative vapor pressures (which can be related to aw in solutions) over which the model can be applied.2329 These modifications can be thought of as representing an intermediate region of more weakly bound water molecules, with one2326 or more2729 additional energies of adsorption for the new layers. Although they result in more model parameters, this does mean that the model can be used to represent the properties of a wider range of solution concentrations (i.e., to higher aw) than before. The most commonly used of these modified models, especially in food science, is the GuggenheimAndersonde Boer (GAB)2325 model, which is typically valid for solutions aw of up to about 0.7. The GAB model adds a single additional energy of sorption dA (equal to EA,i EL). This can be seen as accounting for an extended or incomplete hydration shell surrounding the solute molecule, or perhaps the fact that in concentrated solutions the small number of solvent molecules relative to those of the solute mean that even the solvent not bound in the monolayer is affected by the solute. Sorption by the GAB model is illustrated in Figure 1b, where monolayers with energy parameters of adsorption KACA,1 (layer 1) and KA (layers 2 to ∞, where KA e 1.0) are adsorbed on some fraction of the rA surface sorption sites, and where the energy parameter of the bulk or multilayer region (KA) is proportional to the exponent of

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dA. The GAB isotherm is a three-parameter model, where the meanings of the BET parameters, CA,1, and rA, are retained and an additional parameter, KA, is added. When KA is equal to unity, the GAB model reduces to the BET model. Timmermann26 combines the concepts of the BET and GAB models to create a sorption isotherm with three distinct energetic regions. This is shown schematically in Figure 1c, where monolayers with energy parameters KACA,1 (layer 1), KA (layers 2 to n 1), and 1.0 (layers n to ∞), are adsorbed on a surface with rA sites. The isotherm of Timmermann is a four-parameter model, where the meanings of the BET and GAB parameters, rA, CA,1, and KA, are retained and an additional parameter, n, is added. As n f ∞, the model of Timmermann reduces to the GAB equation. Likewise, when n = 2 and KA = 1.0, the model reduces to the BET equation. An important limitation, however, is that KA, which in this model relates to the sorption energy of the intermediate region of hydration and not the bulk region, is constrained to be 1.0 to avoid a mathematical singularity. This constraint means that predictions of the model are bounded by those of the GAB and BET equations. For example, Figure 2 of Timmerman26 shows an example in which the predicted isotherms for n = 5, 10, 20, and 50, plotted as a function of sorbate activity, lie between the BET and GAB isotherms for KACA,1 = 25 and KA = 0.8. Because the GAB model is able to represent the fraction of surface coverage to higher relative sorbate vapor pressures (or higher water activities in the solution case) than BET, greater numbers of distinct energies for individual monolayers have the potential to yield further benefits. A few authors have attempted this, but unsatisfactory isotherms resulted.2730 The standard statistical mechanical methodology23,31 employed in these studies27,28 has also been used to derive the BET and GAB models, and others that capture other possible effects such as lateral interactions.32 However, these methods have not yet been used to obtain a closed-form (bounded), unrestricted (arbitrary number of layers) adsorption isotherm for the general case where the sorption energy is different for each layer. Ally and Braunstein33 derived the BET isotherm, and expressions for the Gibbs energy, water activity, and solute activity for a two-solute system, using a statistical mechanical approach that combines the Lagrangian multiplier method34 and a BET-specific combinatorial treatment by Hill35 for the number of distinguishable arrangements of molecules outside the first monolayer. Combinatorially dividing the BET isotherm into “monolayer adsorption” (layer 1) and “multilayer adsorption” (layers 2 to ∞), Hill’s approach has been used to derive the BET expressions for water activity (eq 14 of Hill35) and used by Abraham36 to obtain solute activity (eq 20 of Abraham36) in solutions containing single solutes. In this work, we follow the method of Ally and Braunstein33 to derive the GAB isotherm and, for the first time, the GAB expressions for Gibbs energy, water activity, and solute activity. Second, we modify the BET and GAB models to develop a new isotherm that includes distinct energies of sorption of the solvent for an arbitrary number of sorbed layers. The new model is shown schematically in Figure 1d, where an infinite number of monolayers with energy parameters KACA,i (layers 1 to n 1) and KA (layers n to ∞) are adsorbed to the rA available surface sites. Derivations of expressions for the Gibbs energy, water activity, solute concentration, and solute activity are given. The application of the model to represent the relationship between water activity and concentration for several solutes over extended ranges of water activity is demonstrated. Figure 2b shows how the model (top) applies to the solution (bottom) for the case where n = 4. 16475

dx.doi.org/10.1021/jp203879d |J. Phys. Chem. C 2011, 115, 16474–16487


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Figure 2. Schematic of two adsorption isotherm models for systems containing three solute molecules: (a) a two-layer model (BET21 or GAB2325) and (b) an n-layer model (with n = 4). The top sections illustrate conventional lattice adsorption and the bottom ones illustrates the corresponding solution case. In the solution, the three solute entities (numbered circles) are surrounded by hydrated layers (rings) of water molecules (small circles).

Table 1. Symbols and Quantities Used in This Work, Compared with Those of Hill33,35,36 and of Anderson24 a source

parameter

this study

ref 35

ref 33

ref 36

ref 24

total water molecules

Nw

A

H

no

n/a

total solute A molecules

NA

n/a

A

ns

n/a

sites available per solute A

rA

n/a

r

r

n/a

total A sites

rANA

B

rA

nsr

n/a

water bound on first

XA,1

X

XA

n

n/a

monolayer of A water bound on ith

XA,i

n/a

n/a

n/a

n/a

water bound in multilayer of A

XA,n

n/a

n/a

n/a

n/a

energy of first monolayer

EA,1

ε1

UA

E

E1

EA,i

n/a

n/a

n/a

n/a

EL + d EL

εL εL

U1 U1

EL EL

EL + d EL

monolayer of A

adsorption energy of ith monolayer

substrate by recognizing that θ = (rAMwmA)1, where θ is the total fractional surface coverage of the solid substrate by adsorbed gas molecules, mA is the molality of the solute (sorbent) A, and Mw (kg mol1) is the molecular weight of water (the adsorbate).

2. STATISTICAL MECHANICAL DERIVATION Consider a system of Nw water molecules, NA solute molecules, rA sorption sites per solute molecule, rANA total adsorption sites, and n sorption layers surrounding the solute, which consists of n 1 monolayers and an nth layer, the outermost multilayer or bulk solvent, as shown in Figure 1. Following the procedures outlined by Ally and Braunstein,33 we derive the Gibbs free energy of sorption by extending their work to n layers with unique energies, where n = 1, 2, 3, ..., ∞. In the first monolayer, some of the rANA sites are occupied by XA,1 sorbed water molecules, with only one water molecule allowed per site, leaving rANA XA,1 unoccupied sites. The number of distinguishable ways in which the XA,1 water molecules can adsorb to the rANA available sites is given by Ω1 ¼

adsorption energy of multilayer adsorption heat of liquefaction (water)

Some characters (e.g., “A” and subscripted “1”) have different meanings, depending on the author. “n/a” means not applicable.

a

Tables 1 and 2 list the model variable names used in this study together with their relationships to those used in other studies. For clarity, we use our variable names in all cases, even when discussing other models. We specifically tailor our discussion and derivations to solutions, with water molecules considered to be sorbed to solutes such as electrolytes or neutral (uncharged) molecules. However, the model can easily be generalized to gas adsorption on a solid

ðrA NA Þ! ðrA NA XA, 1 Þ!ðXA, 1 Þ!

ð1Þ

This equation is similar to eq 2 in Ally and Braunstein,33 but here is for a single solute only. For subsequent monolayers, the XA,i water molecules occupying each ith monolayer are sorbed on top of the XA,i1 water molecules of the (i 1)th monolayer, with a maximum of one water molecule allowed per site per layer, leaving XA,i1 XA,i unoccupied sites in any layer i. The number of distinguishable ways in which the XA,i water molecules in layer i can sorb to the available sites is given by Ωi ¼ 16476

ðXA, i1 Þ! ðXA, i1 XA, i Þ!ðXA, i Þ!

ð2Þ



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Table 2. Symbols and Quantities Used in the Adsorption Isotherms Developed in Different Studiesa source this study

a

ref 21

ref 35

ref 33

ref 36

ref 24

ref 26

ref 27b

energy variable, first monolayer

KACA,1

c

c

cA

c

kc

fc

d1

energy variable, ith monolayer

CA,i

n/a

n/a

n/a

n/a

n/a

f

di

number of monolayers with distinct energies

n1

1

1

1

1

1

h

∞

energy variable, multilayer

KA

1

1

1

1

k

1

n/a

“n/a” means not applicable. b The isotherm in ref 27 is unclosed, meaning that there is an infinite geometric series.

where i = 2, 3, ..., n 1. Those water molecules not sorbed in the n 1 monolayers are assigned to the outermost or nth layer, the multilayer. Equation 2 does not apply to the BET or GAB sorption isotherm models, since those models assume only one monolayer and the multilayer (i.e., n = 2). The XA,n water molecules occupying the multilayer are sorbed on top of XA,n1 water molecules from the (n 1)th monolayer, there being no limit to the number of water molecules per site. The number of distinguishable ways that XA,n water molecules can sorb to XA,n1 available sites is given by ðXA, n1 þ XA, n 1Þ! ðXA, n1 þ XA, n Þ! ≈ ðXA, n1 1Þ!ðXA, n Þ! ðXA, n1 Þ!ðXA, n Þ!

Ωn ¼

ð3Þ

This equation is similar to eq 3 in Ally and Braunstein.33 An important aspect of this derivation is the assumption that all of the Nw water molecules, even those in the multilayer, are statistically associated with the solute. Each water molecule is either present in the n 1 monolayers or in the outer multilayer. That is Nw ¼

n

∑ XA, i

ð4Þ

i¼1

Therefore, eq 3 can be rewritten as ðNw Ωn ¼ ðNw

n1

∑

i¼1

n2

∑

i¼1

n Y

E¼

E¼

ð5Þ

XA, i Þ!ðXA, n1 Þ!

Ωi

E¼

ð6Þ

The maximization of entropy in the system, S, which is equal to k ln Ω, where k is Boltzmann’s constant, 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 n

∑ XA, i εA, i i¼1

ð7Þ

where εA,i* = EA,i* EL. In the multilayer adsorption of gas onto a solid substrate, EL is the energy of condensation from the gas to the liquid state and EA,i* is the energy of sorption of the gas in the ith layer. Translating to multilayer sorption in solutions, EL is the energy of the free water in the bulk and EA,i* is the energy of the bound water. In the first layer, the water molecule sorbs directly onto solute A, while in subsequent layers, the molecule sorbs onto bound water molecules. Prior sorption isotherm

∑

i¼1

XA, i εA, i XA, n dA

ð8Þ

n1

∑

i¼1

XA, i ðεA, i þ dA Þ XA, n dA

ð9Þ

Using eq 4, the change in energy can be written in its final form as

i¼1

E¼

n1

where the first term is the sum over all the monolayers and the second term is the energy change associated with the multilayer. Finally, similarly to Anderson,24 we incorporate the additional energy of the multilayer adsorption, dA, into the terms for monolayer sorption energy, EA,i*, so that EA,i* = EA,i + dA and εA,i* = εA,i + dA. Hence

XA, i Þ!

and the total number of distinguishable arrangements is given by Ω¼

derivations (e.g., Ally and Braunstein33) assumed that, after the first layer, all εA,i* = 0. However, here we allow distinct sorption energies for all the monolayers, εA,i*, where i corresponds to layers 1 to (n 1). This more closely resembles hydration of the solute in which water molecules at greater distances from the ion or molecule are more weakly bound and eventually merge with the bulk. In the derivation by Anderson,24 the sorption energy of the multilayer, EA,n*, differs from that of liquefaction, EL, by an amount dA for each solute A; thus in our system the multilayer energy EA,n* is equal to EL + dA. With the assumptions of distinct monolayer sorption energies, εA,i*, and incorporating dA, the energy change becomes

n1

∑

i¼1

XA, i εA, i Nw dA

ð10Þ

Using the number of distinguishable arrangements from eq 6 and the energy change due to sorption from eq 10, in the following sections we derive (i) the GAB isotherm, Gibbs free energy, water activity, and solute activity and (ii) a new unrestricted, closed-form isotherm that includes distinct sorption energies of the solvent on n layers. 2.1. Two-Layer Model (n = 2). In this case, water sorbs in either a single monolayer or the multilayer. There are no intermediate monolayers, so the number of distinguishable arrangements from eq 6 is Ω¼

2 Y

Ωi

i¼1

¼ 16477

ðrA NA Þ! ðNw Þ! ðrA NA XA, 1 Þ!ðXA, 1 Þ! ðNw XA, 1 Þ!ðXA, 1 Þ!

ð11Þ



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Using Stirling’s approximation, ln(N!) ≈ N ln(N) N, the statistical thermodynamic entropy is given by S ¼ k ln Ω

!

rA NA ¼ krA NA ln rA NA XA, 1 ðrA NA XA, 1 ÞðNw XA, 1 Þ þ kXA, 1 ln ðXA, 1 Þ2 ! Nw þ kNw ln Nw XA, 1

!

ð12Þ

The energy change due to adsorption from eq 10 becomes E ¼ XA, 1 εA, 1 Nw dA

ð13Þ

The BET adsorption isotherm model was obtained using statistical mechanics of the two-layer system with dA = 0 (i.e., no change in energy for sorption to the multilayer).33,35,36 Here, following the same statistical mechanical procedures as Ally and Braunstein,33 applying eq 4 and using a nonzero dA parameter, we obtain the GAB equation. First, the Lagrangian undetermined multiplier method is used to maximize the entropy of the system given in eq 12, under the energy constraint from eq 13: ð∂ ln Ω=∂XA, i ÞNA , Nw , XA, j , T ð1=kTÞð∂E=∂XA, i ÞNw , XA, j ¼ 0 ð14Þ where i = 1, 2, 3, ..., n 1, j 6¼ i, j 6¼ n, T is temperature, and (kT)1 is the Lagrangian multiplier for the energy constraint.37 Solving eq 14 yields ðXA, 1 Þ2 ¼ expðεA, 1 =kTÞ ¼ CA, 1 ðrA NA XA, 1 ÞðNw XA, 1 Þ

ð15Þ

where CA,i is defined as the energy parameter for monolayer sorption in layer i: CA, i expðεA, i =kTÞ

ð16Þ

By substituting eq 15 into eq 12, the Lagrangian multiplier method is used to determine the most probable distribution, Ω*, of XA,1 water molecules, at temperature T: ! r N A A ln Ω ¼ rA NA ln rA NA XA, 1 ! Nw εA, 1 þ Nw ln XA, 1 Nw XA, 1 kT

Using eqs 13 and 17 in eq 18, the Gibbs energy for a two-layer system is given by rA NA XA, 1 G=kT ¼ Nw dA =kT þ rA NA ln rA NA Nw XA, 1 þ Nw ln ð19Þ Nw The activity of component i (solute or solvent) in the system is found by differentiating the Gibbs energy with respect to the total amount of the component, Ni: ∂G=kT ¼ ln ai ð20Þ ∂Ni Nj , T where j 6¼ i. Substituting eq 19 into eq 20 and differentiating with respect to total water yields the water activity:38 ∂G=kT Nw XA, 1 ¼ ln aw ¼ dA =kT þ ln ∂Nw NA , T Nw ð21Þ 1 Nw XA, 1 aw ¼ KA Nw where KA is the energy parameter for the multilayer KA expðdA =kTÞ

In eq 24, the water activity is proportional to the ratio of the number of water molecules in the multilayer to the total water molecules in the multilayer and the monolayer. For the two-layer case, when KA is equal to unity (i.e., the BET model), this is equivalent to the suggested definition of Stokes and Robinson20,33 that aw is equal to the fraction of “unbound” water (i.e., water not present in a monolayer). We will return to this in section 2.2, where additional layers are explored. Using the expression for water activity from eq 24 and for the energy parameter CA,1 from eq 15, the GAB expression is obtained:2325 NA ð1 aw KA Þð1 aw KA þ CA, 1 aw KA Þ ¼ rA CA, 1 aw KA Nw

ð18Þ

ð25Þ

It should be noted that when KA is equal to 1.0 in eq 25 or, equivalently, dA is 0 in eq 13, the normal BET expression20,21 is recovered. Continuing with the GAB expression from eq 25, and with the definition of the molality (mA) of solute A ð26Þ

where Mw is the molar mass of water (in kg mol1), solute molality is given as a function of water activity by

The Gibbs energy is related to the entropy and enthalpy of the system via G=kT ≈ E=kT ln Ω

ð23Þ

Alternatively, using eq 4, the water activity can be written as ! 1 XA, 2 aw ¼ ð24Þ KA XA, 1 þ XA, 2

mA ¼ NA =ðMw Nw Þ ð17Þ

ð22Þ

mA ¼

ð1 aw KA Þð1 aw KA þ CA, 1 aw KA Þ Mw rA CA, 1 aw KA

ð27Þ

The expression for the activity of solute A is found by differentiating the Gibbs energy with respect to the number of 16478



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0

moles of A: ∂G=kT rA NA XA, 1 ¼ ln aA ¼ rA ln ∂NA rA NA

ð28Þ

0

rA NA XA, 1 rA NA

rA

BNw þ kNw lnB B @ Nw

ð29Þ

Combining the expression for solute activity from eq 29 with the expression for solute molality from eq 26 and the expression for the energy parameter CA,1 from eq 15 gives the solute activity as a function of water activity: !rA 1 a w KA ð30Þ aA ¼ 1 aw KA þ CA, 1 aw KA Equations 27 and 30 can be shown to satisfy the GibbsDuhem relationship for this system. We note that eqs 27 and 30 can be combined in such a way to eliminate the CA,1 parameter. Thus, for the BET isotherm (i.e., when KA is equal to 1.0), the relationship between solute activity, water activity, and solute molality can be given in terms of a single equation: 1 aw ¼ mA Mw rA ð1 aA 1=rA Þ

ðNw ðNw

ð31Þ

ðNw ðNw

GAB related equations which involve additional adsorbed monolayers, and which more closely resemble a partial hydration of the solute extending beyond the first layer. Each additional monolayer has its own distinct energy of adsorption, εA,i. Using eqs 1, 2, 5, and 6, the number of distinguishable arrangements of the system is Ω¼

ðrA NA Þ! ðrA NA XA, 1 Þ!ðXA, 1 Þ!

i¼2

ðXA, i1 Þ! ðXA, i1 XA, i Þ!ðXA, i Þ!

∑

ðNw XA, i Þ! i¼1 n1 XA, i Þ!ðXA, n1 Þ! ðNw

ð32Þ

∑

i¼1

The expression for entropy is obtained using eq 32 and Stirling’s approximation: ! rA NA S ¼ k ln Ω ¼ krA NA ln rA NA XA, 1 ! rA NA XA, 1 þ kXA, 1 ln XA, 1 !! n1 XA, i1 XA, i þk XA, i ln XA, i i¼2 0 0 111 0 n1 ! Nw XA, i CCC B B n2B XA, i i¼1 BX lnB CCC B þk B A, i B CCC B n2 @ XA, i XA, iþ1 @ AAA i¼1 @ Nw XA, i

∑ ∑

∑

∑

i¼1 n1

∑

i¼1

XA, i C C C A XA, i

ð33Þ

n2

∑

XA, i ÞðXA, 1 XA, 2 Þ

∑

XA, i ÞðrA NA XA, 1 Þ

i¼1 n1 i¼1 n2

∑

i¼1 n1

XA, i ÞðXA, i XA, iþ1 Þ

∑ XA, i ÞðXA, i1 XA, iÞ i¼1

εA, 1 ¼ CA, 1 ¼ exp kT

ð34Þ

εA, i ¼ exp ¼ CA, i kT

ð35Þ

ðXA, n1 Þ2 ðNw

n1

∑

i¼1

XA, i ÞðXA, n2 XA, n1 Þ

εA, n1 ¼ exp ¼ CA, n1 kT

!!

n2

1

n2

Next, the procedure from section 2.1 is used to derive an extended BET (dA = 0) and extended GAB (dA ¼ 6 0) expression for sorption to n 1 monolayers. The Lagrangian multiplier method is once again employed to maximize entropy from eq 33 under the energy constraint from eq 10 for each monolayer. The results can be generalized in terms of energy parameters for the first layer (i = 1) in eq 34, intermediate monolayers (2 e i e n 2) in eq 35, and the outermost monolayer (i = n 1) in eq 36:

2.2. Extended Model (n > 2). Here we derive further BET and

nY 1

∑

XA, i C BN w i¼1 C B þ kXA, n1 lnB C XA, n1 A @

and aA ¼

1

n1

ð36Þ

Combining eqs 3436 and eq 33 yields the most probable configuration, Ωi*, of the water molecules sorbed on the solute (XA,i) in all monolayers i, at temperature T: ! rA NA ln Ω ¼ rA NA ln rA NA XA, 1 1 0 n2 BN w þ Nw lnB B @ Nw

∑

i¼1 n1

∑

i¼1

XA, i C C C A XA, i

n1

∑

i¼1

XA, i

εA, i kT

ð37Þ Using eqs 13, 18, and 37, the Gibbs energy is given by rA NA XA, 1 G=kT ¼ Nw dA =kT þ rA NA ln rA NA 1 0 n1 BN w þ Nw lnB B @ Nw

∑

i¼1

16479

∑

i¼1 n2

∑

i¼1

XA, i C C C A XA, i

ð38Þ



ARTICLE

Substituting eq 38 into eq 20 and differentiating with respect to total water content yields the water activity:38 1 0 n1 BN w ∂G=kT dA þ lnB ¼ ln aw ¼ B ∂Nw kT @ Nw 0 Nw 1B B B KA @ Nw

aw ¼

∑ ∑

i¼1

∑

i¼1

XA, i C C C A XA, i

ð39Þ

1

n1 i¼1 n2

∑

i¼1 n2

XA,n1 and XA,n, and on the energy parameter for the multilayer, KA. Thus KAaw is the ratio of the water in the multilayer to the total accessible water, which we define as that in the outermost monolayer plus that in the multilayer. This is consistent with the result for the two-layer problem, that KAaw is the ratio of the water in the multilayer to the total water content (which in that case is water adsorbed in the single monolayer plus that in the multilayer). At equilibrium, the numbers of water molecules in the bulk (the multilayer) and the outermost monolayer are influenced by the distribution of the other water molecules in the system. This can be shown by rearranging the expression for CA,1 from eq 34, substituting the water activity equation from eq 40, and applying eq 4:

XA, i C C C A XA, i

ð40Þ

NA XA, 1 ð1 þ CA, 1 aw KA Þ XA, 2 ¼ n1 Nw rA CA, 1 aw KA ð XA, m þ XA, n Þ m¼1

ð41Þ

In order to express eq 42 purely in terms of the model parameters (rA, CA,i, and KA) and aw, we determine XA,i as a function of CA,i, aw, and KA using the expressions for the energy parameters CA,i from eqs 35 and 36, aw in eq 40, and extensive algebraic manipulation. For n layers, we find

Equation 41 shows that the water activity depends only the numbers of water molecules in the outermost monolayer and multilayer, 0 XA, m

0n 2

∑

ð42Þ

∑

Using eq 4, the water activity can be written as ! 1 XA, n aw ¼ KA XA, n1 þ XA, n

i

i

1

i Q

1

a K C B B C B B i ¼ m w A j ¼ 1 A, j C C C B B C þ 1 C ¼ XA, n ð1 aw KA ÞBð1 aw KA ÞB C nQ 1 a w KA C @ @ A A a w n KA n CA, j

ð43Þ

j¼1

where m = 1, 2, ..., (n 1) in the subscript of XA,m, and: 0 0 n1

∑

m¼1

XA, m þ XA, n ¼

0n 2

∑

n1B B

i

i

i Q

1

11

a w KA CA, j C B B C B B C CC C j¼1 C þ 1 CC þ X BX ð1 a K ÞBð1 a K ÞB i ¼ m w A B w A B A, n B A, n C C nQ 1 a w KA A C @ @ A A m¼1 @ a w n KA n CA, j

∑

ð44Þ

j¼1

where the sum and product of an empty set are 0 and 1, respectively. Combining eqs 4244, the generalized expression for the solute to solvent content ratio as a function of water activity is given by: n1 iQ 1 1 ðaiw KAi ð1 CA, i Þ CA, j Þ NA 1 a w KA j¼1 i¼1 ¼ n2 m nQ 1 Q Nw r A a w KA ð1 aw KA Þ2 ðmaw m 1 KA m 1 CA, j Þ þ ððn 1Þ ðn 2Þaw KA Þaw n 2 KA n 2 CA, i

∑

∑

j¼1

m¼1

which can be rewritten in terms of molality using eq 26: mA ¼

1 aw KA M w r A a w KA

1

ð1 aw KA Þ

2

n2

∑

m¼1

ðmaw m 1 KA m 1

i¼1

iQ 1

n1

∑ ðaiw KAi ð1 CA, i Þ j ¼ 1 CA, jÞ i¼1

m Q j¼1

Substituting eq 38 into eq 20, differentiating with respect to total solute content, and applying the partial derivatives of the energy equalities from eqs 3436, the surviving terms yield a solute activity of rA NA XA, 1 rA ð47Þ aA ¼ rA NA

ð45Þ

CA, j Þ þ

ððn 1Þ ðn 2Þaw KA Þaw n 2 KA n 2

nQ 1 i¼1

ð46Þ CA, i

Equation 47 for the solute activity explicitly includes only the number of sorption sites on the solute molecule and the water sorbed on the first monolayer. From the point of view of the model, the external (i.e., i > 2) monolayers simply act as an extended solvent field. However, the number of water molecules in the first layer, XA,1, is influenced by how the other water molecules are distributed at equilibrium between the other 16480



ARTICLE

Table 3. Expressions for Solute Molality (mA) as a Function of Water Activity (aw) for Numbers of Sorbed Layers n = 1, 2, 3, 4, and 5a n

mA

1

1 aw KA Mw rA aw KA

! 1 aw KA ð1 KA aw ð1 CA, 1 ÞÞ Mw rA CA, 1 aw KA

2

3

1 aw KA Mw rA CA, 1 aw KA

4


5


!

! 1 KA aw ð1 CA, 1 Þ KA 2 aw 2 CA, 1 ð1 CA, 2 Þ ð1 aw KA Þ2 þ ð2 aw KA Þaw KA CA, 2

!

! 1 KA aw ð1 CA, 1 Þ KA 2 aw 2 CA, 1 ð1 CA, 2 Þ KA 3 aw 3 CA, 1 CA, 2 ð1 CA, 3 Þ ð1 aw KA Þ2 ð1 þ 2aw KA CA, 2 Þ þ ð3 2aw KA Þaw 2 KA 2 CA, 2 CA, 3

!

! 1 KA aw ð1 CA, 1 Þ KA 2 aw 2 CA, 1 ð1 CA, 2 Þ KA 3 aw 3 CA, 1 CA, 2 ð1 CA, 3 Þ KA 4 aw 4 CA, 1 CA, 2 CA, 3 ð1 CA, 4 Þ ð1 aw KA Þ2 ð1 þ 2aw KA CA, 2 þ 3aw 2 KA 2 CA, 2 CA, 3 Þ þ ð4 3aw KA Þaw 3 KA 3 CA, 2 CA, 3 CA, 4

There are n 1 monolayers and one multilayer. The nth layer is the multilayer isotherm. For example, when n = 1, all adsorption occurs in the multilayer. The expressions are evaluated from eq 46.

a

Table 4. Solute Activity As a Function of Water Activity for Numbers of Sorbed Layers n = 1, 2, 3, 4, and 5a n

aA

1

ð1 KA aw ÞrA

2

!rA 1 KA aw 1 KA aw ð1 CA, 1 Þ

3

!r A 1 KA aw 1 KA aw ð1 CA, 1 Þ KA 2 aw 2 CA, 1 ð1 CA, 2 Þ

4

!r A 1 KA aw 1 KA aw ð1 CA, 1 Þ KA 2 aw 2 CA, 1 ð1 CA, 2 Þ KA 3 aw 3 CA, 1 CA, 2 ð1 CA, 3 Þ 1 KA aw 1 KA aw ð1 CA, 1 Þ KA 2 aw 2 CA, 1 ð1 CA, 2 Þ KA 3 aw 3 CA, 1 CA, 2 ð1 CA, 3 Þ KA 4 aw 4 CA, 1 CA, 2 CA, 3 ð1 CA, 4 Þ

5

!rA

a In all cases, there are n 1 monolayers and one multilayer. The nth layer is the multilayer isotherm. For example, when n = 1, all adsorption occurs in the multilayer. The expressions are evaluated from eq 48.

monolayers and the multilayer. The influence of sorption in the external layers on solute activity is made apparent by combining eq 47, the CA,1 expression from eq 34, and XA,1 and XA,2 from eq 43: 0 1r A B aA ¼ B B @

1 a w KA n1

iQ 1

i¼1

j¼1

1 ∑ ðaw i KA i ð1 CA, i Þ

CA, j Þ

C C C A

ð48Þ

If all CA,j are equal to unity for j > 1, then eq 48 reduces to eq 30. The special case of CA,j = 1.0 for all j is explored in section 2.3. Expressions for the solute molality (eq 46) and solute activity (eq 48) for the n = 1 to 5 cases are given in Tables 3 and 4, respectively. Equations 46 and 48 satisfy the GibbsDuhem relationship for this system. 2.3. Special Cases. Two limiting cases are considered here. The first is when the energy of sorption to the monolayers is equal to the bulk energy of liquefaction, EA,1 = EA,2 = ... = EA,n1 = EL. The second is a subset of this case, when the constraints of

EA,n = EL and rA = 1.0 are also included. The first results in pure multilayer adsorption, and the second is Raoult’s law. 2.3.1. Pure Multilayer Adsorption. Consider the case when all the monolayer sorption energies are equal to that for the multilayer, that is, CA,i = 1.0 for all i. The isotherm is shown schematically in Figure 3, where an infinite number of monolayers with energy parameter CA,i = 1.0 (layers 1 to n 1) are sorbed onto some fraction of the rA surface sites. Equations 46 and 48 from the multilayer adsorption model and eqs 2730 from the two-layer model reduce to mA ¼

1 a w KA Mw rA CA, 1 aw KA

aA ¼ ð1 aw KA ÞrA

ð49Þ ð50Þ

The number of distinguishable arrangements and the energy equation for pure multilayer adsorption become Ω¼ 16481

ðrA NA þ Nw Þ! ðrA NA Þ!ðNw Þ!

ð51Þ



ARTICLE

NA ¼ Nw

1 aw rA aw

ð59Þ

aA ¼ ð1 aw ÞrA

ð60Þ

Equation 59 can be rewritten in terms of mole fraction of water, eq 61, and mole fraction of solute, eq 62: rA aw ð61Þ xw ¼ rA aw þ 1 aw Figure 3. Schematic of pure multilayer adsorption isotherm. The model assumes a surface with rA adsorption sites, and each layer i contains XA,i water molecules (sites and water molecules are not shown). Each monolayer is sorbed onto the layer below it. The energy parameter CA,i for each layer i is equal to 1.0.

xA ¼

1 aw rA aw þ 1 aw

ð62Þ

Assuming one sorption site per solute (rA = 1.0), eqs 6062 reduce to xw = aw, and xA = aA, which is Raoult’s Law.

and E ¼ Nw dA

ð52Þ

Using the same procedures as before, expressions for the most probable configuration, Gibbs energy, water activity, and solute activity are found: rA NA þ Nw ln Ω ¼ rA NA ln rA NA rA NA þ Nw þ Nw ln ð53Þ Nw

rA NA G=kT ¼ Nw dA =kT þ rA NA ln rA NA þ Nw Nw þ Nw ln rA NA þ Nw aw ¼

1 Nw KA r A N A þ N w

aA ¼

rA NA rA NA þ Nw

rA

ð54Þ ð55Þ ð56Þ

The water activity and solute activities can be rewritten to give the solute molality and solute activity as a function of water activity. mA ¼

1 KA a w Mw rA KA aw

aA ¼ ð1 aw KA ÞrA

3. ALTERNATIVE DERIVATION Nikitas31 derived extensions to the BET adsorption isotherm using an approach in which adsorbed layers are treated as a multicomponent monolayer, and with equations describing the equilibrium between surface fractions occupied by columns containing i and i + 1 adsorbed molecules. The approach of Nikitas31 can be used to derive simplified forms of our eq 47, as follows. First, the key expressions are Nikitas’ eqs 7 and 8: θ1 =ð1 θ2 θ3 ::: θn Þ ¼ βP

ð63Þ

θi =θi1 ¼ βi P

ð64Þ

ði ¼ 2, 3, 4, :::, nÞ

where θi is the fractional coverage of the surface by columns containing i molecules, and P is the adsorbate pressure in the gas phase. The symbol “β” describes the difference between the chemical potentials of a single molecule of adsorbate in the gas and first sorbed layer (ln β = (μo,g μ1o,s)/kT), and βi is the difference between the chemical potential in the gas phase and the difference between that in the ith and (i 1)th layers (ln βi = (μo,g (μio,s μi1o,s))/kT). These equations and others lead to expressions for the total surface coverage (θt) by adsorbed columns of molecules of all lengths (i.e., containing 1, 2, ..., n molecules) θt ¼ θ1 ð1 þ 2β2 P þ 3β2 β3 P2 þ ::: þ nβ2 β3 :::βn Pn 1 Þ

ð65Þ

ð57Þ and the surface coverage by columns containing single molecules (θ1): ð58Þ

Equations 57 and 58 from the derivation of the pure multilayer isotherm are the same as eqs 49 and 50 from the extended monolayer adsorption isotherms with CA,i = 1.0. Therefore, we can conclude that an adsorption isotherm model with no distinct monolayer energy of adsorption is equivalent to a purely multilayer adsorption isotherm with no monolayers present. 2.3.2. Raoult’s Law. Starting with eqs 56 and 58 for pure multilayer adsorption (or, equivalently, CA,i = 1.0), and assuming no Anderson energy of sorption to the multilayer (hence KA = 1.0), we find

θ1 ¼ βP=ð1 þ βPð1 þ β2 P þ β2 β3 P2 þ ::: þ β2 β3 :::βn Pn 1 ÞÞ

ð66Þ

The simplest adsorption case treated by Nikitas is when all of β2 to βn are equal to a constant b. Putting bP = x and βP = cx, eqs 14 and 15 of Nikitas are obtained and, as n tends to infinity: θt ¼ cx=ðð1 xÞð1 x þ cxÞÞ

ð67Þ

Nikitas shows that x is equivalent to P/P°, which is equivalent to water activity, aw, in this study where the adsorbate is water. Next, recognizing that θt = Nw/NArA, that c in eq 67 is the same as CA,1 16482



ARTICLE

Table 5. Fitted Parameters for Two-Layer and Three-Layer Models for Different Aqueous Solutions at 298.15 K two-layer modela

BET model solute (A) NaCl

NH4NO3

NaOH

Glycerol

range fitted (m: mol kg1)

parameter

value

error

parameter

value

three-layer model error

parameter

value

error

m: 3.016.68

rA

2.282

0.016

rA

2.913

0.033

rA

2.003

0.027

aw: 0.890.39

CA,1

22.9

4.2

CA,1

4.86

0.21

CA,1

0.35

0.25

KA

0.9391

0.0025

CA,2

20.6

14.3

KA

0.985

0.002

RMSE

1.60%

RMSEb

7.40%

m: 0.5111.01

rA

1.631

aw: 0.980.27

CA,1

0.593

RMSE

1.83%

m:1.529.0

rA

2.795

aw:0.950.06

CA,1

33.22

RMSE

25.60%

m:0.18961.1

rA

1.0334

aw: 0.9970.026

CA,1

2.085

RMSE

2.94%

RMSE

2.84%

0.012

rA

1.484

0.014

rA

1.638

0.018

0.015

CA,1

0.733

0.019

CA,1

0.980

0.023

KA

1.0036

0.0003

CA,2

0.639

0.021

KA

1.0015

0.0003

RMSE

0.90%

RMSE

1.23%

0.074

rA

3.106

0.052

rA

2.160

0.022

0.56

CA,1

23.5

1.8

CA,1

28.1

1.2

KA

0.9510

0.0045

CA,2

4.84

0.29

KA RMSE

0.9864 3.07%

0.0015

RMSE

11.56%

0.0039

rA

1.047

0.004

rA

1.005

0.027

0.068

CA,1 KA

1.975 0.99967

0.060 0.00006

CA,1 CA,2

1.189 1.696

0.044 0.057

KA

0.99995

0.00003

RMSE

2.59%

RMSE

1.18%

a Two-layer model equation is identical to the BET equation of Anderson.24 b RMSE, root-mean-square error, equal to [∑ (ϕfit ϕmeas)2/np]1/2 100%, where the summation is over all non zero-weighted points, np.

in this study, and that the molality of the solute A (mA) is equal to NA/MwNw, eq 67 yields NA =Mw Nw =ð1 aw Þ ¼ 1=ðCA, 1 rA Þ þ aw ðCA, 1 1Þ=ðCA, 1 rA Þ

ð68Þ

This expression is the BET model relating water activity to molality (eq 11 of Stokes and Robinson20). It is the same as our expression in Table 3 for n = 2, which is for a single adsorbed layer plus a multilayer, and with the assumption that KA is equal to unity. Expressions for adsorption to more than one layer can be derived from the equations of Nikitas by relaxing the constraint that all β2,...,n are equal to a constant. Consider the case where βP = cx, β2P = dx, and β3,...,nP = x. The expression for total fractional surface coverage becomes θt ¼ θ1 ð1 þ 2dx þ 3dx2 þ 4dx3 þ :::Þ

As n f ∞, both xn and xn+1 tend to 0, leaving θt ¼ cxð1 þ 2xðd 1Þ x2 ðd 1ÞÞ =ð1 xÞð1 x þ cx þ cx2 ðd 1ÞÞ

Equations similar to eq 72, for further adsorbed layers, can be obtained by repeating the derivation above. For example, setting βP = cx, β2P = dx, β3P = ex, and β4,...,nP = x yields the three-layer plus multilayer case, setting β4P = fx, and β5,...,nP = x yields the four-layer plus multilayer case, and so on. Using the relationship between θt and the molality of A given above, and recognizing that d in eq 72 is equivalent to CA,2 in this study, eq 72 can be rewritten to give an expression for mA: mA ¼ ½ð1 aw Þ=ðrA CA, 1 aw Mw Þð1 aw þ CA, 1 aw þ CA, 1 aw 2 ðCA, 2 1ÞÞ

ð69Þ

=ð1 þ 2aw ðCA, 2 1Þ aw 2 ðCA, 2 1ÞÞ

Combining eqs 11 and 12 of Nikitas for this case we obtain θt ¼ cxð1 þ 2dx þ 3dx2 þ 4dx3 þ :::Þ =ð1 þ cxð1 þ dx þ dx2 þ dx3 þ :::ÞÞ

ð70Þ

Multiplying top and bottom by (1 x) , and generalizing to n layers yields: 2

θt ¼ cxð1 þ 2xðd 1Þ x2 ðd 1Þ ðn þ 1Þdxn þ ndxn þ 1 Þ=ð1 xÞð1 x þ cx þ cx2 ðd 1Þ cdxn þ 1 Þ

ð71Þ

ð72Þ

ð73Þ

This equation is the same as that given in Table 3 for the n = 3 case, when KA is equal to unity. The study of Nikitas31 also contains treatments of lateral interactions, multisite occupancy, and other effects.

4. MODEL APPLICATIONS As shown in Tables 3 and 4, the model parameters are rA (number of sorption sites per solute molecule), CA,i (the adsorption energy parameters for each monolayer i), and KA 16483



Figure 4. Measured and fitted osmotic coefficients (ϕ) and water activities (aw) of aqueous NaCl at 298.15 K, plotted against the square root of the mole fraction of the salt (x, calculated as nNaCl/(nNaCl + nH2O), where n indicates the number of moles). Solute molality (m) is shown on the upper axis. Key: squares, Archer;42 circles, Tang et al.;43 triangles, Cohen et al.;44 diamonds, Chan et al.;45 solid lines, three-layer model; dashed lines, two-layer model; dotted lines, BET model. The vertical dotted lines mark the range of data fitted (3.016.682 mol kg1 NaCl). The insets show details at low concentrations. (a) Osmotic coefficient. (b) Water activity.

(the adsorption energy of the outermost multilayer). In applications to the thermodynamic properties of aqueous solutions it was hypothesized that the additional adsorbed layers, and extra parameters, would extend the valid range of the model to higher water activity, or lower concentration. Here we test the model equations using measured water activities at 25 °C of aqueous solutions of the following solutes: NaCl, NH4NO3, NaOH, and glycerol. Data for the salts NaCl and NH4NO3 include measurements for supersaturated solutions to water activities as low as 0.39 (NaCl) and 0.27 (NH4NO3). Results are compared for the BET, two-layer (corresponding to the GAB model and equations of Anderson24), and three-layer models. We note that Ally et al.39

ARTICLE

Figure 5. Measured and fitted osmotic coefficients (ϕ) and water activities (aw) of aqueous NH4NO3 at 298.15 K, plotted against the square root of the mole fraction of the salt (x, calculated as nNH4NO3/(nNH4NO3 + nH2O), where n indicates the number of moles). Solute molality (m) is shown on the upper axis. Key: squares, Wishaw and Stokes;46 circles, Chan et al.;47 triangles, Kirgintsev and Lukyanov;48 solid lines, three-layer model; dashed lines, two-layer model; dotted lines, BET model. The vertical dashed lines mark the range of data fitted (0.5111.01 mol kg1). (a) Osmotic coefficient. The inset shows details at low concentrations. (b) Water activity.

have previously applied the BET isotherm, with fitted parameters rA and CA,1, to measurements for the salts NaCl and (NH4)2SO4. They used a linear plot of mAMwaw/(1 aw) against aw to obtain the BET parameters. Using the same method, Marcus40 has reported BET and GAB parameters for a large number of salts. The models were directly fitted to molality-based osmotic coefficients since they more sensitively illustrate deviations of the model from the data. Parameters were determined using the nonlinear-least-squares fitting routine E04FYF, with a subroutine C05ADF used to iterate the water activity equations in Table 3 to obtain the measured molality. Both routines are from the Fortran library of the Numerical Algorithms Group.41 The parameters for 16484



Figure 6. Measured and fitted osmotic coefficients (ϕ) and water activities (aw) of aqueous NaOH at 298.15 K, plotted against the square root of the mole fraction of the salt (x, calculated as nNaOH/(nNaOH + nH2O), where n indicates the number of moles). Solute molality (m) is shown on the upper axis. Key: circles, calculated using the equation of Hamer and Wu;49 solid lines, three-layer model; dashed lines, two-layer model; dotted lines, BET model. The vertical dashed lines mark the range of data fitted (1.529.59 mol kg1). (a) Osmotic coefficient. (b) Water activity.

aqueous NaCl, NH4NO3, NaOH, and glycerol solutions at 298.15 K are listed Table 5, together with the molality range, water activity range, and the root-mean-square error (RMSE). By comparing the RMSE values, we found that the introduction of one additional layer (a two-layer plus multilayer model with one additional parameter, CA,2) significantly improved the accuracy of the fits. Additional adsorbed monolayers (three or more) produced only slight improvements, and are not listed here. Figures 47 compare calculated and measured osmotic coefficients and water activities for these salts. For aqueous NaCl (see Figure 4) the BET model does not provide an accurate fit over the full molality range. The two-layer model (which contains the additional parameter KA) is much

ARTICLE

Figure 7. Measured and fitted osmotic coefficients (ϕ) and water activities (aw) of aqueous glycerol (C3H5(OH)3) solutions at 298.15 K, plotted against the square root of the mole fraction of the solute (x, calculated as nC3H5(OH)3/(nC3H5(OH)3 + nH2O), where n indicates the number of moles). Solute molality (m) is shown on the upper axis. Key: squares, Ninni et al.;50 circles, Rudakov and Sergievskii;51 triangles, Scatchard et al.;52 solid lines, three-layer model; dashed lines, two-layer model; dotted lines, BET model. Data were fitted over the entire concentration range. (a) Osmotic coefficient. (b) Water activity.

Table 6. Lower Limit of Molality (mlow, mol kg1) and Upper Limit of Water Activity (aw,up) for Which the BET, TwoLayer, and Three-Layer Models Have Similar Performance mlow (mol kg1)

aw,up

RMSE solute

(%)

BET two-layer three-layer BET two-layer three-layer

∼1.60 11.2

11.0

3.0

0.53

0.54

0.89

NH4NO3 ∼0.90 5.5 NaOH ∼3.10 11.0

1.3 4.5

0.5 1.5

0.87 0.49

0.96 0.81

0.98 0.95

3.0

0.18

0.84

0.94

0.997

NaCl

glycerol 16485

∼1.20

9.0


The Journal of Physical Chemistry C better, but it is still not sufficient to reproduce the data well for low concentrations (as can be seen in the insets of Figure 4). However, fitted osmotic coefficients and water activities of the three-layer model agree with the measured values very well over a wide concentration range. For NH4NO3 (shown in Figure 5), whose osmotic coefficients decrease monotonically with increasing concentration, all three models represent the data better than for NaCl, and no significant difference between measurements and the fit can be observed from the plot of water activity in Figure 5b. However, the plot of osmotic coefficients (Figure 5a) clearly shows that the three-layer model is better than the two-layer one, and both are better than BET. One further electrolyte example (aqueous NaOH) is depicted in Figure 6, where very good consistency between measured values and those fitted by the three-layer model is obtained up to aw equal to 0.95, but the two-layer and BET models fit the data poorly at low mole fractions. The models were also tested using glycerol/water mixtures as an example of a nonelectrolyte solution; see Figure 7. Excellent performance is observed across almost the entire concentration range using the three-layer model, which is better than both the two-layer model and BET models which give quite similar results in this case. Another way to compare the models is to determine the concentration ranges over which the RMSE values are similar. Better models will be accurate over a wider concentration range. Table 6 lists the lower limits of molality (mlow) and upper limits of water activity (aw,up) for which the BET and two-layer models have the same RMSE as the three-layer model does over the full concentration range of the available data. These results show that the three-layer model has a useful effective range that is wider than both the two-layer and BET models, especially for aqueous NaCl and NaOH. All the examples presented here indicate that the three-layer model can accurately represent solution properties across a wide concentration range for both electrolyte and nonelectrolyte solutions.

5. SUMMARY Using statistical mechanics, we have derived equations for molality, and solvent and solute activities, in single-solute solutions, based upon extensions of standard adsorption isotherms to include arbitrary numbers of monolayers of adsorbed solvent. The equations are an extension of the BrunauerEmmettTeller (BET) isotherm21 but also incorporate elements of the GuggenheimAndersonde Boer (GAB) equations,2325 notably the additional parameter KA first proposed by Anderson.24 The equations for solute molality and water activity are also derived using the equilibrium approach described by Nikitas32 (for the case where KA is equal to unity). In systems containing multiple adsorbed monolayers the solvent activity is proportional to the ratio of the amount of water not adsorbed in the monolayers (termed the multilayer in this work) to the sum of the water in the multilayer and the outermost adsorbed monolayer only. 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. In solutions where all the monolayer sorption energies are equal to that for the multilayer (i.e., unity), and both KA and the number of sites per mole (rA) are also equal to 1.0, Raoult’s law is obtained.

ARTICLE

Applications of the isotherm model to osmotic coefficients and water activities of aqueous solutions of NaCl, NH4NO3, NaOH, and glycerol at 25 °C demonstrate that a three-layer model incorporating KA, i.e., one that includes two adsorbed monolayers plus a multilayer, is able to represent the data to significantly lower concentration (higher water activity) than the simpler models and is therefore likely to have a wider practical application.

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected] (C.S.D.); [email protected] (S.L.C.).

’ ACKNOWLEDGMENT The authors gratefully acknowledge the support of the National Oceanic and Atmospheric Administration (Grant NA07OAR4310191), the Natural Environment Research Council of the U.K. (Grant NE/E002641/1), and the U.S. Department of Energy (Grant DE-FG02-08ER64530). ’ REFERENCES (1) Anderson, R. P.; Hinckley, C. E. J. Ind. Eng. Chem. 1920, 12, 735–738. (2) Mehta, S. K.; Gaur, J. P. Crit. Rev. Biotechnol. 2005, 25, 113–152. (3) Langmuir, I. J. Am. Chem. Soc. 1918, 40, 1361–1403. (4) Roos, Y. H. J. Food Process. Preserv. 1993, 16, 433–447. (5) Timmermann, E.; Chirife, J.; Iglesias, H. A. J. Food Eng. 2001, 48, 19–31. (6) Almuhtaseb, A.; McMinn, W.; Magee, T. Food Bioprod. Process. 2002, 80, 118–128. (7) Kammerer, J.; Carle, R.; Kammerer, D. R. J. Agric. Food. Chem. 2011, 59, 22–42. (8) Parde, S. R.; Jayas, D. S.; White, N. D. G. Sci. Aliments 2003, 23, 589–622. (9) Hinz, C. Geoderma 2001, 99, 225–243. (10) Bradl, H. B. J. Colloid Interface Sci. 2004, 277, 1–18. (11) Breus, I. P.; Mishchenko, A. A. Eurasian Soil Sci. 2006, 39, 1271–1283. (12) Koretsky, C. J. Hydrol. 2000, 230, 127–171. (13) McKay, G.; Ho, Y. S.; Ng, J. C. Y. Sep. Purif. Methods 1999, 28, 87–125. (14) Dabrowski, A. Adv. Colloid Interface Sci. 2001, 93, 135–224. (15) Sing, K. Colloids Surf., A: Physicochem. Eng. Aspects 2001, 187, 3–9. (16) Kim, D. Korean J. Chem. Eng. 2000, 17, 156–168. (17) McLaren, A. D.; Rowen, J. W. J. Polym. Sci. 1951, 7, 289–324. (18) van der Wel, G. K.; Adan, O. C. G. Prog. Org. Coat. 1999, 37, 1–14. (19) Haworth, A. Adv. Colloid Interface Sci. 1990, 32, 43–78. (20) Stokes, R. H.; Robinson, R. A. J. Am. Chem. Soc. 1948, 70, 1870–1878. (21) Brunauer, S.; Emmett, P. H.; Teller, E. J. Am. Chem. Soc. 1938, 60, 309–319. (22) Impey, R. W.; Madden, P. A.; McDonald, I. R. J. Phys. Chem. 1983, 87, 5071–5083. (23) Guggenheim, E. A. Applications of Statistical Mechanics; Clarendon Press: Oxford, U.K., 1966. (24) Anderson, R. B. J. Am. Chem. Soc. 1946, 68, 686–691. (25) de Boer, J. H. The Dynamical Character of Adsorption; Clarendon Press: Oxford, U.K., 1968. (26) Timmermann, E. O. J. Chem. Soc., Faraday Trans. 1989, 85, 1631. 16486



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(27) Cook, M. A. J. Am. Chem. Soc. 1948, 70, 2925–2930. (28) Dole, M. J. Chem. Phys. 1948, 16, 25–30. (29) Balazs, N. L. Physica 1951, 17, 865–875. (30) The resulting isotherm in the work of Balazs (eq 25 in ref 29) does not reduce to GAB when n = 2 or to BET when n = 2, KA = 1, despite what is claimed. (31) Fowler, R.; Guggenheim, E. A. Statistical Thermodynamics: A Version of Statistical Mechanics for Students of Physics and Chemistry; University Press: Cambridge, U.K., 1952. (32) Nikitas, P. J. Phys. Chem. 1996, 100, 15247–15254. (33) Ally, M. R.; Braunstein, J. J. Chem. Thermodyn. 1998, 30, 49–58. (34) Mayer, J. E.; Mayer, M. G. Statistical Mechanics; John Wiley & Sons: New York, 1977. (35) Hill, T. L. J. Chem. Phys. 1946, 14, 263–267. (36) Abraham, M. J. Chim. Phys. Phys.-Chim. Biol. 1981, 78, 57– 59. (37) Casas-Vazquez, J.; Jou, D. Braz. J. Phys. 1997, 27, 547–559. (38) When taking the partial derivative with respect to total water content of the number of water molecules in layer i of solute A, additional nonzero contributions emerge. However, it can be shown that by applying the partial derivative with respect to total water content to the energy parameter equations and substituting the results into the water activity expression, the additional nonlogarithmic, non-Anderson, energy related terms combine to equal 0. For eq 21 in the n = 2 problem, the energy equation is eq 15 . For eq 39 in the n > 2 problem, the energy equations are eqs 3436. (39) Ally, M. R.; Clegg, S. L.; Braunstein, J.; Simonson, J. M. J. Chem. Thermodyn. 2001, 33, 905–915. (40) Marcus, Y. J. Solution Chem. 2005, 34, 297–306. (41) The NAG Fortran Library; The Numerical Algorithms Group Ltd.: Oxford, U.K., 2011. (42) Archer, D. G. J. Phys. Chem. Ref. Data 1992, 21, 793–829. (43) Tang, I. N.; Munkelwitz, H. R.; Wang, N. J. Colloid Interface Sci. 1986, 114, 409–415. (44) Cohen, M. D.; Flagan, R. C.; Seinfeld, J. H. J. Phys. Chem. 1987, 91, 4563–4574. (45) Chan, C.; Liang, Z.; Zheng, J.; Clegg, S. L.; Brimblecombe, P. Aerosol Sci. Technol. 1997, 27, 324–344. (46) Wishaw, B. F.; Stokes, R. H. Trans. Faraday Soc. 1953, 49, 27–31. (47) Chan, C. K.; Flagan, R. C.; Seinfeld, J. H. Atmos. Environ. 1992, 26, 1661–1673. (48) Kirgintsev, A. N.; Lukyanov, A. V. Russ. J. Phys. Chem. 1965, 39, 653–655. (49) Hamer, W. J.; Wu, Y.-C. J. Phys. Chem. Ref. Data 1972, 1, 1047–1100. (50) Ninni, L.; Camargo, M. S.; Meirelles, A. J. A. J. Chem. Eng. Data 2000, 45, 654–660. (51) Rudakov, A. M.; Sergievskii, V. V. Russ. J. Phys. Chem. 2006, 80, 1804–1808. (52) Scatchard, G.; Hamer, W. J.; Wood, S. E. J. Am. Chem. Soc. 1938, 60, 3061–3070.

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Statistical Mechanics of Multilayer Sorption: Extension of the Brunauer

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