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J. Phys. Chem. B 2008, 112, 4680-4686
Interactions of Na-Salts and 1-Propanol in 1-Propanol-Na-Salt-H2O Systems: Toward an Understanding the Hofmeister Series (IV) Kumiko Miki Department of Liberal Arts and Basic Sciences, College of Industrial Technology, Nihon UniVersity, Narashino, Chiba, Japan 275-8575
Peter Westh NSM, Research Unit for Functional Biomaterials, Roskilde UniVersity, Roskilde DK-4000, Denmark
Yoshikata Koga* Department of Chemistry, The UniVersity of British Columbia, VancouVer, British Columbia, Canada V6T 1Z1, and Research Center for Molecular Thermodynamics, Graduate School of Science, Osaka UniVersity, Toyonaka, Osaka, Japan 560-0043 ReceiVed: December 3, 2007; In Final Form: January 24, 2008
The excess chemical potential of 1-propanol (1P), µE1P, was evaluated in ternary 1P-Na-salt(S)-H2O at 25 °C. The counter anions of the Na-salts studied are SO42-, F-, Cl-, I-, and ClO4-. The effect of the anion on µE1P follows the Hofmeister ranking, in that the more kosmotropic ions make the µE1P value more positive. We then evaluate the effect of the Na-salt (S) on µE1P, the 1P-S interaction in terms of excess chemical potential, at a semi-infinite dilution. The results indicate that the 1P-S interaction in terms of excess chemical potential is unfavorable (repulsive) for all of the ions studied. The degree of repulsive interaction decreases in the order of the Hofmeister ranking from the kosmotropic to the chaotropic end. Namely, salting-out samples make the excess part of the chemical potential of 1P more unfavorable, while the salting-in counterparts make it less unfavorable. From earlier calorimetric studies on the same ternary systems, the enthalpic 1P-S E E interaction function, H1P-S , was calculated. Hence, the entropy analogue, S1P-S , was also obtained, and a E detailed thermodynamic signature of 1P-S interactions became available. This revealed that both H1P-S and E S1P-S decrease from the kosmotropic ion to the middle of the ranking (Cl-), whereupon they turn to increase toward the chaotropic end. Hence, the build up of unfavorable 1P-S interactions in Hofmeister salts (signified by µE1P) relies on a pronounced enthalpy-entropy compensation, which must be accounted for in attempts to understand the molecular mechanisms underpinning Hofmeister effects.
Introduction The Hofmeister series has proven relevant to a wide range of salt effects in aqueous systems, perhaps most notably a number of structural and functional factors for proteins and other biomolecules.1 Its omnipresence suggests that an important factor is the effects of salts on the molecular organization of H2O. Much progress has emerged in this regard from many decades of work on the properties of binary aqueous salt solutions (see e.g. refs 2-9). While this type of information is crucially important, we argue that understanding Hofmeister effects also requires systematic investigations of ternary systems, since there is an equally important issue; how such modified H2O affects the structures and functions of biopolymers. The simplest way toward this is to add a relevant probe mimicking biopolymers and to study the interactions of the probe with the salts. One interesting example of this “ternary approach” has been gel permeation chromatography,6,7,10-12 which has generally shown that chaotropes (“salting in agents” such as I-, SCN-, and ClO4-) interact with the stationary phase and thus elude * To whom correspondence should be addressed. Phone: (604) 8223491. Fax: (604) 822-2847. E-mail:
[email protected].
much slower than expected on the basis of the ionic radius. Kosmotropes (“salting out agents” such as F- and SO42-) show the opposite correlation. It was concluded that chaotropes “stick” to the gel material as a result of a weak hydration sphere, which is readily replaced with a different (weakly hydrated) interface. Conversely, the strong hydration of kosmotropic anions keeps them away from the interface of the stationary phase and hence makes then elude early. We have previously dealt with this “ternary approach” by choosing 1-propanol (1P) and applying a differential approach in solution thermodynamics on 1P-salt (S)-H2O ternary systems. 1P is known to have a comparable ratio of hydrophobic and hydrophilic moieties to that of some soluble hydrophobic proteins.13 By studying ternary 1P-S-H2O systems, we have concluded that the effects of Na-salts of typical anions listed in the Hofmeister series are not only quantitatively but also qualitatively different from others in the series.14-18 In particular, the kosmotropes work in aqueous solution as either hydration centers or hydrophobes, while the chaotropes were found to behave as hydrophiles. For hydration centers, the more kosmotropic the ion, its hydration number is larger consistent with Hofmeister’s original concept of the H2O withdrawing power.19
10.1021/jp7113829 CCC: $40.75 © 2008 American Chemical Society Published on Web 03/25/2008
1P-S-H2O Systems
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We showed experimentally that a hydration center, while accommodating a number of H2O molecules in the hydration shell, leaves the bulk H2O away from the hydration shell unaffected.14-16 A hydrophobe, on the other hand, enhances the hydrogen bond network of H2O in the immediate vicinity similar to the classical picture of “iceberg formation” (hydration in a broader sense), but the hydrogen bond probability of bulk H2O away from “icebergs” is reduced progressively.20-22 This is true only in the H2O-rich concentration range up to the threshold composition where the hydrogen bond probability of the bulk H2O is reduced to the hydrogen bond percolation threshold, and the integrity of liquid H2O is lost.20-23 A hydrophile forms hydrogen bonds to the existing hydrogen bond network of H2O and becomes an impurity center in the otherwise widely fluctuating network. This affects the nature of bulk H2O in two ways: to retard the degree of fluctuation inherent in liquid H2O and to cause the hydrogen bonds to cleave in the region between the impurity centers. However, the integrity of H2O is retained also within the limited dilute composition range.14,15,24,25 Furthermore, the interactions among solutes were shown to be operative via the so-modified bulk H2O within this H2O-rich region, whether the solutes are of the same kinds (hydrophobes or hydrophiles) or different.14,15,20-25 The experimental method used to reach the above conclusions is an advanced thermodynamic work that we call the 1P probing methodology. The details have been given elsewhere.14-17 Briefly, we study by calorimetry a ternary system, 1P-S- H2O E , (W) and evaluate the 1P-1P enthalpic interaction, H1P-1P defined as E H1P-1P ≡N
( ) ∂HE1P ∂n1P
system. An advantage of using 1P for a model for biopolymers is that it has a comparable vapor pressure as that of H2O and µE1P could be evaluated by vapor pressure measurements. We note that µE1P is the first derivative of G. Here, we determine the excess chemical potential of 1P in dilute ternary systems with the Na salts of SO42-, F-, Cl-, I-, and ClO4-. The ranking of these anions from the kosmotropic to the chaotropic end is known as1
SO42- > F- > Cl- > I- > ClO4-
We then evaluate the effect of S on the excess chemical potential of 1P, or the 1P-S interaction in terms of excess chemical potential, µE1P, defined as E ≡N µ1P-S
where n1P, nS, and nW are the amount of 1P, S, and W, respectively, and hence, the mole fraction of 1P, x1P ) n1P/(n1P + nS + nW). HE1P is the excess partial molar enthalpy of 1P that signifies the actual enthalpic situation of 1P in the highly E shows therefore how an infinitesiinteracting mixture. H1P-1P mal increase in n1P changes the enthalpic situation of the existing 1P. Thus, we call this the 1P-1P interaction in terms of enthalpy. We note that HE1P contains the second derivative of G E and, hence, H1P-1P the third derivative.14-18 For the binary E 1P-H2O system, the x1P-dependence pattern of H1P-1P shows a unique peak type anomaly in the H2O-rich region, reflecting qualitative changes in mixing scheme from the H2O-rich region E to the intermediate range.14-18,20-25 If S is present, the H1P-1P pattern, while keeping a peak pattern, shifts in the x1P-direction E -direction depending on the nature and/or the ordinate H1P-1P of S. Having calibrated the induced changes with the samples with the known propensity toward H2O, we deduce the nature E . We note that, of S by the changes that it induces to H1P-1P while we studied ternary system, 1P-S-H2O, we learned the effect of S on H2O, the first part of the inquiry into the Hofmeister effects mentioned above, and did not address the 1P-S interactions. The latter is the issue we deal with in this work. The original work by Hofmeister deals with the solubility of egg white globulin in a given aqueous salt solution.19 The more kosmotropic the ion, the less the solubility of protein, i.e., the higher its excess chemical potential. If an important factor of the Hofmeister effect is the manner in which H2O is modified by a salt, then the excess chemical potential of 1P, µE1P, might also be higher for a more kosmotropic salt in the 1P-S-H2O
( ) ∂µE1P ∂nS
(3)
n1P,nW
This is the second derivative quantity, which is a measure of the excess chemical potential of 1P-S interactions (we note in E is akin to the so-called preferential interacpassing that µ1P-S tion parameters,26 which are used to specify net interaction from dialysis equilibrium or osmometry experiments). From the earlier calorimetric studies for the same ternary systems,14-18 E we calculate the 1P-S enthalpic interaction function, H1P-S , E and hence its entropy analogue, S1P-S, defined as
( ) ( )
E ≡N H1P-S
(1)
nS,nW
(2)
E S1P-S ≡N
∂HE1P ∂nS
∂SE1P ∂nS
(4)
n1P,nW
(5)
n1P,nW
Both quantities are the third derivatives of G. Due to the E E thermodynamic stability criteria, when µ1P-S and H1P-S are negative the 1P-S interaction can be said to be favorable in E terms of excess chemical potential and enthalpy. Positive S1P-S indicates favorable interaction in terms of excess entropy. E decrease following We show below that the values of µ1P-S the Hofmeister series, eq 2, from the kosmotropic to the E E chaotropic end, as expected. Those of H1P-S and S1P-S , however, decrease to Cl- in eq 2 and turn to increase toward the chaotropic end. This indicates that the entropy-enthalpy compensation27,28 is operative and more importantly that the mechanism for the Hofmeister effect differs between the kosmotropes and the chaotropes with the boundary at Cl-, which we have already pointed out by using the 1P-probing methodology ealier.14-18 Experimental Section Vapor pressures of ternary systems 1P-S-H2O were determined29-31 using a home-built apparatus. Briefly, the over head gas-phase pressure of a given solution was determined by a capacitance manometer, Baratron gauge, at 25.02 ( 0.005 °C. While the sensitivity of the gauge was ( 0.001 Torr, the overall uncertainty was estimated to ( 0.01 Torr. The composition of the liquid mixture was altered by successively adding a known amount of 1P or H2O using a gas handling manifold with a known volume, pressure, and temperature. The virial correction was applied in determining the exact amount of 1P and H2O. The Baratron gauge used seems to have a part that adsorbs 1P and H2O, the amount of which was previously
4682 J. Phys. Chem. B, Vol. 112, No. 15, 2008
Figure 1. Excess chemical potential of 1P and H2O, µE1P/RT, and µEW/RT, in binary 1P-H2O at 25 °C. Open square symbols indicate the results by the Boissonnas analysis, while open triangles show those assuming the Raoult’s law for H2O. See text.
Figure 2. Excess chemical potential of H2O in S-H2O at 25 °C. Note that the ordinate scale is much smaller than those in Figures 3-7.
determined as a function of pressure. The relative uncertainty in the mole fraction, x1P, for the most dilute solution became sizable coming from the uncertainties in the adsorption calibration, as it will become evident in Figures 4 and 6. Na2SO4 anhydrous (Fluka, > 99.0%) was heated overnight under vacuum (99.8%) is known to be extremely hygroscopic. Although we used this salt from a fresh bottle, at the end of the vapor pressure measurements, we evaluated its purity to be actually 96.54 mol % by drying completely until the vapor pressure becomes zero within the sensitivity of the Barathron gauge. NaCl (Aldrich, 99+ % ACS Reagent) was dried at 130 °C for 19 h and kept in a desiccator. NaF (Sigma, 99%) and NaI (Merck, GR > 99.5%) were used from fresh bottles. In calculating x0S, we took into account of these values of purity. 1-Propanol (Fluka, >99. 8% GC) was dried over molecular sieves (0.3 nm, 2 mm beads, Merck, Darmstadt, Germany) and vacuum distilled within the vapor
Miki et al.
Figure 3. Excess chemical potential of 1P, µE1P/RT, in 1P-S-H2O at 25 °C as a function of the initial mole fraction of S, x0S. S ) Na2SO4.
Figure 4. Excess chemical potential of 1P, µE1P/RT, in 1P-S-H2O at 25 °C as a function of the initial mole fraction of S, x0S. S ) NaF.
pressure apparatus. H2O was purified by MilliQ (Millipore, Bedford, MA) immediately prior to use. Results and Discussion Excess Chemical Potential of 1P. The measured vapor pressures of 1P-S-H2O at 25.02 ( 0.005 °C for S ) Na2SO4, NaF, NaCl, NaI, and NaClO4 are given as supplementary data, together with binary 1P-H2O in a more dilute composition range than the previous measurement.31 For each series of measurements, a known amount of stock solution of a given salt was freeze-dried in the cell to which a required amount of H2O was vacuum-distilled from the gas handling manifold. To this mixed solvent of S-H2O with the initial mole fraction of S, x0S, was successively added 1P, and the equilibrium vapor pressure, p, was measured. For binary 1P-H2O, the previous vapor pressure data started at about x1P ) 0.00331 and were analyzed by the Boissonnas method,29-31 which singles out the partial pressures of each component on the basis of the Gibbs-
1P-S-H2O Systems
Figure 5. Excess chemical potential of 1P, µE1P/RT, in 1P-S-H2O at 25 °C as a function of the initial mole fraction of S, x0S. S ) NaCl.
J. Phys. Chem. B, Vol. 112, No. 15, 2008 4683
Figure 7. Excess chemical potential of 1P, µE1P/RT, in 1P-S-H2O at 25 °C as a function of the initial mole fraction of S, x0S. S ) NaClO4.
equally by the Boissonnas method, and calculated the excess chemical potential of i as
µEi /RT ≡ ln
( ) pi xip*i
(6)
ignoring the nonideality in the gas phase.29-31 p* i is the vapor pressure of the pure component i at the same temperature. Figure 1 shows the plots of µE1P/RT and µEW/RT for binary 1P-H2O. The error bars are large only for the first few points in µE1P/RT. It is evident that the values of µEW/RT are zero within (0.0001 in this concentration range. Hence we recalculated p1P by
p1P ) p - p*W(1 - x1P)
Figure 6. Excess chemical potential of 1P, µE1P/RT, in 1P-S-H2O at 25 °C as a function of the initial mole fraction of S, x0S. S ) NaI.
Du¨hem equation. As discussed at some length earlier,29-31 Boissonnas analysis carries an intrinsic systematic error in the first few data points of dilute solutions which is automatically corrected for as the analysis proceeds point by point. Thus, those data are not reliable in the range x1P < 0.01. Here we limit our study to a more dilute region and hence redetermined the vapor pressures in the range 0.0003 < x1P < 0.01. The results are also listed in the table in the Supporting Information. While the uncertainty in p is estimated as (0.01 Torr out of the value of p of about 24 Torr, that in x1P amounts typically to (0.00004 due to the error in adsorption correction mentioned above, depending on the total amount of mixed solvent S-H2O in the cell. This gives inevitably a sizable relative error in x1P for the first few data points of the present measurements. As will become evident, this is carried into a sizable absolute uncertainty in µEi /RT due to the logarithm of x1P, as shown in eq 6 below. From the present set of p data for binary 1P-H2O, we calculated the partial pressure of the ith component, pi (i ) 1P or W),
(7)
Namely, µEW is taken to be identically zero, and the mixture is in the Raoult’s law range for W. Using the values of p1P by eq 7, we recalculated µE1P by eq 6 and plotted it in Figure 1. The discrepancy between “proper” results by the Boissonnas method and those under the assumption of µEW ) 0 is negligible in the composition range shown in the figure. This finding provides some confidence in using the following approximation in analyzing the vapor pressure data for ternary 1P-S-H2O. Since the vapor pressure of S is zero, the total pressure, p, is the sum of p1P and pW. We make then an approximation as
p1P ) p - p*W+S(1 - x1P)
(8)
where p*W+Sis the vapor pressure of a mixed solvent S-H2O before adding 1P. Figure 2 shows the values of µEW/RT calculated by eq 6 for a given mixed solvent S-H2O against the initial mole fraction of S, x0S. The value of µEW/RT lies between 0 and -0.07 in the x0S range studied here. The values of µE1P/RT, on the other hand, are at least 2.6. While this difference between µEW/RT and µE1P/RT does not guarantee that the assumption that µEW ) 0 and hence eq 8 is acceptable for the present ternary systems, we proceed here by assuming eq 8. Thus, µE1P/RT was calculated by eq 6 for all the ternary systems studied and plotted in Figure 3 for S ) Na2SO4, Figure 4 for NaF, Figure 5 for NaCl, Figure 6 for NaI and Figure 7 for NaClO4. In view of the large error bars for the most dilute data
4684 J. Phys. Chem. B, Vol. 112, No. 15, 2008
Miki et al.
Figure 8. Excess chemical potential of 1P, µE1P/RT, at x1P ) 0.005 in 1P-S-H2O at 25 °C. x0S is the initial mole fraction of S for various salts. The straight lines connecting the data points in the same group are guides for eye. E E E TABLE 1: Interaction Functions, µ1P-S , H1P-S , and S1P-S , in 1P-S-H2O
S
slope in Figure 8
E µ1P-S /RT
E H1P-S /RT
E TS1P-S /RT
Na2SO4 NaF NaCl NaI NaClO4
46.2 ( 0.5 25 ( 5 17.8 ( 0.5 10 ( 1 10 ( 2
46.3 ( 0.5 25 ( 5 17.9 ( 0.5 10 ( 1 10 ( 2
88 ( 7 38 ( 2 31 ( 1 42 ( 3 76 ( 5
41 ( 8 13 ( 7 13 ( 2 32 ( 4 66 ( 7
points for some salts, it seems safer not to extrapolate the µE1P/RT value to the infinite dilution, as commonly done. We rather interpolate using reliable data points shown in Figures 3-7 to the most dilute yet reliable common point for all salts, which turned out to be at x1P ) 0.005. We read off the value of µE1P/RT at x1P ) 0.005, and plotted them in Figure 8 as a function of x0S. It is clear from the figure that µE1P/RT increases as x0S increases and the x0S dependences are mostly linear. The slopes in Figure 8 indicates how each salt increases the excess chemical potential of 1P (at constant x1P ) 0.005) and their positive values signifies that the salts generate an unfavorable situation for the alcohol. Table 1 lists the slopes in Figure 8. The effect of a salt is in the order of anions
SO42- > F- > Cl- > I- ≈ ClO4-
(9)
co-incident with the Hofmeister ranking, eq 2. This adds yet another example of ubiquity of the Hofmeister ranking.1 We point out that 1P has a comparable ratio of polar and nonpolar surfaces as some soluble proteins.13 Thus, the behavior of 1P in aqueous salts may be similar to the interactions in protein solutions. We now evaluate the 1P-S interaction in terms of excess E chemical potential, µ1P-S , by eq 3 using the present data. E Furthermore, the enthalpic 1P-S interaction, H1P-S (eq 4), and E hence the entropy analogue, S1P-S (eq 5), can be calculated from the previous calorimetric data for the same ternary systems.16-18 1P-S Interaction Functions in 1P-S-H2O. The definition E E E , H1P-S , and S1P-S , eqs 3-5, is based on the (n1P, nS, of µ1P-S
Figure 9. Excess partial molar enthalpy of 1P, HE1P/RT at x1P ) 0.005 in 1P-S-H2O at 25 °C. The data are taken from refs 3 and 4. x0S is the initial mole fraction of S for various salts. The straight lines connecting the data points in the same group are guides for eye.
and nW) variable system keeping p and T constant. Experimentally, however, it is more convenient to use the (x1P, x0S, and N) variable system, with N ) n1P + nS +nW, x1P ) n1P/N, and x0S ) nS/(nS + nW). The conversion of the variable system is straight forward as shown in the Appendix. The result is written as E µ1P-S )
E Similarly for H1P-S E H1P-S )
( ) ( )
(10)
( ) ( )
(11)
(1 - x0S) ∂µE1P (1 - x1P) ∂x0S
(1 - x0S) ∂HE1P (1 - x1P) ∂x0S
- x1P
- x1P
∂µE1P ∂x1P
∂HE1P ∂x1P
Then E E E TS1P-S ) H1P-S - µ1P-S
(12)
As mentioned above, we evaluate these interaction functions at x1P ) 0.005 and x0S ) 0.005. The latter value for x0S was dictated by the fact that the highest available data point for NaF was 0.0066, Figure 4. (The saturation for NaF-H2O at 25 °C is at x0S ) 0.017.3) The first term on the right of eq 10 is approximated by the slope at x0S ) 0.005 in Figure 8, and the second term, while its contribution is minimal due to the factor x1P ()0.005), can be interpolated of the slopes at x1P ) 0.005 in Figures 3-7 to that at x0S ) 0.005. The additional uncertainty by this interpolation does not affect the resulting E µ1P-S /RT due to the factor x1P ()0.005) in the second term. The results are listed in Table 1 together with the estimated uncertainty. Using the HE1P data determined earlier for a given x0S shown E in Figure 9,16,17 the values of H1P-S /RT were calculated in a similar fashion by eq 11 at x1P ) x0S ) 0.005 were determined, listed in Table 1 and plotted in Figure 10. The entropy analogue, E /RT, was calculated by eq 12 and shown in Table 1 and TS1P-S Figure 10. On the abscissa of Figure 10, the salts are placed in the order of the Hofmeister ranking, eq 2, from the left to the right. This figure, which is the main result of the present work,
1P-S-H2O Systems
E E Figure 10. 1P-S interaction functions, µ1P-S /RT, H1P-S /RT, and E 0 TS1P-S/RT, for 1P-S-H2O at x1P ) xS ) 0.005, 25 °C. The abscissa indicates anions from kosmotropic SO42- to chaotropic ClO4- in the order of typical Hofmeister ranking, eq 2. The 1P-S interaction in terms of excess chemical potential is more unfavorable for kosmotropes and less so for chaotropes. The excess enthalpic and entropic interactions show different trends for kosmotropes and chaotropes. See text for detail. The straight lines connecting the data points in the same group are guides for eye.
suggests that, while the effect of salt on 1P follows the Hofmeister ranking in the excess chemical potential level, E µ1P-S , (the second derivative of GE), the enthalpic and the E E entropic interactions, H1P-S and TS1P-S , (the third derivative of E G ) show different behaviors in the kosmotropic and the E chaotropic ends of the series. Namely, H1P-S decreases from E 2SO4 to Cl , eq 2, and then turns to increase to ClO4-. TS1P-S follows the same course. However, the net 1P-S interaction in E terms of excess chemical potential, µ1P-S , decreases monoto2nously from SO4 all of the way to ClO4-, as shown in Figure 10. While the molecular mechanisms for strong positive (unfaE E vorable) H1P-S and favorable TS1P-S for both kosmotropic SO42- and chaotropic ClO4- are yet to be elucidated, the compensation between the enthalpy and entropy effects for E chaotropes are almost complete providing weak net µ1P-S values. For the kosmotropes, on the other hand, the entropic contribution is less than half of the enthalpic effect and this E gives rise to sizable 1P-S repulsion in terms of µ1P-S (which in turn underlies the “salting out” generated by these compounds). We suggest that this shift in the balance between enthalpic and entropic effects is a characteristic of “Hofmeister effects”. As mentioned in the introduction, the effects of each salts on H2O have been studied by the induced changes of the E pattern.14-18 By this method, we learned that kosmoH1P-1P tropes are either hydration centers or hydrophobes. Both F- and Cl- were found to hydrate 14 ( 2 and 2.3 ( 0.6 H2O molecules, respectively, without changing the nature of the bulk H2O away from the hydration shell.14-16 SO42-, on the other hand, works as a hydrophobe in a moderate concentration as well as a hydration center with the hydration number of 17 ( 3.14,15,17 Thus, the 1P-S interaction for S ) SO42-, occurring via the modified bulk H2O by 1P and SO42-, shows the largest E E and favorable S1P-S among kosmotropes unfavorable H1P-S studied here. For F and Cl , both being purely hydration centers, the 1P-S (S ) F- or Cl-) interaction occurs via
J. Phys. Chem. B, Vol. 112, No. 15, 2008 4685 E unmodified H2O network. Hence the absolute values of H1P-S E and S1P-S are small and comparable between them, as is evident in Figure 10. For chaotropes, I- and ClO4- (having hydrophilic propensities16,17), the 1P-S interaction occurs via a more rigidified bulk H2O modified by hydrophiles. Hence, E E the increase in the values of H1P-S and S1P-S toward the chaotropic end may reflect the enthalpic difficulty and the entropic ease (or less-difficulty) in accommodating 1P; a hydrophobe in the bulk H2O previously modified by a stronger hydrophile. We gave similar arguments on the heterogeneous interactions between a hydrophobe and a hydrophile in aqueous ternary systems elsewhere.32-34 If any more detailed molecular mechanism is to be suggested, it should be consistent with the variations of the 1P-S E E E interaction functions, µ1P-S , H1P-S , and S1P-S , shown in Figure 10. The latter information displayed for the first time will therefore be useful as guidance for further studies and the understanding of the molecular mechanism. In this connection, we plan to repeat the same sort of studies using a more hydrophilic probe, glycerol for example, to mimic more hydrophilic biopolymers, and to provide the same interaction functions for a hydrophile-salt in addition to what are shown in Figure 10. After all, biopolymers are colloidal, and their hydrophobic and hydrophilic parts could interact with H2O and salts separately and concurrently. Hence, such information for both hydrophobe and hydrophile would be useful toward a fuller understanding of the Hofmeister effects.
Appendix Variable Conversion from (n1P, nS, and nW) to (x1P, x0S, and N) Systems. A useful and general formulation for variable conversion is given in ref 21. We apply it here for the present variable systems. For any thermodynamic function Φ
Φ (n1P, nS, nW) ) Φ (x1P, x0S, N)
(A1)
with N ) n1P + nS +nW, x1P ) n1P/N, and x0S ) nS/(nS + nW) (A2)
For the total differential for both systems should be the same as long as the relations eq A2 hold. Hence dΦ ) )
( )
( )
( )
∂Φ ∂Φ ∂Φ dn1P + dnS + dnW ∂n1P ∂nS ∂nW
( )
( ) ( )
∂Φ ∂Φ ∂Φ dx + dx0S + dN 0 ∂x1P 1P ∂N ∂xS
(A3)
E If Φ ) µE1P and we wish to evaluate µ1P-S defined by eq 3 in the text, we impose n1P and nW to be constant, and rewrite eq A3 as
E µ1P-S ≡
( ) ( )( )
E E 1 ∂µ1P 1 ∂µ1P ∂x1P ) N ∂nS N ∂x1P ∂nS
+
n1P,nW
( )( )
E 0 1 ∂µ1P ∂xS N ∂x0 ∂nS S
(A4)
n1P,nW
Since µE1P is an intensive quantity, the third term on the extreme right of eq A3 is identically zero. With the relations in eq A2, eq A4 is rewritten as
( )
E µ1P-S )
( )( )
∂µE1P ∂µE1P 1 - x0S (-x1P) + ∂x1P ∂x0S 1 - x1P
which is given as eq 10 in the text.
(A5)
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