Deriving Second Osmotic Virial Coefficients from Equations of State

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Deriving Second Osmotic Virial Coefficients from Equations of State and from Experiment Kenichiro Koga, Vincent Holten, and Benjamin Widom J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.5b07685 • Publication Date (Web): 17 Sep 2015 Downloaded from http://pubs.acs.org on September 22, 2015

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The Journal of Physical Chemistry

Deriving Second Osmotic Virial Coefficients from Equations of State and from Experiment K. Koga,∗,† Vincent Holten,‡ and B. Widom‡ †1 Department of Chemistry, Faculty of Science, Okayama University, Okayama 700-8530, Japan ‡Department of Chemistry, Baker Laboratory, Cornell University, Ithaca, New York, 14853-1301, USA E-mail: [email protected]

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Abstract The osmotic virial coefficients, which are measures of the effective interactions between solute molecules in dilute solution, may be obtained from expansions of the osmotic pressure or of the solute activity in powers of the solute concentration. In these expansions the temperature is held fixed and one additional constraint is imposed. When the additional constraint is that of fixed chemical potential of the solvent, the coefficient of the second-order term yields directly the second osmotic virial coefficient itself. Alternative constraints, such as fixed pressure, fixed solvent density, or the specification of liquid-vapor equilibrium, yield alternative measures of the solute-solute interaction, different from but related to the osmotic virial coefficient. These relations are summarized and, where new, are derived here. The coefficient in question may be calculated from equations of state in which the parameters have been obtained by fitting to other experimental properties. Alternatively, the coefficients may be calculated from direct experimental measurements of the deviations from Henry’s law based on measurements of the activity of the solute in a coexisting gas phase. It is seen for propane in water as a test case that with the latter method, even with what appear to be the best available experimental data, there are still large uncertainties in the resulting second osmotic virial coefficient. With the former method, by contrast, the coefficient may be obtained with high numerical precision, but then depends for its accuracy on the quality of the equation of state from which it is derived.

1

Introduction

The osmotic virial coefficients are measures of the solvent-mediated solute-solute interactions in dilute solution. When the interactions are predominantly repulsive, the second virial coefficient is large and positive; when the attractions predominate, the coefficient is large and negative. The virial coefficients are the coefficients in expansions of the osmotic pressure Π or of the solute activity z2 in powers of the solute number density ρ2 at fixed temperature T and 2 ACS Paragon Plus Environment

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solvent chemical potential µ1 : Π = ρ2 kT (1 + Bρ2 + · · · )

(fixed T, µ1 )

(1)

1 ρ2 (1 + 2Bρ2 + · · · ) Σ

(fixed T, µ1 )

(2)

z2 =

where k is the Boltzmann constant, B is the second osmotic virial coefficient, and the proportionality coefficient 1/Σ, a function only of the thermodynamic state of the pure solvent and reflecting the solute-solvent interactions, is the infinite dilution limit of the ratio z2 /ρ2 . By definition, Π = p(T, µ1 , ρ2 ) − p(T, µ1 , 0) with p the pressure as a function of T, µ1 , ρ2 . Here, in (2), z2 is defined so as to become equal to the number density ρ2 in an ideal gas. Thus, with µ2 the solute chemical potential in the solution, z2 = ρ2 e [µ2 −µ2 (i.g.)]/kT

(3)

where µ2 (i.g.) is the chemical potential of a hypothetical ideal gas of those same solute molecules; i.e.,when those molecules, at the density ρ2 , are a gas with no solvent and no intermolecular interactions. The difference µ2 − µ2 (i.g.) is often called the excess chemical potential of the solute. The leading terms in (1) and (2), Π ∼ ρ2 kT and z2 ∼ ρ2 /Σ, are van’t Hoff’s law and Henry’s law, respectively. The coefficient 1/Σ in (2) is closely related to the conventionally defined Henry’s-law constant and Ostwald absorption coefficient. The dimensionless Σ would be exactly the Ostwald coefficient ρ2 /ρG 2 , the ratio at infinite dilution of the solute number densities in the liquid solution and in a coexisting gas phase, if the gas were ideal. The Henry’s-law coefficient kH , which has the dimensions of pressure, is similarly defined as the infinitely dilute limit of p2 /x2 , with p2 the partial pressure of solute in the equilibrium vapor and x2 its mole fraction in the liquid solution. It is thus approximately related to Σ by Σ ≃ ρ1 kT /kH 3 ACS Paragon Plus Environment

(4)


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where ρ1 is the number density of the pure solvent. This approximation assumes the vapor in equilibrium with the liquid solution is an ideal gas. The relation (4) then follows because in the infinitely dilute limit the mole fraction x2 is the same as the mole ratio ρ2 /ρ1 . The number density ρ1 , it should be noted, is generally not well defined (only ρ2 is, because the solution is dilute); it is defined only by an arbitrary assignment of what is taken to be the “molecule” in the dense, strongly interacting liquid solvent. That convention is thus present also in kH , but not in Σ, which is a well defined property of the solute and solvent, independently of what is taken to be the solvent molecule. If the constraint of fixed µ1 in (2) were replaced by another, say fixed pressure p, or fixed solvent density ρ1 (as defined by any measure), or coexistence with an equilibrium vapor, the result would be an analogous expansion, also carrying information about the solventmediated solute-solute interactions, but with coefficients different from those in (2). Thus, in place of B in (2) there are coefficients we denote respectively by B ′ , B ′′ , and Bcoex : 1 ρ2 (1 + 2B ′ ρ2 + ...) Σ 1 z2 = ρ2 (1 + 2B ′′ ρ2 + ...) Σ 1 z2 = ρ2 (1 + 2Bcoex ρ2 + ...) Σ z2 =

(fixed T, p)

(5)

(fixed T, ρ1 )

(6)

(liquid-vapor coexistence at fixed T )

(7)

with the same Σ in all. But it is uniquely the osmotic B in (2) that is related to the integral of the infinitely dilute limit of the solute-solute pair correlation function, h22 (r), by 1 B=− 2

∫ h22 (r)dτ,

(8)

where the integration is over all space with element of volume dτ and where r is the distance between (arbitrarily assigned 1–3 ) molecule centers. The relation (8) is as in the McMillanMayer solution theory 4,5 and is also the infinite-dilution limit of one of the Kirkwood-Buff relations. 6–9


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The relation of B ′ to B is known 10–12 and will be recalled in Section 2, where the relation of B ′′ to B, which we believe to be new, is then derived. Likewise, in Section 3 the relation of Bcoex to B is derived. Any of these relations allows B to be calculated from an equation of state in which the parameters are fixed by other experimental properties of the solute-solvent pair, 12,13 as is illustrated in Section 4. Alternatively, B may be extracted from direct experimental measurements. This may be done from measurements of freezing-point depression for solutions of non-volatile solutes, 14,15 or from measurements of gas solubility, 11,16 which is our focus here. We illustrate this in Section 5, where we use the method of Watanabe and Andersen 11 with data on the solubility of propane in water. The results are summarized in the concluding Section 6.

2

B , B ′, B ′′

The four coefficients B, B ′ , B ′′ , Bcoex are associated with distinct constraints on the expansions in Eqs. (2-5). In addition to fixed temperature in all, these additional constraints are respectively fixed solvent chemical potential µ1 , fixed pressure p, fixed solvent density ρ1 (with any measure of that density), and liquid-vapor equilibrium.These coefficients are all interrelated. In this section we note the relations among the first three, B, B ′ , and B ′′ , and then in Section 3 we add Bcoex to the list. The relation between B and B ′ is already known: 10–12 1 B = B ′ + (v2 − kT χ) 2

(9)

where v2 is the partial molecular volume of the solute at infinite dilution in the solvent and χ is the isothermal compressibility of the pure solvent. Next, the relation of B ′′ to B is derived. From a partial derivative identity, with the derivatives then evaluated at infinite dilution



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as denoted by superscript◦ , [

∂(z2 /ρ2 ) ∂ρ2

]◦ T,ρ1

[

∂(z2 /ρ2 ) = ∂ρ2

]◦

[

∂(z2 /ρ2 ) + ∂p T,p

]◦

(

T,ρ2

∂p ∂ρ2

)◦ .

(10)

T,ρ1

The left-hand side, from (6), is 2B ′′ /Σ. The first term on the right-hand side, from (5), is 2B ′ /Σ. The ratio of the partial molecular volume of the solute to the compressibility of the solution is (∂p/∂ρ2 )T,ρ1 , so the factor (∂p/∂ρ2 )◦T,ρ1 on the right-hand side of (10) is v2 /χ with v2 and χ as in (9). It remains to evaluate the factor [∂(z2 /ρ2 )/∂p]◦ T,ρ2 . From (5), and then the definition of χ, this is

[

∂(z2 /ρ2 ) ∂p

]◦ T,ρ2

[

∂(1/Σ) = ∂p

]

[

T

∂(1/Σ) = ∂ρ1

]◦ ρ1 χ.

(11)

T

But from Eq. (A. 10) of ref. 17, for example, one has [∂(1/Σ)/∂ρ1 ]◦T = (v2 − kT χ)/kT χρ1 Σ.

(12)

Then assembling these various pieces of (10), one has B ′′ = B ′ +

1 v2 (v2 − kT χ) 2 kT χ

(13)

B = B ′′ −

1 (v2 − kT χ)2 2kT χ

(14)

so, from (9),

as the relation of B to B ′′ that was sought. As expected and required, the arbitrarily defined density ρ1 does not appear in the final result. Factors Σ have also canceled, so B − B ′′ from (14), like the earlier B − B ′ from (9), depends only on v2 and kT χ, with v2 the partial molecular volume of the solute at infinite dilution and χ the compressibility of the pure solvent. In the present contexts (although not always), v2 is positive and much greater than kT χ because the dense liquid solvents are so


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nearly incompressible. It is also seen that B ′ is then less than (more negative than) B while B ′′ is much greater than B, so much so that it is most often large and positive even when B is negative.

3

B, Bcoex

In this section, the relation between the coefficients B and Bcoex in (2) and (7) is derived. The analog of (10) in Section 2, with the infinitesimal increment in the solute density now taken to occur while maintaining liquid-vapor coexistence rather than at fixed ρ1 , is [

∂(z2 /ρ2 ) ∂ρ2

]◦ T,coex

[

∂(z2 /ρ2 ) = ∂ρ2

]◦

[

∂(z2 /ρ2 ) + ∂p T,p

]◦

(

T,ρ2

∂p ∂ρ2

)◦ (15) T,coex

The left-hand side, from (7), is (2/Σ)Bcoex . The first term on the right-hand side, from (5), is again (2/Σ)B ′ , as in Section 2. The factor [∂(z2 /ρ2 )/∂p]◦T,ρ2 is itself again, as in Section 2, the product of the two factors (v2 − kT χ)/kT χρ1 Σ and ρ1 χ; so (15) is

Bcoex

1 (v2 − kT χ) =B + 2kT ′

(

∂p ∂ρ2

)◦ (16) T,coex

With the relation of B ′ to B known from (9), it now remains only to evaluate (∂p/∂ρ2 )◦T,coex . Near infinite dilution, the equilibrium gas phase is almost all gaseous solvent; the solute in the gas phase contributes very little to the equilibrium pressure p. But the increase in p with increasing concentration ρ2 of solute in the liquid solution is due almost entirely to the corresponding increase in its concentration in the gas, and it is that rate of increase of p, evaluated at infinite dilution, that is measured by (∂p/∂ρ2 )◦T,coex . The concentrations of solvent and solute in the liquid solution are still to be denoted G ρ1 and ρ2 ; their concentrations in the equilibrium gas phase will be denoted ρG 1 and ρ2 .

The pressure p and chemical potentials µ1 and µ2 all have common values in the coexisting phases. There are then two Gibbs-Duhem equations, one for each phase, which, at fixed T ,



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are G dp = ρG 1 dµ1 + ρ2 dµ2 .

dp = ρ1 dµ1 + ρ2 dµ2 ,

(17)

Therefore, (

(

∂p ∂ρ2

∂p ∂ρ2

)

( = ρ1

T,coex

)

( =

ρG 1

T,coex

∂µ1 ∂ρ2

∂µ1 ∂ρ2

)

( + ρ2

T,coex

)

∂µ2 ∂ρ2

( +

ρG 2

T,coex

)

∂µ2 ∂ρ2

(18a) T,coex

) (18b) T,coex

From (3), µ2 = kT ln(Λ32 z2 ) with a length Λ2 that is a function of T alone (conventionally, for point particles, the thermal de Broglie wavelength) , with µ2 (i.g.) = kT ln(Λ32 ρ2 ). Then (∂µ2 /∂ρ2 )T,coex = (kT /z2 )(∂z2 /∂ρ2 )T,coex . But in the limit of infinite dilution, from (7), this is just kT /ρ2 . Then, from (18), (

(

∂p ∂ρ2

∂p ∂ρ2

)◦

( = ρ1

T,coex

)◦

( =

ρG 1

T,coex

∂µ1 ∂ρ2

∂µ1 ∂ρ2

)◦ + kT

(19a)

T,coex

)◦ + kT T,coex

ρG 2 . ρ2

(19b)

On multiplying (19a) by ρG 1 and (19b) by ρ1 , and subtracting one from the other and then rearranging, one has the required (

∂p ∂ρ2

)◦ = kT T,coex

G ρG 2 /ρ2 − ρ1 /ρ1 . 1 − ρG 1 /ρ1

(20)

Then from (9), (16), and (20) there follows the sought-for relation between B and Bcoex : ρG /ρ2 − 1 1 . B = Bcoex − (v2 − kT χ) 2 G 2 1 − ρ1 /ρ1


(21)

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It is to be noted that the densities of component 1 occur only as the ratio ρG 1 /ρ1 , and (21) is thus well defined even with an arbitrary (common) factor in ρG 1 and ρ1 separately; i.e., with an arbitrary assignment of the “molecule” of solvent. The ratio ρG 2 /ρ2 is the reciprocal of the Ostwald absorption coefficient (very nearly 1/Σ), and is thus much greater than 1 when, as is the case of interest here, the gaseous solute is of very low solubility in the liquid solvent. By contrast, ρG 1 /ρ1 is almost always much less than 1. Then since v2 in most applications is greater (usually much greater) than kT χ, one has Bcoex greater than B and usually positive even when B is negative, as was true also of B ′′ .

4

B from equations of state

Once one has a reliable equation of state, B can be calculated from any of (1), (2), (5), (6), or (7), the latter three supplemented with (9), (14), or (21), respectively. The result could be numerically essentially exact; its quality would depend only on that of the equation of state from which it was derived. Such calculations were illustrated earlier 12,18 with the 2-component van der Waals equation of state, 19 which is relatively crude but had the advantage that all of z1 , z2 , Σ, v2 , χ could then be calculated analytically from a prescribed p(T, ρ1 , ρ2 ). 12 It was applied as an example to a solution of propane in water at 298.15 K. The parameters in the assumed p(T, ρ1 , ρ2 ) were determined so as to reproduce the experimental density and vapor pressure of water and the experimental solubility (essentially Σ) and partial molecular volume of propane in water. It also incorporated the known sizes of the propane and water molecules. The result, consistently from (1), (2), or (5), the latter supplemented with (9), was B = −570 cm3 /mol. For the same system, with the more elaborate UNIQUAC 20 and Wang-Chao 21 equations of state, Liu and Ruckenstein 13 had earlier found B = −668 cm3 /mol. Those values for the osmotic second virial coefficient of propane in water are notable for being more negative than −393 cm3 /mol, which is the experimental value of the second virial



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coefficient of pure gaseous propane at the same temperature. It is in accord with the idea that propane in water would exhibit hydrophobic attraction; i.e., that through the mediation of the solvent, the effective attraction between propane molecules in water would be stronger than in vacuum. 22–26

5

B from experimental gas-solubility measurements

An effective way of obtaining the osmotic second virial coefficient from experimental gassolubility measurements, as opposed to calculating it from an assumed equation of state, is that of Watanabe and Andersen. 11 Their formula with which the data may be analyzed is re-derived here in the Appendix using the present notation and in a slightly simpler form in which the solute concentration is expressed as number density ρ2 instead of mole fraction x2 . Then the number density ρ1 of the pure solvent, or its reciprocal, the molecular volume v1 , no longer appears. The resulting form of the Watanabe-Andersen formula, in the present notation, is ln(z2 /ρ2 ) − (p − p◦ )v2 /kT = ln(1/Σ) + (2B − v2 + kT χ)ρ2 .

(22)

Here, 1/Σ is again the proportionality coefficient in (1), (2), and (5)−(7), related to the Henry’s-law constant and the Ostwald absorption coefficient as noted in the Introduction; ρ2 is the number density of the solute in the solution; and v2 and χ are again the partial molecular volume of the solute at infinite dilution and the isothermal compressibility of the pure solvent. In the present applications, p◦ may be taken to be the vapor pressure of the pure solvent at temperature T , or 1 atm, or any other convenient reference pressure, and p and z2 are the total pressure and solute activity when the liquid solution at T, ρ2 is in equilibrium with its vapor phase. In Fig. 1 is shown a schematic of the associated Watanabe-Andersen plot. One plots the left-hand side of (22) vs. ρ2 . The intercept on the vertical axis is ln(1/Σ). One finds B from the slope of the plot together with the known experimental values of v2 and χ, and with z2 10 ACS Paragon Plus Environment

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obtained from independent experimental measurements of the properties of the equilibrium vapor. As a test of (22), the model van der Waals 2-component equation of state referred to above was used to give model p vs. ρ2 “experimental” data for propane in water at 298.15 K. The model was already known 12 to be in good accord with the experimental liquid-gas equilibrium pressure p vs. ρ2 measurements of Chapoy et al. 27 For this model calculation, which is essentially analytical, everything was obtained with high numerical precision. When care was taken to account for the water content in the gas phase when the gas is primarily propane, which is the case in the range of concentrations in the real experiments, the model calculations via the Watanabe-Andersen plot gave nearly the exact result as known from the earlier calculations (Section 4). (When the water content in the vapor is not properly accounted for, the plot is noticeably curved and the resulting estimate of B is more negative than its known exact value by 40% or more, depending on how one estimates the slope.) Now we construct the corresponding Watanabe-Andersen plot from experimental propane/water data at 298.12 K (the temperature of the data of Chapoy et al. 27 ), both p vs. ρ2 solubility data and the gas-phase data required in calculating the activity z2 for use in the formula (22). It will be seen that even with the best available data there remains substantial uncertainty in the resulting value of B. The reference pressure p◦ , which is the fixed pressure in the expansion (5) that defines B ′ and thus also B via (9), is here chosen to be the vapor pressure of pure water at 298.12 K. That means that the solutions in the expansion (22) are formally in metastable states. 18 But it was shown in the model calculation in ref. 18 that the resulting B is hardly different from what it would have been had p◦ been chosen to be 1 atm. For the plot corresponding to the expansion in ρ2 , as noted above, one plots the left-hand side of (22) (to be denoted ξρ ) vs. ρ2 , finds B from the slope via (22), and ln(1/Σ) from the intercept on the ξρ axis. The original formulation of Watanabe and Andersen 11 was as an expansion in mole fraction x2 and thus also contained the molecular volume v1 of the pure



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solvent at T, p◦ . In the present notation, that formulation is

ln

1 1 v1 z2 (p − p◦ )v2 = ln + (2B − 2v2 + v1 + kT χ)x2 . − x2 kT Σ v1

(23)

Then one plots the left-hand side (to be denoted ξx ) vs. x2 , finds B from the slope via (23), and ln(1/Σ), again, from the intercept on the ξx axis. It was noted in the Introduction how the Ostwald absorption coefficient and Henry’s-law constant kH are related to Σ. From tests of these expansions with the van der Waals two-component equation of state, for which the exact value of B is known, 12,18 it was found that the value of B from these plots is sensitive to the value of the solute activity z2 in the gas phase. In particular, calculating z2 under the assumption that the gas phase is pure solute at the pressure p leads to a significant deviation of the calculated B from its known value. Two alternative corrections to this approximation were found to improve the accuracy of the value for B. First, z2 can be approximated as the activity of the pure gaseous solute, to be denoted ζ2 , at the approximate partial pressure p − p◦ of the solute in the mixture. Thus, for z2 in the gas phase, z2 ≃ ζ2 (T, p − p◦ ).

(24)

Alternatively, one may approximate it by

z2 ≃ (1 − p◦ /p)ζ2 (T, p).

(25)

Either correction would ensure that z2 vanishes in the limit of infinite dilution, where p = p◦ . It was found from tests with the van der Waals equation of state that the two corrections are about equally accurate. Measurements of the solubility of propane in water at a partial propane pressure of 1 atm have been evaluated by Battino 28 and his recommended fit gives the value x2 = (0.2734±0.0033)×10−4 at the temperature considered. The partial molar volume of propane


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was estimated from data of Masterton 29 as NA v2 = 66.8 cm3 /mol. The vapor pressure and compressibility of pure water were calculated from the IAPWS-95 equation of state. 30,31 The mole fraction of propane in water at 298.12 K and pressures up to 0.92 MPa has been measured by Chapoy et al. 27 Kobayashi and Katz 32 have measured the mole fraction of propane in the water-rich liquid phase along the liquid-liquid-vapor triple-point line from 285 K to 369 K. The mole fraction and pressure at 298.12 K were obtained from polynomial fits to their data. Mole fractions x2 were converted to number densities ρ2 with the approximation ρ2 ≃

x2 (1 − x2 )v1 + x2 v2

(26)

where the molecular volume v1 of water was calculated from the IAPWS-95 equation of state. [Eq. (26) would have been exact if the v1 and v2 had been the actual partial molecular volumes in the solution. Here they are approximated by their values at infinite dilution.] The activity of pure gaseous propane was obtained from the equation of state of Lemmon et al. 33 It was found in passing that it would have been sufficient to use a virial expansion for propane up to the third virial coefficient, but for the results shown here, the full equation of state was used. The resulting Watanabe-Andersen plots are shown in Fig. 2 and are seen to be essentially identical. Because of the uncertainty in the solubility at atmospheric pressure, and the scatter of the data at higher pressures, it is not possible to obtain an accurate value for B from the plots. What can be concluded is that B probably lies in the range −300 cm3 /mol to −1000 cm3 /mol.

6

Summary and Conclusions

Methods for extracting the osmotic virial coefficients from equations of state or directly from experimental measurements are discussed. When a reliable equation of state for a two-component liquid mixture, solvent plus solute, 13 ACS Paragon Plus Environment


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is known, the solute activity z2 in the solution as a function of its number density ρ2 can be calculated essentially analytically and with almost any desired numerical precision. Then the second osmotic virial coefficient B follows from an expansion of z2 in powers of ρ2 at fixed temperature T and chemical potential µ1 of the solvent [Eq. (2)]. Equivalently, if the expansion is at fixed pressure p instead of fixed µ1 , one obtains the expansion coefficient B ′ [Eq. (5)], from which B follows via (9), with v2 and χ the partial molecular volume of the solute at infinite dilution and the isothermal compressibility of the pure solvent, respectively. Alternatively, the expansion may be at fixed density ρ1 of the solvent [Eq. (6)], in which case the expansion coefficient is B ′′ , from which B follows via (14). Finally, instead of the constraint of fixed µ1 , p, or ρ1 , the expansion may be of z2 in powers of the solute density ρ2 subject to the constraint of the liquid solution being in equilibrium with a vapor phase [Eq. (7)]. The expansion coefficient is then Bcoex , from which B follows via (21), where ρG 2 /ρ2 is the limiting ratio of the gas-phase to the liquid-phase density of the solute in the limit of infinite dilution, which is the reciprocal of the Ostwald absorption coefficient of the solute, and ρG 1 /ρ1 is the corresponding limit of the gas-phase to the liquid-phase density of what is then the pure solvent at the temperature T . In practice, the Ostwald coefficient is nearly the same as the Σ that occurs in the coefficient in the expansions (2) and (5)−(7), and would be exactly the same if the gas phase were ideal so that z2 were the same as ρG 2 . The Ostwald coefficient and Σ are in turn approximately related to the conventionally defined Henry’s-law coefficient kH by (4), but in this kH there is the choice of an often ill-defined “molecule” of solvent in the dense, strongly interacting liquid, and this is reflected in the number density ρ1 in (4). By contrast, there is no arbitrariness in the ratio ρG 1 /ρ1 above, which could equally well be the ratio of the mass densities of the solvent in the gas and liquid phases. In practice, ρG 1 /ρ1 is so much less than 1 that it may be neglected in (21). In obtaining B from an equation of state one thus depends on the reliability of the equation with the assigned values of its parameters. As an alternative, B may be obtained from


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a Watanabe-Andersen plot, via (22), as illustrated schematically in Fig. 1 and realistically in Fig. 2. It is then obtained from direct experimental measurements of the pressure and of the solute density when the liquid solution is in equilibrium with its vapor. That may be successful when, as is often the case just beyond the infinitely dilute limit, the gas phase is almost all solute. The equilibrium activity z2 required in (22) is then derivable from independent measurements on the pure solute vapor, with a small but necessary correction for the content of solvent in it. The method is illustrated in Section 5 for propane in water at 298.12 K (not appreciably different from 298.15 K). It results in a large range of possible values of the virial coefficient B, −300 to −1000 cm3 /mol, even with what are likely the best available experimental data. By contrast, the value −570 cm3 /mol for B found earlier from a model equation of state 12,18 and the value −668 cm3 /mol found from more elaborate equations, 13 are obtained with high numerical precision, and lie in that range, but then depend for their accuracy on the quality of the equations. It is to be noted that in principle B may also be obtained from the integral of the infinitely dilute limit of the solute-solute pair-correlation function via Eq. (8). The required pair correlation function is obtainable from computer simulations with realistic molecular parameters. 34 When that was done for propane/water, the result 3 was an osmotic second virial coefficient less negative by a factor of about 15 than that previously obtained from equations of state. 12,13 This large discrepancy remains unresolved and is the subject of current study.



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Appendix This is the derivation of (22), which is a variation and simplification of the original formula of Watanabe and Andersen. 11 The chemical potential µ2 of the solute in the solution may be taken to be a function of temperature T , pressure p, and the ratio ρ2 /ρ1 of the solute and solvent densities, with ρ2 here the number density of the solute and ρ1 any measure of the solvent density. Alternatively, the same µ2 may be taken to be a function of T, p, and ρ2 alone, since ρ1 itself may be expressed as a function of T, p, ρ2 . First, take µ2 to be the function of T, p, ρ2 /ρ1 . Then from a known identity [e.g., Eq. (A.9) of ref. 17], (∂µ2 /∂p)T,ρ2 /ρ1 is the partial molecular volume of the solute in the solution. Then with any convenient choice of a reference pressure p◦ , one has, approximately, µ2 (T, p, ρ2 /ρ1 ) ≃ µ2 (T, p◦ , ρ2 /ρ1 ) + (p − p◦ ) v2

(A.1)

for pressures p that are not too great and with the concentration of solute in the solution small enough for the partial molecular volume in the solution to be replaced by its value v2 at infinite dilution at T, p◦ . It is not required that |p − p◦ | be much less than p◦ ; only that it be small enough to have only a small effect on the chemical potential of the solute in this nearly incompressible liquid solution, so that terms of order (p − p◦ )2 and higher may be neglected. In practice, p◦ may be taken to be the vapor pressure of the pure solvent at T , or to be 1 atm, or any other pressure not too greatly different from the pressures p of the measurements from which B will ultimately be derived. Next, take µ2 alternatively to be the function of T, p, ρ2 ; it is the same µ2 as in (A.1), only differently expressed. Thus, (A.1) is also

µ2 (T, p, ρ2 ) ≃ µ2 (T, p◦ , ρ2 ) + (p − p◦ ) v2 .


(A.2)

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Then from (3) and (A.2), and since µ2 (i.g.)/kT in (3) is ln(Λ32 ρ2 ) with Λ2 a function of T alone (conventionally, for point particles, the thermal de Broglie wavelength),

ln

z2 z2 p − p◦ = ln + v2 . ρ2 T,p,ρ2 ρ2 T,p◦ ,ρ2 kT

(A.3)

But from the definition of B ′ in (5) and its relation to B in (9),

ln

z2 = ln(1/Σ) + (2B − v2 + kT χ)ρ2 + · · · ρ2 T,p◦ ,ρ2

(A.4)

with Σ, B, v2 , χ now all evaluated at T, p◦ . Then from (A.3) and (A.4),

ln

z2 p − p◦ − v2 = ln(1/Σ) + (2B − v2 + kT χ)ρ2 , ρ2 T,p,ρ2 kT

(A.5)

which is (22), as required. This is equivalent to the expression in ref. 11 to leading order but expanded here in ρ2 instead of in mole fraction x2 , so the result is a little simpler and without reference in it to the solvent density ρ1 or its reciprocal v1 . Granting the truncation of the expansion to first order in ρ2 , the only other approximation in (22) and (A.5) is in the term (p − p◦ )v2 /kT , where terms of order (p − p◦ )2 have been neglected. If the solution is in equilibrium with a vapor phase, p is the common total pressure and z2 the common activity of the solute in the two phases. Thus p as well as z2 varies with the solute density ρ2 in the liquid solution. One then plots the experimental values of the lefthand side of (22) or (A.5), as obtained from gas-solubility measurements and measurements of the properties of the gas phase, as functions of the varying ρ2 in the solution. If the solution is dilute enough over the range of values of ρ2 in the measurements, and if the approximation (p − p◦ )v2 in (A.1) and thus in (A.5) is sufficient, the plot will be linear within the experimental scatter. Then Σ will be found from the intercept on the vertical axis and B from the slope, as shown schematically in Fig. 1.



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References (1) Hansen, J.-P.; McDonald, I. R. Theory of Simple Liquids, 2nd ed.; Academic Press, 1986; pp 442, 447. (2) Lockwood, D. M.; Rossky, P. J. Evaluation of Functional Group Contributions to Excess Volumetric Properties of Solvated Molecules. J. Phys. Chem. B 1999, 103, 1982–1990. (3) Koga, K.; Widom, B. Thermodynamic Functions as Correlation-Function Integrals. J. Chem. Phys. 2013, 138, 114504. (4) McMillan, W. G.; Mayer, J. E. The Statistical Thermodynamics of Multicomponent Systems. J. Chem. Phys. 1945, 13, 276–305. (5) Vafaei, S.; Tomberli, B.; Gray, C. G. McMillan-Mayer Theory of Solutions Revisited: Simplifications and Extensions. J. Chem. Phys. 2014, 141, 154501. (6) Kirkwood, J. G.; Buff, F. P. The Statistical Mechanical Theory of Solutions. I. J. Chem. Phys. 1951, 19, 774–777. (7) Zimm, B. H. Simplified Relation Between Thermodynamics and Molecular Distribution Functions for a Mixture. J. Chem. Phys. 1953, 21, 934–935. (8) Ben-Naim, A. Water and Aqueous Solutions: Introduction to a Molecular Theory; Plenum, 1974; p 142. (9) Ben-Naim, A. Inversion of the Kirkwood-Buff Theory of Solutions: Application to the Water-Ethanol System. J. Chem. Phys. 1977, 67, 4884–4890. (10) Hill, T. L. Introduction to Statistical Thermodynamics; Addison-Wesley, 1960; pp 362– 368. (11) Watanabe, K.; Andersen, H. C. Molecular Dynamics Study of the Hydrophobic Interaction in an Aqueous Solution of Krypton. J. Phys. Chem. 1986, 90, 795–802. 18 ACS Paragon Plus Environment

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(12) Widom, B.; Underwood, R. C. Second Osmotic Virial Coefficient from the TwoComponent van der Waals Equation of State. J. Phys. Chem. B 2012, 116, 9492–9499. (13) Liu, H.; Ruckenstein, E. Aggregation of Hydrocarbons in Dilute Aqueous Solutions. J. Phys. Chem. B 1998, 102, 1005–1012. (14) Stigter, D. Interactions in Aqueous Solutions. II. Osmotic Pressure and Osmotic Coefficient of Sucrose and Glucose Solutions. J. Phys. Chem. 1960, 64, 118–124. (15) Kozak, J. J.; Knight, W. S.; Kauzmann, W. Solute-Solute Interactions in Aqueous Solutions. J. Chem. Phys. 1968, 48, 675–690. (16) Tucker, E. E.; Lane, E. H.; Christian, S. D. Vapor Pressure Studies of Hydrophobic Interactions. Formation of Benzene-Benzene and Cyclohexane-Cyclohexanol Dimers in Dilute Aqueous Solution. J. Solution Chem. 1981, 10, 1–20. (17) De Gregorio, P.; Widom, B. Effect of Solutes on the Structure and Energetics of a Model Solvent. J. Phys. Chem. C 2007, 111, 16060–16069. (18) Widom, B.; Koga, K. Note on the Calculation of the Second Osmotic Virial Coefficient in Stable and Metastable Liquid States. J. Phys. Chem. B 2013, 117, 1151–1154. (19) Konynenburg, P. H. V.; Scott, R. L. Critical Lines and Phase Equilibria in Binary Van Der Waals Mixtures. Phil. Trans. Roy. Soc. London A 1980, 298, 495–540. (20) Prausnitz, J. M.; Richtenthaler, R. N.; Gomes de Azevedo, E. Molecular Thermodynamics of Fluid Phase Equlilbria, 2nd ed.; Prentice-Hall, 1986; pp 238–245. (21) Wang, W.; Chao, K.-C. The Complete Local Concentration Model Activity Coefficients. Chem. Eng. Sci. 1983, 38, 1483–1492. (22) Rossky, P. J.; Friedman, H. L. Benzene-Benzene Interaction in Aqueous Solution. J. Phys. Chem. 1980, 84, 587–589. 19 ACS Paragon Plus Environment


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(23) Wood, R. H.; Thompson, P. T. Differences between Pair and Bulk Hydrophobic Interactions. Proc. Natl. Acad. Sci. U.S.A. 1990, 87, 946–949. (24) Blokzijl, W.; Engberts, J. B. F. N. Hydrophobic Effects. Opinions and Facts. Angew. Chem., Int. Ed. Engl. 1993, 32, 1545–1579. (25) L¨ udemann, S.; Abseher, R.; Schreiber, H.; Steinhauser, O. The TemperatureDependence of Hydrophobic Association in Water. Pair versus Bulk Hydrophobic Interactions. J. Am. Chem. Soc. 1997, 119, 4206–4213. (26) Liu, L.; Michelsen, K.; Kitova, E. N.; Schnier, P. D.; Klassen, J. S. Evidence that Water Can Reduce the Kinetic Stability of Protein–Hydrophobic Ligand Interactions. J. Am. Chem. Soc. 2010, 132, 17658–17660. (27) Chapoy, A.; Makraoui, S.; Valtz, A.; Richon, D.; Mohammadi, A. H.; Tohidi, B. Solubility Measurement and Modeling for the System Propane Water from 277.62 to 368.16 K. Fluid Phase Equilib. 2004, 226, 213–220. (28) Battino, R. In Propane, Butane and 2-Methylpropane; Hayduk, W., Ed.; IUPAC Solubility Data Series; Pergamon, Oxford, 1986; Vol. 24. (29) Masterton, W. L. Partial Molal Volumes of Hydrocarbons in Water Solution. J. Chem. Phys. 1954, 22, 1830–1833. (30) Wagner, W.; Pruß, A. The IAPWS Formulation 1995 for the Thermodynamic Properties of Ordinary Water Substance for General and Scientific Use. J. Phys. Chem. Ref. Data 2002, 31, 387–535. (31) Revised Release on the IAPWS Formulation 1995 for the Thermodynamic Properties of Ordinary Water Substance for General and Sceintific Use, International Association for the Properties of Water and Steam (2014), available from www.iapws.org (accessed June 26, 2015). 20 ACS Paragon Plus Environment

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(32) Kobayashi, R.; Katz, D. L. Vapor-Liquid Equilibria for Binary Hydrocarbon-Water Systems. Ind. Eng. Chem. 1953, 45, 440–446. (33) Lemmon, E. W.; McLinden, M. O.; Wagner, W. Thermodynamic Properties of Propane. III. A Reference Equation of State for Temperatures from the Melting Line to 650 K and Pressures up to 1000 MPa. J. Chem. Eng. Data 2009, 54, 3141–3180. (34) Koga, K. Osmotic Second Virial Coefficient of Methane in Water. J. Phys. Chem. B 2013, 117, 12619–12624.

Acknowledgments The authors thank Professor Claudio Cerdeiri˜ na for discussions, for proposing the notation B ′′ in Eq. (6), and for having already noted years ago that to derive the activity of gaseous propane from its equation of state it is insufficient for the present purpose to stop at the second virial coefficient. We thank Naho Iwagaki and Tomonari Sumi for their help with the manuscript and the figures. KK acknowledges support for this work by a Grant-in-Aid for Scientific Research from MEXT, Japan. VH and BW acknowledge support from the National Science Foundation, grant no. CHE-1212543.



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ln(z2 / 2 )

(p

Page 22 of 25

p )v2 /kT

ln(1/ )

2

slope = 2B

v2 + kT

Figure 1: Sketch of a Watanabe-Andersen plot, from which Σ is obtained from the intercept on the vertical axis and B from the slope.

ξx

3.3

ξρ

Kobayashi and Katz (1953) Battino (1986) Chapoy et al. (2004)

3.29

3.29

3.28

3.28

-1 0 3.27

00

cm 3 /mo

-300 cm 3 /mol

-1 0

00

3.27

l

cm 3 /mo

l

- 3 0 0 cm 3 /mol

3.26

3.25

3.3

3.26

0

0.5

1

1.5

2

2.5

3.25 0

x2 /10-4

2

4

6

8

10

ρ2 (mol/m3)

Figure 2: Watanabe-Andersen plots for propane in water. Left: ξx versus mole fraction of propane; right: ξρ versus number density of propane. The activity of gaseous propane was computed from Eq. (25). The results would have been little different if we had used Eq. (24). Lines that correspond to B values of −300 cm3 /mol and −1000 cm3 /mol are shown.


12

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ln(z2 / 2 )


(p

p )v2 /kT

ln(1/ )

2

slope = 2B ACS Paragon Plus Environment

v2 + kT


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ξx

3.3

ξρ

Kobayashi and Katz (1953) Battino (1986) 3.29

Page 24 of 25

3.3

3.29

Chapoy et al. (2004)

3.28

3.28

-1 0 3.27

00

-1 0

cm 3 /mo l

00

3.27

-300 cm 3 /mol

cm 3 /mo

l

- 3 0 0 cm 3 /mol

3.26

3.26

3.25

3.25 0

0.5

1

1.5

2

2.5

0

x2 /10-4

2

4

6

ρ2 (mol/m3)

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8

10

12

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B from Equations of State

B from Experiments

1 ⇢2 (1 + 2B⇢2 + · · · ) (fixed T, µ1 ) ⌃ 1 z2 = ⇢2 (1 + 2B 0 ⇢2 + ...) (fixed T, p) ⌃ 1 z2 = ⇢2 (1 + 2B 00 ⇢2 + ...) (fixed T, ⇢1 ) ⌃ 1 z2 = ⇢2 (1 + 2Bcoex ⇢2 + ...) ⌃ (liquid-vapor coexistence at fixed T )

ln(z2 / 2 )

z2 =

(p

p )v2 /kT

ln(1/ )

with relations among B, B 0 , B 00 , Bcoex

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2

slope = 2B

v2 + kT

Deriving Second Osmotic Virial Coefficients from Equations of State

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