J. Phys. Chem. B 2006, 110, 24077-24082
24077
A Fluctuation Theory Analysis of the Salting-Out Effect Robert M. Mazo* Institute of Theoretical Science and Department of Chemistry, UniVersity of Oregon, Eugene, Oregon 97403 ReceiVed: September 1, 2006
An analysis of the salting-out, or Sechenow, effect is given in terms of Kirkwood-Buff, or fluctuation, integrals. The analysis is formally exact but cannot easily be applied in its original form. When the solute that is being salted out is sparingly soluble, simplifications arise and the theory can be used to compute one of the Kirkwood-Buff integrals which is otherwise difficult to obtain.
I. Introduction When an electrolyte is added to a saturated solution of a nonelectrolyte in water, the solubility of the nonelectrolyte (usually) decreases. This is know as the “salting-out” effect or Sechenow effect. The effect is also present when the solute is a weak electrolyte instead of a nonelectrolyte. The name is a bit of a misnomer, since in occasional cases the solubility increases; i.e., there is a “salting-in” effect. Furthermore, Sechenow,11 apparently the first person to investigate this effect, confined his investigations to the case of a single gas dissolved in water, namely carbon dioxide. However, the names are well established and it would serve no rational purpose to attempt to alter them. Much of the early experimental work has been reviewed by Randall and Failey22 and is described in the monograph by Harned and Owen.33 The effect is usually ascribed to the preferential solvation of the electrolyte by the solvent with respect to that of the solute. Debye44 and Debye and McCauley55 presented theories of the salting-out effect based on this idea. Further theoretical considerations based on scaled particle theory have been published by Masterton.66 A treatment for gaseous solutes based on perturbation theory has been given by Tiepel and Gubbins.7 The object of this paper is to present a formally exact theory of the salting-out effect based on the fluctuation, or Kirkwood-Buff8 theory of solutions. Fluctuation theory has been previously used by Shulgin and Ruckenstein9 for gaseous solutes. These authors also considered the solubility of proteins in mixed solvents from this point of view;10 however, electrolytes were not considered as cosolvents. The Kirkwood-Buff theory (hereafter abbreviated as K-B) shows how to express the thermodynamic functions of an r-component solution in terms of the elements of an r × r matrix B
BRβ ) cRδRβ + cRcβGRβ GRβ )
∫[gRβ(r) - 1] dV
(1)
cR is the concentration of species R, and gRβ is the pair correlation function between species R and β. The relation to thermodynamics is given by * To whom correspondence should be addressed. E-mail: mazo@ uoregon.edu. Phone: (541) 346-5224. Fax: (541) 344-5217.
( )
1 ∂µR kT ∂Nβ
1 ) (B-1)Rβ V T,V,N′
(2)
A physical meaning may be given to the elements of B
1 BRβ ) (〈NRNβ〉 - 〈NR〉〈Nβ〉) V
(3)
That is, BRβ is the relative fluctuation of the particle numbers of the two species R and β in an open system (the theory is based on the grand canonical ensemble). Some applications and discussion of K-B theory can be found in the book of Matteoli and Mansoori.11 We shall see below that it is possible to express the Sechenow coefficient (to be defined below) exactly in terms of the G integrals, or, as they are sometimes called, fluctuation integrals or Kirkwood-Buff integrals. This yields a formally exact theory of salting-out. Although Smith et al.12-14 have acheived considerable success in computing K-B integrals using the molecular dynamics method, it is fair to say that the K-B integrals are not easy to compute from molecular properties. Hence the formal elegance of the result hides the difficulty of comparison with experiment. However, one can take a different point of view. One of the goals of a chemical theory is, of course, to predict experimental results. But another goal is to relate macroscopic information, gained in the laboratory, to microscopic information about the structure and dynamics of the system on the molecular scale. In the present case we are concerned with thermodynamic information which will teach us nothing about dynamics, but we can learn about statics. What we shall emphasize in this paper is how a measurement of the Sechenow coefficient can lead us to valuable information on a microscopic picture of preferential interaction. Ben Naim11,15 has shown that one can invert K-B theory. That is, instead of regarding the equations of K-B as giving the thermodynamic functions in terms of the fluctuation integrals, one can regard the theory as giving the fluctuation integrals as functions of the thermodynamic quantities. He carried this out explicitly for a two component system; it can also be done in the general case, although the algebra becomes more complicated the greater the number of components. Our systems have three components, and we are suggesting that the Sechenow coefficient may be conveniently used in place of one of the more conventional chemical potential derivatives in the inversion of K-B theory.
10.1021/jp0656936 CCC: $33.50 © 2006 American Chemical Society Published on Web 10/26/2006
24078 J. Phys. Chem. B, Vol. 110, No. 47, 2006
Mazo
We conclude this introduction by noting that K-B theory in its original form8 is not applicable to solutions of electrolytes. This was, to the best of our knowledge, first remarked by Friedman and Ramanathan.16 It comes about because the electroneutrality condition can be expressed as ∑βzβBRβ ) 0. (zβ is the charge on an ion of type β.) This implies that the matrix B is singular; its inverse does not exist. It is not clear from the original paper8 whether the authors were aware of this problem. However, it has been successfully overcome by several authors.16-19 There are other methods that may also be used. The problem is also discussed in a recent monograph by Ben Naim20
to discuss the modification of K-B theory necessary when one or more of the components are charged. This manifests itself in the singular nature of the B matrix which arises because of the electroneutrality conditions, ∑zβBRβ ) 0. The singular K-B matrix B operates on a four-dimensional linear space (in our case). The four dimensions correspond to the four species present. Because of electroneutrality there are only three components and one must construct a threedimensional linear subspace corresponding to these three components. When restricted to this subspace, B becomes a 3 × 3 matrix which we call B h . Then the new B matrix is
(
B11 B h ) B21 B1+/ν+
II. The Sechenow Coefficient We shall use the following notation, which is common in biochemical applications. The solvent will be called component 1, the solute, component 2, and the cosolvent (electrolyte in our case), component 3. The positive and negative ions of the electrolyte will be given the obvious symbols + and respectively. If the salt has the formula Cν+Aν-, it will be convenient to denote by cs the total concentration of ions, cs ) c+ + c- ) (ν+ + ν-)c3. It is found that the solubility of component 2 in component 1, in the presence of component 3 is described empirically by the equation log(m2/m20) ) kmI where here I is the ionic strength and m20 is the solubility of component 2 in the absence of component 3. This empirical equation holds over an appreciable range of ionic strength. The concentration units used here are molality. The Sechenow coefficient, km, is then defined by
( ( ))
m20 1 d km ) log λ dm3 m2
µ2
|m3)0
1 λ ) (ν+ z+2 + ν-z-2) 2
(4)
and λ is just the conversion factor between ionic strength and molality (for a single electrolyte). The evaluation of the derivative in eq 4 at infinite dilution of electrolyte comes about because the quoted empirical observation, while quite accurate, is not expected to be exact. Equation 4 directs us to take the limiting slope which, empirically, is the actual slope over a wide concentration range. There are several other representations of the Sechenow relationship in the literature. We have used the one given by Randall and Faily2 (see also Harned and Owen3). These all express the same experimental facts, but differ in the concentration units used. It is interesting to compare the Sechenow coefficient, eq 4, with the expression for the preferential interaction coefficient, Γ, which appears in the biochemical literature21
Γ ) ((∂m3/∂m2))µ1,µ3
(5)
Clearly Γ encapsulates a similar qualitative idea to k. The difference is that eq 4 refers to solubility equilibria whereas eq 5 refers to dialysis equilibria. There are additional quantities found in the literature which are derivatives similar to eq 4 but with different quantities held constant in the differentiation. These also capture aspects of the preferential interaction of solutes. They are discussed in the monograph by Eisenberg.22 Our object is to express km in terms of K-B integrals. Before doing so, since our system contains electrolytes, it is necessary
B12 B22 B2+/ν+
B1+/ν+ B2+/ν+ B++/ν+2
)
(6)
The derivation of this form for B h is given in the appendix. The third row and third column of this matrix refer to component 3, the neutral salt. We have used the electroneutrality conditions to eliminate all matrix elements with “-” labeled matrix elements in terms of those with “+” labels.12 This is an arbitrary choice. Kirkwood-Buff theory can now be applied to the three component solution just as in the original K-B paper. It must be kept in mind, however, that the third component, the neutral salt, is not an actual molecule in the solution. It is a perfectly valid component in the thermodynamic sense, but in the statistical mechanical sense, its B integrals, and in particular, its G integrals, must be defined in terms of actually existing particles (whose coordinates appear in the Hamiltonian). This is fundamentally why + subscripts appear in eq 6 even though they are not labels for the matrix B h. Returning to eq 4, we may write
( ) ∂m2 ∂m3
µ2,p
( ) ( )
)-
∂m2 ∂µ2
m3,p
∂µ2 ∂m3
)
m2,p
-
( ) ( ) ∂N2 ∂µ2
N1,N3,p
∂µ2 ∂N3
(7)
N1,N2,p
Here and henceforth, for brevity we shall omit T as a subscript to partial derivatives since T is held constant throughout all the considerations of this paper. The second of eq7 arises because of the definition of molality. The particle number, N, and the molality, m, are simply proportional; The molality or particle number of the solvent is held constant. We can now apply the equations of K-B theory directly. From eq 12 of K-B we have that
km ) -
( )( )(
)
c10M1 2 1 + G2+0 + G110 - G1+0 - G120 (8) 1000 2.303λ c 0 1
To get this equation, we have already taken the limit as m3 f 0; the zero superscripts refer to that limit. Furthermore we have used the fact that csGss f -1/2 as cs f 0,18 and the relation between the ionic fluctuation integrals (the only ones actually defined by the grand partition function) and the fluctuation integrals of the fictitious component s in terms of which B h was originally defined; see the appendix. Eq 8 is an exact formal result. For the general case of arbitrary c2, this is as far as one can go. It gives another measurable quantity, km, in addition to those listed by Kirkwood and Buff8 and Ben Naim,15 in terms of which one can express the microscopic fluctuation integrals.
Analysis of the Salting-Out Effect
J. Phys. Chem. B, Vol. 110, No. 47, 2006 24079
One can, however, go further by making reasonable approximations in the special case when the solute, component 2, is sparingly soluble. III. The Case of a Sparingly Soluble Solute K-B theory tells us that, in a two component solution
V2 )
1 + (G11 - G12)c1 c1 + c2 + c1c2(G11 + G22 - 2G12)
(9)
1 + G1100 - G1200 c10
(10)
1 + G1100 - G1300 0 c1
(11)
Similarly we may write
V30 )
Equation 10 refers to component 2 at infinite dilution in 1, while eq 11 refers to component 3 at infinite dilution in 1. Both of these can be determined experimentally by extrapolation of binary solution data. Using the well-known compressibility relation for a pure fluid, 1/c10 + G110 ) RTκT0, eq 8 can finally be written as
km ) -
( )
c10M1 2 (V 0 + V30 - RTκT0 + G2+00) (12) 2.303λ 1000 2
This completes our goal of expressing G2+00 in measurable quantities, km, the partial volumes, and pressibility of the solvent. It must be remembered, that eq 12 is an approximate result, only derived for soluble solutes.
salt concentration
value (cc)
method
infinite dilution 1M
-107 -100
this paper approximate integration simulation-newer potential
2M
Since c3 has already been sent to zero, we may apply this to eq 9, taking c2 as c20. If c20 is small, then as an approximation it may be set equal to zero in eq 9 and the G’s become G00’s. The double zero superscript denotes that both components 2 and 3 are at infinite dilution. That is, we may approximate V2 by
V20 )
TABLE 1: GHe,+ Integral Estimates
terms of the comhowever, sparingly
IV. Comparison with Experiment It would be useful to confront the above theory with an independent determination of G2+00. In principle one could do this by making the appropriate thermodynamic measurements, inverting K-B theory to obtain the four G’s appearing in eq 8, and comparing the result with an experimentally measured km. Unfortunately, to the best of my knowledge there is no ternary system where there is enough thermodynamic data available to perform this calculation. Alternatively, one could compute the relevant K-B integrals by molecular dynamics and then calculate km. Such computations of integrals are feasible and, in fact, have been done successfully by Chitra and Smith.12 They did not calculate km although Smith23 has calculated k (using a definition based on different concentration units) using numerical methods but by a route not involving K-B theory. Equation 12 is proposed as a method for determining G2+00, which is not directly measurable, from km, which is. It is a possible ingredient in the inversion of K-B theory. A direct test requires some independent method for determining G2+00. Smith23 has reported some radial distribution functions in his study of the rare gases and methane as solutes. Although these compounds fall in the category of slightly soluble solutes and
-65
reference 23 24
so would be suitable for our purpose, unfortunately he does not report G2+00. He only gives rather small graphs showing the radial distribution functions themselves in 1 M cosolvent solution. According to Professor Smith,24 the original data from which the small graphs were drawn are no longer available. However, he very kindly offered to recompute the Na+-He and Cl--He radial distribution functions. He has done this for a 2 M solution of NaCl. The result, obtained by numerical integration of the tabulated rdf provided by Smith is -65 cm3 mol-1 for both G3+0 and G3-0. Of course, the general theory says that they should be equal. The earlier results23 of Smith pertain to 1 M NaCl solution, not infinite dilution. There are doubtless good numerical reasons for this. Even though the graph (Figure 2 of ref 23, top left panel) is quite small, it is possible to make a rough estimate of G2+0 for NaCl-He in water from the graph. From the top left curve in Smith’s Figure 2, we see that gHe+(R) is zero for R f 0.2 nm and then rises roughly like tanh[n(R - 0.2)] reaching unity at about 0.8 nm. The curve actually has more structure than the very smooth hyperbolic tangent curve, but the hyperbolic tangent approximation captures the overall shape reasonably well. The n is an adjustible parameter that regulates the steepness of the tanh curve. We have taken n as 6 although there is considerable arbitrariness here, especially considering the small scale of the curve we are trying to “fit”. There is no theoretical reason for using the tanh function to represent the data. It is merely a function that has qualitatively the right shape and can be conveniently handled. We now estimate GHe+ by evaluating numerically 4π∫1.0 0.2 (tanh[6(R - 0.2)] - 1)R2 dR. The result must be multiplied by Avogadro’s number and by 10 21 to convert the answer to cm3 mol-1. Finally we find that GHe+= -100 (to two significant digits). We must now compare this with the value predicted by eq 12. The value of km for the He-NaCl-H2O system is 0.053.25 V20 is 29.7 cm3,26 and V30 is 16.64 cm3.27 Putting these values into eq 12, we find that G2+0 ) -107 cm3. This is in excellent agreement with the estimated value of the previous paragraph, remarkably so in view of the very rough nature of the simulation estimate made here. Unfortunately the value of G2+ computed by numerical integration from the data of Smith at 2 M salt is appreciably smaller (in absolute magnitude). These computations are summarized in Table 1. Is this true concentration dependence or is it due to some other cause? As far as I know, the only available data on the concentration dependence of K-B integrals is for two component systems with one of the components, the solvent, present in excess. Here we are interested in the K-B integral for two dilute components in a solvent. Chitra and Smith12 have calculated the G values for a number of cosolvent-water systems by the method of inversion of K-B theory.15 The concentration dependence in these cases is slight. In the three component data, there is an obvious qualitative difference between the 1 and 2 M g2-(r) functions reported by Smith. These are very marked for the Heanion distribution function, and less so for the He-cation function.
24080 J. Phys. Chem. B, Vol. 110, No. 47, 2006 Weerasinghe and Smith28 have computed a force field for NaCl in water by adjusting parameters to fit the K-B integrals derived by inversion of empirical thermodynamic data on the binary NaCl-H2O system. It was this force field that was used in the computation of the 2 M results. Professor Smith reports24 that K-B integrals, at least those involving ions, are very sensitive to the intermolecular potential used in their computation by simulation. Consequently he stated that he was not surprised by the difference between the 1 and 2 M results, which were computed with different potentials. Consequently one must acknowledge that the good agreement between the result of eq 12 and the 1 M simulation estimate loses much of its strength. It must be confessed that the question of correspondence between theory and simulation results is extremely murky at the present time and remains to be resolved. Nevertheless, it is my current opinion that there is no reason to reject the value of -107 cm3 mol-1 for G2+00, for it comes from a sound theory and experimental data on partial molar volumes. I believe the result can be used with some confidence to determine the solute-cosolvent K-B integrals. Part of G2+ is contributed by the almost hard core of the Na+-He pair. It is of interest to see the magnitude of this contribution which is just -(4π/3)Rc3, where Rc is the effective core radius, 0.2 nm for Na+-He.23 This yields (G2+)core ) -20 cm3 mol-1. Assuming the value of -100 for the total G2+, this leaves -80 cm3 mol-1 for the contribution from the matter outside the core. The Na+ concentration was 1 M for the graph from which the core radius was taken. Hence the excess of Na+ ions around a He atom over that of a random distribution, outside the core, is about 0.08. That is, the solvation of a He atom by Na+ ions is quite small, about 1/12 of a sodium ion. It is worth noting that G1+ in NaCl solution27 is appreciably smaller (in absolute value) than G2+ under the conditions considered. The two species pairs have roughly the same core radii. The contribution from outside the core is positive for water while negative for helium. Thus water solvates Na+ while helium “antisolvates” Na+, albeit weakly. This is certainly in accord with chemical intuition, but here we have a quantitative measure of the effect. Another way of looking at eq 12 is to see what it predicts for a series of cosolvent electrolytes and a single solute. There are no microscopic calculations of this type for comparison, but McDevit and Long29 have presented a series of measurements suitable for this purpose. These authors measured the Sechenow coefficient for benzene in a number of aqueous electrolytes. The solubility of benzene in water at 25 °C is 1.77 mol L-1. This is on the verge of what might be considered sparingly soluble; nevertheless, we shall use eq 12 to analyze the data. Table 3 of ref 29 gives the measured km values, as well as the relevant partial molar volumes of the cosolvents. In our calculations we shall take V20 ) 89.4 cm3, the molar volume of pure benzene, as did McDevit and Long. Table 2 shows all of the 1-1 electrolytes from Table 3 of ref 29, rearranged in order of increasing partial molar volume of cosolvent (instead of decreasing value of Sechenow coefficient). The experimntal value of km and the values of G2+00 calculated according to eq 12 are also given. There are three series of results in the table: a series of salts with a common cation (Na+), a series with common anion (Cl-), and two salts which salt in rather than salt out. One can see from the common cation part of the table that G2+00 ranges from approximately the same size as for He-
Mazo TABLE 2: G2+ for Benzene and Various Salts salt
km29
V30 (cm3)
G2+0 (cm3/mol)
NaOH NaF NaCl NaBr NaNO3 NaI NaClO4
0.256 0.254 0.195 0.155 0.119 0.095 0.106
anion series -6.7 -2.5 16.61 23.48 27 35.10 42
-111 -115 -127 -130 -129 -134 -143
NaCl LiCl HCl KCl RbCl NH4Cl CsCl
0.195 0.141 0.048 0.166 0.140 0.103 0.088
cation series 16.61 17.06 18.07 26.8 31.9 35.98 39.2
-127 -134 -162 -134 -126 -136 -138
HClO4 (CH3)4NH4
-0.041 -0.24
salting-in 43.5 114.1
114 84
NaCl solutions to about 45% larger. Note that the G2+00 values increase monotonically with the partial molar volumes, that is, presumably with the size of the cation. This can be interpreted as, the larger the anion of a given electrolyte is, the less it is solvated by benzene (since G2+ ) G2-). The common anion part of the table is less regular. The values for the small but highly hydrated cations H+ and Li+ are anomalously high and that for Rb+ is lower than expected, about the same as that for Na+. Weersinghe and Smith28 found very strong first shell Cl-Cl coordination in their numerical simulations of NaCl solutions. Perhaps that is reflected in the irregularities just mentioned. The radial distribution functions, g2+, are not available for benzene in water so the core contribution to G2+ cannot be clearly identified. However, an estimate can be made by assuming that the core radius is the sum of the ionic radius of Na+ (0.98 nm) and half the Lennard-Jones σ for benzene (0.395 nm), or 0.29 nm. For comparison, a similar estimate for Na+He yields 0.22 nm, while the value read from the radial distribution function is 0.2 nm.23 This estimate gives a core contribution for Na+-benzene of -63 cm3 mol-1, about three times larger than for He. Since the values of G2+ are not too different one may infer that the excess solute in the coordination region surrounding a sodium ion is less for benzene than for helium. This conclusion seems counter intuitive. V. Discussion The basic results of this paper are eqs 8 and 12. Equation 8 expresses the Sechenow coefficient in terms of Kirkwood-Buff integrals. Equation 12 is an approximation to eq 8 valid for the case of a sparingly soluble solute. In this limiting case three of the four K-B integrals that enter the complete formula can be expressed easily in terms of available thermodynamic data. Thus, eq 12 may be looked on as a tool for expressing G2+0 in terms of experimentally accessible data. In the general case it will be necessary to use simulation data to get three of the G integrals and use eq 12 to get the fourth. Alternatively one could use eq 8 as one of the equations needed for the inversion11,15 of K-B theory; this could be convenient because the G values enter linearly in (8). Of course, if one wants to predict km instead of using it as an input datum, then it would be necessary to have some way of determining all four G values.
Analysis of the Salting-Out Effect
J. Phys. Chem. B, Vol. 110, No. 47, 2006 24081
When this work was begun, it was hoped that the final result, which turned out to be eq 8, would be in a form which made the physical interpretation obvious. Unfortunately this does not seem to be the case. The term 1/c1 + G110, which is numerically small compared to the other terms, does have a simple interpretation. It is a compressibility correction due to the fact that km is defined at constant pressure, while the G values are computed in the grand canonical ensemble, that is, at constant volume. But the author has not been able to give a simple physical interpretation to the combination of the other G values that appear. We want to make especially clear what we have accomplished here. In particular, we have not derived Sechenow’s law, log(m2/m20) ) kmI. For example, we have not proved (or indeed, even attempted to prove) that log(m2/m20) should be linear in electrolyte concentration. Rather we have accepted it as an established empirical observation and tried to give a theoretical interpretation to km. We have been able to express km in terms of Kirkwood-Buff integrals, but this expression does not immediately suggest any obvious physical interpretation. Acknowledgment. I thank Professor John Schellman for many useful discussions on the subject of this paper and Professor Paul E. Smith for providing me with the radial distribution functions discussed in Section IV. Appendix In this appendix we derive the form for B h given in eq 6 of the main text. The grand partition function for our system is
eΩ )
∑N e-βA ΠR λRN N
R
(A1)
Here Ω is the grand partition function, pV/kT, and λR is the absolute activity of species R; λR ) exp(µR/kT). The sum is over all (but see below) values of N1, N2, N+, N-, and the product is over all species, 1, 2, +, -. However, the condition of electroneutrality requires that z+N+ + z-N- ) 0. Consequently only one of the pair of variables (N+, N-) is independent. We choose N+ as the independent variable. Sometimes it is more convenient to use the neutral component N3 ) N+/ν+ as the independent variable. Although there are four species present, there are only three components in the thermodynamic sense. As a parenthetical remark on the electroneutrality condition, in principle it should not be necessary to require electroneutrality in the construction of the grand partition function. One should sum over all compositions of the system, those which violate as well as those which obey electroneutrality.20 The compositions that violate electroneutrality will have such high energies that they will make negligible contributions to the sum over states. Furthermore, in a system with excess mobile charge, the excess charge will accumulate on the boundary of the system for the overwhelming majority of configurations. Thus the rare, high energy charges states are inhomogeneous, and require different methods of analysis than homogeneous states. For these reasons it is usual, and accurate, to impose the electroneutrality conditions from the start. But then, as we have just discussed, one must use special methods to account for the constraint. We have also referred, in the main text, to the equation ∑βzβBRβ ) 0 as an electroneutrality condition. Indeed, if R refers to a neutral species, then this equation says that, when looked at from a coordinate system centered on a particle of that species, the system is electrically neutral. When R is a charged species,
it says that when looked at from a coordinate system centered on such a particle, the system has a charge equal and opposite to that of the atom at the origin. We adopt the notational convention that Greek subscripts denote species, 1, 2, +, -. Latin letters will denote components, 1, 2, 3. The original K-B development assumed that all components were also species. As we have already discussed, in electrolyte solutions this is not the case. Also, when chemical reactions can occur between the species similar complications occur, but we shall not discuss this problem in this paper. Equation A1, expressing Ω in terms of species, with the implied electroneutrality restriction on the sum, can then be transformed to
eΩ )
∑N e-βA Πj λjN N
j
(A2)
which expresses Ω in terms of components. We can now follow to the Appendix of the paper of Kirkwood and Buff8 for the ensuing steps of the development. The steps indicated in eqs A2 through A5 in K-B can be carried through exactly as written, keeping in mind that all mathematical operations must be carried out on components rather than species. One ends with matrixes A h and B h , which are 3 × 3 matrixes with latin subscripts. It only remains to find the relationship between the Bij and the BRβ matrix elements. This is easily obtained from the electroneutrality relationships. It is important to express the B h matrix elements in terms of the B matrix elements because the BRβ values are expressed in terms of the G integrals pertaining to the actual physical species present in the solution. If species β is a charged species, then B h j3 ) Bjβ/νβ for j ) 1, h 13 2 and B h 33 ) B++/ν+2. This follows because, for example, B ) 〈N1N3〉 - 〈N1〉〈N3〉 and N3 ) N+/ν+. These relations yield eq 6 of the main text. Note that, because N3 can also be expressed as N-/ν-, there are equivalent alternative forms for the matrix elements of B h. It should be realized that the development sketched here is valid for a single electrolyte, Aν+Xν-. For a mixed electrolyte the computations are more complex; see Friedman and Ramanathan16 for the case of an AX-BX system and O’Connell et al.17 for the general case. The single electrolyte case suffices for the application discussed in this paper and is much simpler than the more general cases. References and Notes (1) Sechenow, M. Z. Physik. Chem. 1889, 4, 117. (2) Randall, M.; Failey, C. F. Chem. ReV. 1927, 4, 27, 285, 291. (3) Harned, H.; Owen, B. B. The Physical Chemistry of Electrolytic Solutions, 2nd ed.; Reinhold Publishing Co.: NY, 1950. (4) Debye, P. Z. Phyzik. Chem. 1927, 130, 56. (5) Debye, P.; McCauly, G. V. Physik. Z. 1925, 22, 22. (6) Masterton, W. L.; Lee, T. P. J. Phys. Chem. 1970, 74, 1776. Masterton, W. L. J. Solution. Chem. 1975, 4, 523. (7) Tiepel, E. W.; Gubbons, K. E. Ind. Eng. Chem. Fundam. 1973, 12, 18. (8) Kirkwood, J. G.; Buff, F. P. J. Chem. Phys. 1951, 19, 774. (9) Ruckenstein, E.; Shulgin, I. Ind. Eng. Chem. Res. 2002, 41, 4674. Shulgin, I.; Ruckenstein, E. Ind. Eng. Chem. Res. 2002, 41, 1689. (10) Shulgin, I.; Ruckenstein, E. J. Chem. Phys. 2005, 123, 054909. (11) Matteoli, E.; Mansoori, G. A. Fluctuation Theory of Mixtures; Taylor & Francis: New York, 1990. (12) Chitra, R.; Smith, P. E. J. Phys. Chem. B 2002, 1491, 1491. (13) Weerasinghe, S.; Smith, P. E. J. Chem. Phys. 2003, 119, 1342. (14) Smith, P. E. J. Phys. Chem. B 2000, 104, 5854. (15) Ben Naim, A. J. Chem. Phys. 1977, 67, 4884. (16) Friedman, H. L.; Ramanathan, P. S. J. Phys. Chem. 1970, 74, 3756. (17) Perry, R. L.; O’Connell, J. P. Mol. Phys. 1984, 52, 137. Perry, R. L.; Cabezas, H.; O’Connell, J. P. Mol. Phys. 1988, 63, 189. O’Connell, J. P.; DeGance, A. E. J. Solution. Chem. 1975, 4, 763. (18) Kusalik, P. G.; Patey, G. N. J. Chem. Phys. 1987, 86, 5110.
24082 J. Phys. Chem. B, Vol. 110, No. 47, 2006 (19) Behera, R. J. Chem. Phys. 1998, 108, 3373. (20) Ben Naim, A. Molecular Theory of Solutions; Oxford University Press: Oxford, U.K., 2006. (21) Schellman, J. Biophys. Chem. 1990, 37, 121; cf. the appendix. (22) Eisenberg, H. Biological Macromolecules and Polyelectrolyte Solutions; Oxford University Press: Oxford, U.K., 1976. (23) Smith, P. E. J. Chem. Phys. 1999, 103, 525.
Mazo (24) Smith, P. E., University of Kansas. Private communication, 2005. (25) Reference 3, p 565. (26) Hildebrand, J. H.; Scott, R, L. The Solubility of Nonelectrolytes; Reinhold: NY, 1950. (27) Reference 3, p 253. (28) Weersinghe, S.; Smith, P. E. J. Chem. Phys. 2003, 119, 11342. (29) McDevit, W. F.; Long, F. A. J. Am. Chem. Soc. 1952, 74, 1689.
