Osmotic Second Virial Coefficients of Aqueous Solutions from Two

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Osmotic Second Virial Coefficients of Aqueous Solutions from Two-Component Equations of State Claudio A Cerdeirina, and Benjamin Widom J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.6b09912 • Publication Date (Web): 01 Dec 2016 Downloaded from http://pubs.acs.org on December 12, 2016

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The Journal of Physical Chemistry

Osmotic Second Virial Coecients of Aqueous Solutions from Two-Component Equations of State Claudio A. Cerdeiri

†

n ˜ a∗ † ‡ , ,

and B. Widom

‡

Departamento de Física Aplicada, Universidad de Vigo Campus del Agua, Ourense 32004, Spain ‡Department of Chemistry, Baker Laboratory, Cornell University, Ithaca, New York 14853-1301 E-mail: [email protected]

Phone: +34 (0) 988 387217

Abstract Osmotic second virial coecients in dilute aqueous solutions of small non-polar solutes are calculated from three dierent two-component equations of state. The solutes are ve noble gases, four diatomics, and six hydrocarbons in the range C1 to C4 . The equations of state are modied versions of the van der Waals, Redlich-Kwong, and Peng-Robinson equations, with an added hydrogen-bonding term for the solvent water. The parameters in the resulting equations of state are assigned so as to reproduce the experimental values and temperature dependence of the density, vapor pressure, and compressibility of the solvent, the gas-phase second virial coecient of the pure solute, the solubility and partial molecular volume of the solute, and earlier estimates of the solutes' molecular radii. For all fteen solutes the calculations are done for 298.15 K, while for CH4 , C2 H6 , and C3 H8 in particular, they are also done as functions

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of temperature over the full range 278.15 K to 348.15 K. The calculated osmotic virial coecients are compared with earlier calculations of these coecients for these solutes and also with the results derived from earlier computer simulations of model aqueous solutions of methane. They are also compared with the experimental gas-phase second virial coecients of the pure gaseous solutes to determine the eect the mediation of the solvent has on the resulting solute-solute interactions in the solution.

1 INTRODUCTION The osmotic second virial coecient B is the analog for a dilute solution of the gas-phase second virial coecient in a dilute gas. In the gas, the sign and magnitude of the coecient are a measure of the nature and strength of the intermolecular interactions as manifested in, and measured by, deviations from the ideal-gas law. Analogously, the osmotic B is a measure of the solute-solute interactions, now as mediated by the solvent and as measured by the deviations from the ideal-dilute solution laws: the van 't Ho osmotic pressure law and Henry's law. It is these solvent-mediated interactions and the special properties of liquid water as the mediating solvent that are responsible for the widely recognized hydrophobic phenomena. By its denition, B is the coecient of the second-order term in the expansion of the osmotic pressure Π or solute activity z2 in powers of the concentration of the solute at xed temperature T and xed chemical potential µ1 of the solvent:

Π = ρ2 kT (1 + Bρ2 + ...)

(T, µ1 fixed)

(1)

1 ρ2 (1 + 2Bρ2 + ...) Σ

(T, µ1 fixed)

(2)

z2 =

where k is the Boltzmann constant, ρ2 is the number density of the solute in the solution, z2 is dened so as to be the number density if the solute were an ideal gas at that density, Σ is a T and µ1 -dependent proportionality factor closely related to the Ostwald absorption coecient 2


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and Henry's-law constant for the solute in the solution, and B is the virial coecient in question. The solute-solute interactions measured by B are predominantly repulsive if B is positive, attractive if B is negative. That follows also from its connection to the solute-solute pair correlation function at innite dilution, h22 (r), as seen in the McMillan-Mayer 1,2 and Kirkwood-Bu 36 solution theories:

1 B=− 2

Z h22 (r)dτ

(3)

where the integration is over all space with r the varying distance between solute molecules in the integration and with dτ the element of volume. Thus, positive solute-solute correlations contribute negatively to B , negative correlations contribute positively. In deriving the osmotic second virial coecient from eq 3 one evaluates h22 (r), as has been done recently via simulations to obtain B(T ). 710 It may also be calculated from a two-component equation of state, solvent and solute. The present work illustrates such calculations of B(T ) for solutions of non-polar solutes in water, with each of three dierent equations of state. To do so, one rst evaluates a coecient B 00 , which is the analog of B in eq 2 with the expansion at xed density ρ1 of the solvent instead of xed µ1 : 11

z2 =

1 ρ2 (1 + 2B 00 ρ2 + ...) Σ

(T, ρ1 fixed)

(4)

That is done because the equations of state are usually expressed as functions of T , ρ1 , ρ2 rather than T , µ1 , ρ2 . Then B is obtained from B 00 by the thermodynamic identity 11

B = B 00 −

1 (v2 − kT χ)2 2kT χ

(5)

where v2 and χ are, respectively, the partial molecular volume of the solute at innite dilution and the compressibility of the solvent, both evaluated at the given T and ρ1 . In

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most applications, v2 is positive and much greater than kT χ, and the two terms that are subtracted on the right-hand side of eq 5 are both positive and may largely cancel, while their dierence B may be of either sign. 11 In the following section 2, we introduce the equations of state with references to the literature and we outline the calculations. The details of the equations of state and of the calculations are relegated to the Appendix.

2 METHODS The equations of state we use here are 2-component variants of those of van der Waals, Redlich-Kwong, and Peng-Robinson. In their form as equations for a single component, 12 they contain parameters a and b. They are then converted to two-component equations, as has been done in the past, 13,14 where with the two components 1 and 2, the two original a and

b then become six: the molecular size and repulsion parameters b11 , b12 , b22 and attraction parameters a11 , a12 , a22 . Additionally, we replace the form of the repulsive terms in the equations of state by the Carnahan-Starling 3-dimensional hard-sphere term, 12 which is an improvement over its original 1-dimensional van der Waals form. In particular, its account of the compressibility, with reasonable size parameters, is more realistic than that with the original 1-dimensional version; or, alternatively, the size parameters required in matching the experimental compresibility are more reasonable. Finally, for the solvent water, as in ref 15, we add to the original free energy in each equation of state the hydrogen-bonding term of Poole et al. 16 Such a term (see Appendix) is characterized by four parameters, two for volumetric eects, ρHB and σHB , and another two for energetic and entropic eects, εHB and Ω, respectively. 15 For the solvent, the parameters a11 , b11 , and Ω are obtained, with εHB , ρHB , and σHB xed, by requiring the equation of state to yield the experimental density, vapor pressure, and compressibility of water at four selected temperatures, 278.15 K, 298.15 K, 323.15 K,

4


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and 348.15 K, such data 17 being displayed in Table 1. One thus gets values for the three adjustable parameters at each temperature (see Supporting Information). Interestingly, b11 proves to be nearly constant for all the equations of state used. For example, with the van der Waals equation of state, modied as described above, the resulting b11 in units of

cm3 mol−1 is 7.99, 8.04, 8.04, and 8.00 at T = 278.15 K, 298.15 K, 323.15 K, and 348.15 K, respectively. With the identication b11 = πσ13 /6, where σ1 denotes the hard-core molecular diameter of the solvent, this translates to σ1 = 2.9 Å for water, which may be compared, for example, with the value 2.75 Å in ref 18. By contrast, a11 proves to be noticeably temperature dependent: in units of J m3 mol−2 , it is 0.37, 0.41, 0.43, and 0.45. Constant b and a temperature-dependent a are common features of equations of state such as those of Redlich-Kwong (as modied by Soave) and Peng-Robinson. 12 Table 1: First column, the temperature T (K). Second, third, and fourth columns, literature values 17 for the density ρ (mol m−3 ), vapor pressure pvap (105 Pa), and compressibility χ [(1012 Pa)−1 ] of pure water.

T 278.15 298.15 323.15 348.15

ρ 55504 55342 54842 54110

pvap 0.0087 0.0317 0.1234 0.3856

χ 491.8 452.6 441.9 456.4

the equations of state referred to as van der Waals, Redlich-Kwong, and Peng-Robinson will always mean the twocomponent versions of those equations with their hard-sphere terms modied as described above and with the hydrogen bonding term incorporated for the solvent . Hereafter, unless otherwise specied, it is to be understood that

The solute-solute interaction parameter b22 is related to the solute diameter σ2 . We take it to be temperature-independent and x it from the work by Bondi 19 on van der Waals volumes (vvdW ) and radii, except for the noble gases, for which we use the molecular diameters given by Pierotti. 20 In the former case we use b22 = vvdW , in the latter b22 = πσ23 /6. Then we x a22 as a function of temperature by tting the experimental gas-phase second 5



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Table 2: First column, the solutes. Second column, the temperature T (K). Third and fourth columns, literature values for the partial molecular volume v2 (cm3 mol−1 ) and solubility (dimensionless) of solute in solvent. Sources of v2 data are indicated individually with the reference numbers, while Σ data were taken from ref 23. Solute He Ne Ar Kr Xe H2 N2 O2 CO CH4

C2 H6

C2 H4 C3 H8

n-C4 H10 i-C4 H10

T 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 278.15 298.15 323.15 348.15 278.15 298.15 323.15 348.15 298.15 278.15 298.15 323.15 348.15 298.15 298.15

v2 24.626 28.0* 32.726 38.028 43.028 23.126 33.126 32.126 37.327 34.724 35.624 36.724 37.824 50.424 51.724 53.324 54.924 45.426 54.025 66.825 75.525 80.525 76.626 83.126

Σ 0.0094 0.0110 0.0341 0.0609 0.1052 0.0191 0.0159 0.0311 0.0234 0.0504 0.0340 0.0257 0.0232 0.0823 0.0454 0.0295 0.0249 0.1163 0.0735 0.0367 0.0225 0.0189 0.0298 0.0199

*Interpolated from the v2 values of He and Ar.

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virial coecient 21,22 of the pure solute to

vdW,PR Bgas (T ) = 4b22 − a22 /kT,

RK Bgas (T ) = 4b22 − a22 /kT 3/2

(6)

where superscripts vdW, RK, and PR refer to van der Waals, Redlich-Kwong and PengRobinson. Thus, we have a xed b22 and a temperature-dependent a22 . For example, with the van der Waals equation, a22 in units of J m3 mol−2 is 1.39, 1.34, 1.28, and 1.24 at 278.15 K,

298.15 K, 323.15 K, and 348.15 K, respectively. The cross parameters a12 and b12 , once a11 and b11 have been determined, are obtained, as has been done previously, 14 by tting the solubility and partial molecular volume of the solute as calculated from the 2-component equation of state, which expresses them in terms of a11 , b11 , a12 , b12 , to the experimental solubility 23 Σ (essentially the Ostwald absorption coecient) and partial molecular volume 2428 v2 as functions of temperature. Table 2 shows the values of Σ and v2 used in the calculations while the results for the cross parameters can be found in the Supporting Information. The seven parameters a11 , b11 , Ω, a12 , b12 , a22 , b22 at each temperature for each of the three equations of state considered are thus found, and each equation of state is now guaranteed to reproduce, as functions of the temperature, the experimental density, vapor pressure, and compressibility χ of water, the solubility (related to Σ) and partial molecular volume v2 of the solute in water, and the gas-phase second virial coecient of the pure solute, over the temperature range 278.15 K to 348.15 K. The coecient B 00 in eq 4 may thus be found from each of the equations of state considered, and then nally the osmotic second virial coecient B , as sought, from eq 5. The details of the calculations are in the Appendix. The results are summarized in the following section.

3 RESULTS The results are rst illustrated for propane in water. 7



In Figure 1, the lowest set of points is as calculated from the original two-component van der Waals equation in which the 1-dimensional hard-sphere form of the repulsive term has been retained. They were obtained using the calculation procedure outlined in section 2 and are shown here for illustration. A second set, systematically less negative than the former by about 300 cm3 mol−1 , corresponds to the van der Waals equation in which the 1-dimensional term has been replaced by its Carnahan-Starling form. The next set, less negative than the previous ones, is obtained from the Redlich-Kwong equation while the uppermost set of points is from Peng-Robinson. This gure illustrates the sensitivity of the resulting B to the particular form of the equation of state. 0

B (cm3 mol-1)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

-200

-400

-600

-800 275

300

325

350

T (K)

Figure 1: The osmotic second virial coecient B for propane in water as a function of temperature, calculated with the two-component van der Waals equation in which the term for the hard-core repulsion is of its original one-dimensional form (diamonds), with that term replaced by a term of Carnahan-Starling form (circles), with the Redlich-Kwong equation of state (squares), and with the Peng-Robinson equation of state (triangles). For all cases, the hydrogen-bonding term has been included for the water. The solid line is the gas-phase second virial coecient Bgas of propane.

We have observed that the calculated osmotic second virial coecient B for the lighter hydrocarbons methane, ethane, and propane, for which experimental partial molecular volumes over a wide temperature range are available, 24,25 decreases with increasing temperature

T . This is true with all three of the equations of state for which B was calculated, and was illustrated in Figure 1 for propane but now also for methane in Figure 2, which contains 8


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results from the van der Waals, Redlich-Kwong, and Peng-Robinson equations. There is some suggestion that B may reach a minimum and then increase at higher T , reecting that which we observe in the term −(1/2kT χ)(v2 − kT χ)2 in eq 5. This is concluded from Figure 3, where that term is called B − B 00 , from eq 5, and is plotted along with B 00 . These B 00 and B − B 00 are the two contributions to B . The rst is positive and the second negative, so B calculated in this way results from a balance between them. Since the resulting B was seen in Figure 2 to become more negative with increasing T , it shows that it is the negative

B − B 00 term, with its temperature minimum, that dominates the balance. It is this that suggests that the temperature minimum in B − B 00 (Figure 3) may ultimately be reected in that of B itself at some higher T . It is to be recalled that since all the equations of state used here have been designed to reproduce the experimental v2 and χ as functions of the temperature, the B − B 00 in Figure 3 is an experimental quantity; it is B 00 that depends on the choice of equation of state. To further explore the origin of the temperature minimum in B − B 00 , we rst note from eq 5 that B − B 00 ≈ −(1/2kT χ)v22 since v2 is typically much greater than kT χ. On the other hand, Figure 3 shows that the factor v22 /T decreases regularly with T for aqueous solutions of methane. It is then the fact that χ(T ) displays a minimum which leads to the B − B 00 minimum. Further, under the assumption that v22 /T continues to decrease with T in a regular manner, one expects no B −B 00 minimum for the hypothetical case of a solvent with standard behavior, that is, one with a monotonously increasing χ(T ). Therefore, it is concluded that the B − B 00 minimum shown in Figure 3 is rooted in the unusual thermodynamics of water. Other connections of this sort between the thermodynamics of water and that of hydrophobic hydration have been noted previously (see, e.g. ref 15). In Figure 2, with B calculated with the van der Waals, Redlich-Kwong, and PengRobinson equations, the pattern of behavior noted for propane in Figure 1 is now seen to hold also for methane: the values of B calculated with the van der Waals, Redlich-Kwong, and Peng-Robinson equations, increase (become less negative) in that order. In addition,

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30

B (cm3 mol-1)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

0

-30

-60 275

300

325

350

T (K)

Figure 2: B for methane in water as a function of temperature, calculated with the van der Waals equation (circles), with the Redlich-Kwong equation (squares), and with the PengRobinson equation (triangles). Also shown are values obtained by Koga 7 by computer simulation (diamonds) and the gas-phase second virial coecient Bgas of methane (solid line).

in Figure 2, we compare our values of B calculated with the Redlich-Kwong equation of state with those obtained by Koga from computer simulations. 7 Computer simulations by Ashbaugh et al. 9 led to values that roughly parallel those of Koga but are generally higher (less negative) by 20 − 30 cm3 mol−1 . In Table 1 are the values of B calculated for 15 solutes from each of the three equations of state considered (all modied as described in section 2). (An earlier such tabulation of

B calculated with other equations of state, mostly for higher hydrocarbons, is in Liu and Ruckenstein. 29 ) The data from Table 1 at 298.15 K that were calculated with the PengRobinson equation of state are plotted in Figure 4. In analogy with a plot by Watanabe and Andersen, 30 the present B is plotted vs. the logarithim of the inverse solubility, ln(1/Σ); but by contrast with the former, where they refer to the same solutes, the B data here show little correlation with Σ, span a narrower range, and are of smaller magnitude. For the last three entries in the table, which are the only ones with conspicuously negative B (and are not among the data in ref 30), the osmotic coecient calculated here is indeed more negative the less soluble the solute, which does agree with the pattern seen in ref 30. As discussed earlier, 11 each of the two dierent methods of deriving B from experimental data has its own 10


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600 4.32 300 4.23

0 -300

4.14

-600

v22 / T (cm6 mol-2 K-1)

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B'', B - B'' (cm3 mol-1)

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4.05 280 320 360 280 320 360

T (K)

T (K)

Figure 3: B 00 (squares), B − B 00 (circles), and v22 /T (triangles) for methane in water as calculated with the Peng-Robinson equation of state.

uncertainties. The table provides further support for the earlier observation (Figures 1 and 2) that the results for B from the Peng-Robinson, Redlich-Kwong, and van der Waals equations generally become more negative in that order. As to the relation between the osmotic second virial coecient B in aqueous solutions of non-polar solutes and the pure solute gas-phase second virial coecient Bgas , one generally expects B to be the more negative of the two, at least at high temperatures where the solutes are then most hydrophobic, and that the opposite would be the case at low enough temperatures (see Figures 1 and 2 for illustration). It is seen from the entries in the table that when B is calculated with the van der Waals equation, it is generally more negative than Bgas , with only a few exceptions for the heaviest solutes, while the exceptions are generally more numerous for the calculations with Redlich-Kwong and more so with Peng-Robinson, in accord with the order noted above. Finally, for all three of the equations of state, B is more negative than Bgas for He, Ne, and H2 . These three have relatively small solute-solute and solute-solvent attractive interactions (see Tables S.2 and S.3 in the Supplementary Information). Indeed, in that respect, since there is no dierence at all among those three equations as concerns the solute when a22 = a12 = 0, it suggests that the water-mediated interactions of hard-sphere solutes would be attractive, which is in accord with what is found in simulations. 8 It agrees also with 11



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Table 3: First column, the solutes. Second column, the temperature T (K). Third, fourth, and fth columns, B (cm3 mol−1 ) as calculated with the van der Waals, Redlich-Kwong, and Peng-Robinson 2-component equations of state, respectively, with the Carnahan-Starling hard-sphere term and hydrogen-bonding term for the water, incorporated in all. Sixth column, the gas-phase second virial coecient Bgas (cm3 mol−1 ) of the pure solutes at temperature T . Sources of Bgas data are indicated individually with the reference numbers. Solute

T

He Ne Ar Kr Xe H2 N2 O2 CO CH4

298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 278.15 298.15 323.15 348.15 278.15 298.15 323.15 348.15 298.15 278.15 298.15 323.15 348.15 298.15 298.15

C2 H6

C2 H4 C3 H8

n-C4 H10 i-C4 H10

B vdW RK -74 -73 -92 -91 -60 -42 -84 -56 -130 -82 -43 -39 -23 4 -50 -29 -52 -28 -39 -2 -48 -14 -55 -23 -57 -27 -179 -100 -185 -111 -188 -121 -185 -123 -98 -37 -311 -163 -406 -281 -465 -359 -479 -386 -695 -494 -720 -539

12

Bgas PR -72 -90 -33 -42 -57 -38 17 -19 -16 16 4 -7 -12 -60 -74 -87 -91 -6 -88 -218 -306 -338 -394 -448


1422 1122 −1622 −5122 −12822 1522 −522 −1622 −922 −5121 −4321 −3421 −2721 −21321 −18421 −15521 −13221 −14122 −45321 −39021 −32821 −27921 −70622 −62822

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200

B (cm3 mol-1)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0

-200

-400

-600 2

3

4

5

ln (1/Σ)

Figure 4: B as a function of the logarithm of the reciprocal solubility, ln(1/Σ), obtained with the Peng-Robinson equation at 298.15 K for aqueous solutions of the fteen solutes listed in Table 1. Circles correspond to noble gases (from right to left, He, Ne, Ar, Kr, and Xe), squares to diatomics (from right to left, N2 , O2 , CO, and H2 ), and triangles to hydrocarbons (from right to left, i- C4 H10 , n-C4 H10 , CH4 , C3 H8 , C2 H6 , and C2 H4 ). Data for the solubility were taken from ref 23.

recent numerical studies of the eect of solute-solute attractive interactions 10 in making the forces between solutes in water less attractive or even repulsive, along lines suggested also by Ben-Amotz. 31

4 SUMMARY AND DISCUSSION The osmotic second virial coecients B for non-polar solutes in water were calculated as functions of temperature from three dierent equations of state. The results are tabulated in Table 3. They are also compared with the gas-phase second virial coecients Bgas for the pure gaseous solutes. The dierences B − Bgas would then be quantitative measures of the eect of the solvent water in mediating the solute-solute interactions. In this respect there are some dierences among the equations of state, but generally B is more negative than Bgas for the smallest solutes, which are those with the weakest solute-solute and solute-solvent attractions. It is noted that this is then consistent with the nding that hard-sphere solutes in water attract 8 while direct solute-solute attractions may even be opposed through the

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mediation of the water solvent. 10,31 There is a suggestion that followed from Figure 3 that the osmotic B may prove to have a temperature minimum at some higher temperature, reecting that in the compressibility of water. This is still speculative but would be of interest if it proved true, as another example of that property of water manifesting itself. 15 It is perhaps surprising that there have proved to be such signicant quantitative dierences among the values of B calculated from the three dierent equations of state (Table 3), since all three reproduce the experimental density, vapor pressure, and compressibility of water, the solubility and partial molecular volume of the various solutes in water, and the experimental Bgas of the pure solutes, all as functions of the temperature over the range 278.15 K to 348.15 K. It must be concluded that the osmotic B is very sensitive to the ner details of the equation of state from which it is derived. This means the sensitivity of B 00 , since the dierence B − B 00 is an experimental quantity that the equations of state reproduce exactly. This sensitivity of B 00 may be understood from its denition in eq 4 as the limit of (z2 Σ − ρ2 )/2ρ22 as ρ2 → 0 at xed T and ρ1 . All three of the equations of state adopted here yield the same values of Σ derived from the experimental solubilities, and may be expected to yield nearly the same values for z2 , both of them as functions of ρ2 . But even very small dierences among the values of z2 Σ as functions of ρ2 would be greatly magnied in B 00 because z2 Σ approaches exact cancellation with ρ2 in the limit. There are indeed other methods of obtaining B from experiment, other than from equations of state; for example, from gas-solubility measurements as functions of the pressure at liquid-vapor coexistence combined with the independently measured properties of the vapor phase; but these are subject to their own errors and uncertainties. 11

14


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A APPENDIX: THERMODYNAMIC DERIVATION OF B FROM EQUATIONS OF STATE We outline here the thermodynamics relevant to the derivation of B from the van der Waals, Redlich-Kwong, and Peng-Robinson equations of state, modied as specied in section 2. The procedure has been set out earlier 11,14 and leads to eq 5, which indicates that B is obtained from the isothermal compressibility of the pure solvent χ, the partial molecular volume of the solute at innite dilution v2 , and the coecient B 00 in eq 4. They are dened as:

−1 1 ∂ρ 1 ∂p χ= = ρ ∂p T,N ρ ∂ρ T,N ∂V ∂p v2 = =χ (ρ2 → 0) ∂N2 T,p,N1 ∂ρ2 T,ρ1 Σ ∂(z2 /ρ2 ) 00 B = (ρ2 → 0) 2 ∂ρ2 T,ρ1

(A.1) (A.2) (A.3)

where ρ = N/V stands for the number density, 1 and 2 refer to solvent and solute, Σ is a measure of the solubility of the solute in the solvent, and z2 = eµ2 /kT /Λ32 is solute's activity, √ with µ and Λ = h/ 2πmkT the chemical potential and (by convention, for monatomic solutes) the thermal de Broglie wavelength, respectively. The p = p(ρ, T ) 1-component equation of state of the pure solvent is required to obtain χ. The partial molecular volume then follows from the 2-component equation of state, p = p(ρ1 , ρ2 , T ). As for B 00 we take into account that 14

z2 ∂(f − fi.g. ) kT ln = ρ2 ∂ρ2 T,ρ1

(A.4)

where f = F/V is the Helmholtz free energy per unit volume and fi.g. its ideal-gas part. This implies that, as expressed in A3, B 00 is obtained from the second ρ2 -derivative of f − fi.g. and from

1 ∂(f − fi.g. ) Σ = exp − kT ∂ρ2 15

(ρ2 → 0) T,ρ1


(A.5)


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Page 16 of 29

which follows from eqs 4 and A4. Clearly, f −fi.g. for the two-component solution is required. On the other hand, as noted in section 2, the gas-phase second virial coecient of the pure solute, Bgas , is also needed. For the van der Waals equation of state with the 1-dimensional hard-sphere term replaced by the Carnahan-Starling 3-dimensional form, Bgas comes from a low-density expansion of

p kT

vdW

4y − 2y 2 aρ2 =ρ 1+ − (1 − y)3 kT

(A.6)

where a = a22 and y = b22 ρ, with b22 = πσ23 /6, σ2 being the hard-core molecular diameter of the solute. This leads to the result in eq 6. For the solvent water we have, in addition, a hydrogen-bonding contribution: 15

pvdW

4y − 2y 2 Ω+q 2 2 0 = kT ρ 1 + − aρ − 2kT ρ g0 (ρ) ln (1 − y)3 Ω+1

(A.7)

2 , and where now a = a11 , y = b11 ρ, q = exp(εHB /kT ), g00 (ρ) = −2(ρ − ρHB )g0 (ρ)/σHB

g0 (ρ) = exp{−[(ρ − ρHB )/σHB ]2 }, with εHB the energy of hydrogen bonding, Ω accounting for orientational constraints, and ρHB and σHB characterizing volumetric eects. These are the four characteristic parameters of the hydrogen-bonding contribution, which, naturally, also enter in the free energy: 15

4y − 3y 2 − aρ2 (1 − y)2 n o − 2kT ρ g0 (ρ) ln(Ω + q) + [1 − g0 (ρ)] ln(Ω + 1)

fvdW − fi.g. = kT ρ

(A.8)

From eqs A1 and A7 one nds the compressibility of pure solvent water:

χ−1 vdW

8y − 2y 2 Ω+q 2 0 2 00 = kT ρ 1 + − 2aρ − 2kT ρ(2ρg0 + ρ g0 ) ln (1 − y)4 Ω+1

(A.9)

2 2 with g000 (ρ) = −2g0 (ρ)[1 − 2(ρ − ρHB )2 /σHB ]/σHB . (This corrects the expression provided in

ref 15.) 16


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For 2-component solutions, a and b in the two rst terms of the right-hand side of eqs A7 and A8 are considered functions of composition. Specically, as has been adopted earlier, 13,14

a=

a11 ρ21 + 2a12 ρ1 ρ2 + a22 ρ22 (ρ1 + ρ2 )2

(A.10)

b=

b11 ρ21 + 2b12 ρ1 ρ2 + b22 ρ22 (ρ1 + ρ2 )2

(A.11)

For the last, hydrogen-bonding term in eqs A7 and A8 we just replace ρ by ρ1 . 15 Then, from eqs A2, A3, and A5 one nds:

v2 kT χ

=1+ vdW

ΣvdW

00 BvdW =

∂y 4y + 4y 2 − 2y 3 2a12 ρ 4y − 2y 2 + − (1 − y)3 ∂ρ2 b11 (1 − y)4 kT

3y 3 − 5y 2 + 4b12 ρ(2 − y) 2a12 ρ = exp − + (1 − y)3 kT

(A.12)

(A.13)

∂y 3y 3 − 5y 2 + 4b12 ρ(2 − y) a22 ∂y 9y 2 − 10y − 4b12 ρ 2b22 (2 − y) + + 3 − (A.14) ∂ρ2 2(1 − y)3 (1 − y)3 ∂ρ2 2(1 − y)4 kT

where y = b11 ρ1 and ρ = ρ1 since these properties are evaluated in the ρ2 → 0 limit while also in this limit

∂y = 2b12 − b11 ∂ρ2

(A.15)

The algebraic expressions in A12 to A14 do not contain terms arising from the hydrogenbonding contribution. As noted earlier, 15 this is because the part of the free energy due to hydrogen bonding does not depend on ρ2 . Indeed, this contribution, which describes the unusual thermodynamics of water including the maximum of the density as a function of temperature, enters numerically in A12 to A14 via the nonmonotonic temperature dependence of y and ρ. Following the procedure outlined above for the van der Waals equation, results for the

17



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Page 18 of 29

Redlich-Kwong equation of state are obtained. For the solute we have

p kT

RK

4y − 2y 2 aρ2 =ρ 1+ − (1 − y)3 kT 3/2 (1 + y)

(A.16)

For the solvent

pRK

Ω+q 4y − 2y 2 aρ2 2 0 = kT ρ 1 + − 1/2 − 2kT ρ g0 (ρ) ln (1 − y)3 T (1 + y) Ω+1

(A.17)

4y − 3y 2 aρ ln(1 + y) − (1 − y)2 bT 1/2 n o − 2kT ρ g0 (ρ) ln(Ω + q) + [1 − g0 (ρ)] ln(Ω + 1)

(A.18)

fRK − fi.g. = kT ρ

Then

χ−1 RK

aρ2 (2 + y) Ω+q 8y − 2y 2 0 2 00 − 2kT ρ(2ρg0 + ρ g0 ) ln − 1/2 = kT ρ 1 + (1 − y)4 T (1 + y)2 Ω+1

(A.19)

For the two-component solution

v2 kT χ

RK

∂y 4y + 4y 2 − 2y 3 4y − 2y 2 + =1+ (1 − y)3 ∂ρ2 b11 (1 − y)4 2a12 ρ(1 + y) − a11 (∂y/ρ2 )ρ2 − kT 3/2 (1 + y)2

3y 3 − 5y 2 + 4b12 ρ(2 − y) 1 ΣRK = exp − + 3/2 3 (1 − y) kT

a11 ∂(a/b) +ρ b11 ρ2

(A.20)

a11 (∂y/ρ2 )ρ ln(1+y)+ b11 (1 + y) (A.21)

∂y 9y 2 − 10y − 4b12 ρ 2b22 (2 − y) + ∂ρ2 2(1 − y)3 (1 − y)3 ∂y 3y 3 − 5y 2 + 4b12 ρ(2 − y) +3 ∂ρ2 2(1 − y)4 1 1 ∂(a/b) ∂ 2 (a/b) a11 ∂(a/b) ∂y 1 − 2 +ρ ln(1 + y) + +ρ kT 3/2 2 ∂ρ2 ∂ρ22 b11 ∂ρ2 ∂ρ2 1 + y 2 2 a11 ρ ∂ y 1 ∂y 1 + − 2 2b11 ∂ρ2 1 + y ∂ρ2 (1 + y)2

00 BRK =

18


(A.22)

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where, as before, y = b11 ρ1 , ρ = ρ1 , a and b are given by eqs A10 and A11, while

∂ 2y 2(b11 + b22 − 2b12 ) = 2 ∂ρ2 ρ

(A.23)

2[(a22 b11 − a11 b22 )b11 − 4(a12 b11 − a11 b22 )b12 ] ∂ 2 (a/b) = 2 ∂ρ2 b11 y 2

(A.24)

∂(a/b) 2(a12 b11 − a11 b12 ) = ∂ρ2 b11 y

with the results for these derivatives corresponding to the ρ2 → 0 limit. Finally, for the Peng-Robinson equation of state we have for the solute

p kT

PR

4y − 2y 2 aρ2 =ρ 1+ − (1 − y)3 kT (1 + 2y − y 2 )

(A.25)

For the solvent

pPR

4y − 2y 2 aρ2 Ω+q 2 0 = kT ρ 1 + − 2kT ρ g0 (ρ) ln − (1 − y)3 1 + 2y − y 2 Ω+1

fPR − fi.g.

χ−1 PR

(A.26)

√ aρ 1/y + 1 + 2 4y − 3y 2 √ − √ ln = kT ρ (1 − y)2 (A.27) 2 2b 1/y + 1 − 2 n o − 2kT ρ g0 (ρ) ln(Ω + q) + [1 − g0 (ρ)] ln(Ω + 1)

Ω+q 8y − 2y 2 2aρ2 (1 + y) 0 2 00 − 2kT ρ(2ρg0 + ρ g0 ) ln = kT ρ 1 + − (1 − y)4 (1 + 2y − y 2 )2 Ω+1

(A.28)

And for the two-component solution v2 4y − 2y 2 ∂y 4y + 4y 2 − 2y 3 2a12 ρ(1 + 2y − y 2 ) − 2a11 (∂y/ρ2 )ρ2 (1 − y)] = 1+ + − kT χ PR (1 − y)3 ∂ρ2 b11 (1 − y)4 kT (1 + 2y − y 2 )2 (A.29)

3y 3 − 5y 2 + 4b12 ρ(2 − y) (1 − y)3 √ √ (A.30) 1 a11 ∂(a/b) 1/y + 1 + 2 2 2a11 (∂y/ρ2 )ρ √ + √ +ρ ln + b11 ρ2 b11 (1 + 2y − y 2 ) 2 2kT 1/y + 1 − 2

ΣPR = exp −

19



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∂y 9y 2 − 10y − 4b12 ρ 2b22 (2 − y) + ∂ρ2 2(1 − y)3 (1 − y)3 ∂y 3y 3 − 5y 2 + 4b12 ρ(2 − y) +3 ∂ρ2 2(1 − y)4 √ 1 1 ∂(a/b) ∂ 2 (a/b) 1/y + 1 + 2 √ 2 √ − +ρ ln kT 4 2 ∂ρ2 ∂ρ22 1/y + 1 − 2 a11 ∂(a/b) ∂y 1 + +ρ b11 ∂ρ2 ∂ρ2 1 + 2y − y 2 2 1 2(1 − y) a11 ρ ∂ 2 y ∂y + − 2b11 ∂ρ22 1 + 2y − y 2 ∂ρ2 (1 + 2y − y 2 )2

Page 20 of 29

00 BPR =

(A.31)

SUPPORTING INFORMATION Parameters of equations of state for pure solvent water and pure solute as well as cross parameters.

Acknowledgement The authors are grateful to Dr. Katerina Zemánková, who collaborated in the preparation of the manuscript. C. A. C. acknowledges support from the Spanish Ministry of Education, Culture, and Sports, grant no. PRX15-00152. B. W. acknowledges support from the U. S. National Science Foundation, grant no. CHE-1212543.

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Peng-Robinson

B (cm3 mol-1)

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4.32 300 4.23

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Osmotic Second Virial Coefficients of Aqueous Solutions from Two

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