16886
J. Phys. Chem. B 2005, 109, 16886-16890
Fluctuation Functions in Aqueous NaCl and Urea David Siu and Yoshikata Koga* Department of Chemistry, The UniVersity of British Columbia, VancouVer, British Columbia, Canada V6T 1Z1 ReceiVed: April 2, 2005; In Final Form: July 16, 2005
We earlier devised a set of fluctuation functions that provide relative qualitative differences of the amplitude (intensity) and the wavelength (extensity) of fluctuations in entropy and volume and the entropy-volume cross fluctuations. We discuss the mixing schemes in aqueous NaCl and urea using these fluctuation functions. Our earlier studies by using the second and third derivatives of Gibbs energy indicated that their effects on H2O are qualitatively different. An NaCl hydrates 7.5 molecules of H2O but leaves the bulk H2O away from the hydration shell unperturbed. Urea, on the other hand, connects onto the hydrogen bond network of H2O but retards the degree of fluctuation inherent in H2O. The behavior of the fluctuation functions calculated here are consistent with the above mixing schemes. Furthermore, urea was found to reduce the wavelength of fluctuation more strongly than NaCl.
Introduction Earlier we had devised a set of fluctuation functions that distinguish the intensity (amplitude) and the extensity (wavelength) of fluctuation though qualitatively.1-6 In the Gibbs variable system (p, T, ni), where ni is the amount of the ith component, we control and fix the independent variables and let the respective conjugates, V, S, and µι fluctuate. Here we limit our attention to the fluctuation nature of V and S. Since they are extensive quantities, we divide the entire system into a number of coarse grains, each containing the same amount, Nz, with the same composition as the global average mole fractions. We then evaluate the local values of V and S within each coarse grain. The size of Nz must be small enough so that the fluctuation is recognized but large enough for a thermodynamic quantity to be evaluated.1-6 Thus, the mean-square fluctuations in volume and entropy and the entropy-volume cross fluctuations are related to the response functions following general formulas developed recently.7-9 Namely
〈(∆V)2〉 ) kT(∂V/∂p) ) kTVNzκΤ
(1)
〈(∆S)2〉 ) kT(∂S/∂T) ) kCpNz
(2)
〈(∆V)(∆S)〉 ) kT(∂V/∂T) ) kTVNzRp
(3)
∆ signifies the variation of the quantity in the coarse grain from its average. The partial derivatives are taken keeping all the other independent variables other than that of differentiation. In relating the derivatives to response functions, we must recognize that the derivatives are evaluated for a coarse grain with Nz. Hence, VNz is the average value of the volume, and CpNz is the average total heat capacity of a coarse grain containing Nz. κΤ and Rp are isothermal compressibility and isobaric thermal expansivity, which are intensive quantities and independent from the size of the coarse grain. Thus, all the above * To whom correspondence should be addressed. Tel: (604) 822-3491. E-mail:
[email protected].
mean-square fluctuations, eqs 1-3, depend on the size of the coarse grain. We stress that the coarse grain must be small enough for “fluctuation” to be detected but must be large enough for the extensive thermodynamic quantities V and S to be evaluated. The problem is that we have no way of knowing its size. To circumvent this difficulty, we divide both sides of eqs 1-3 by the average volume of the coarse grain, 〈V〉, and define the mean square fluctuation densities as V
S
δ ≡ 〈(∆V)2〉/(k〈V〉) ) TκT
δ ≡ 〈(∆S)2〉/(k〈V〉) ) 〈Cp〉/〈V〉 ) CpNz/VNz ) Cpm/Vm SV
δ ≡ 〈(∆S)(∆V)〉/(k〈V〉) ) TRp
(4) (5) (6)
where Cpm and Vm are the molar heat capacity and the molar volume. These quantities thus defined do not depend on the size of the coarse grain, and hence they can be interpreted as representing the intensity or the amplitude of fluctuation. In passing, we define the effect of a solute on the fluctuation function, the partial molar mean-square fluctuation density as q
δi ≡ N(∂qδ/∂ni)p,T,nj ) (1 - xi)(∂qδ/∂xi)
(7)
where q ) V, S, or SV. When the fluctuation propensity is to be compared between aqueous solutions and nonaqueous solutions, the extensity or the wavelength of fluctuation could become important. Indeed, molecular dynamic simulation studies10,11 and an X-ray scattering investigation12 indicate that in H2O collective fluctuations involving a large number of molecules are prevalent. This suggests that H2O has a characteristically larger extensity or a longer wavelength of fluctuation than normal liquids. It is thus expected, though not yet rigorously proven, that the size of the coarse grain is necessarily larger for H2O. Is there any way of identifying this effect of H2O? It turned out that the (mean-square) normalized fluctuation q∆ values defined by eqs 8b-10b below serve this purpose although qualitatively.1-6
10.1021/jp0516792 CCC: $30.25 © 2005 American Chemical Society Published on Web 08/12/2005
Aqueous NaCl and Urea
J. Phys. Chem. B, Vol. 109, No. 35, 2005 16887
First, we normalize the deviation of a local value (a value in a coarse grain) from its global average by the mean volume of the coarse grain34 and then take its mean square.
〈(∆V/〈V〉)2〉 ) kTκT/〈V〉 ) kTκT/NzVm
(8a)
〈(∆S/〈V〉)2〉 ) k〈Cp〉/〈V〉2 ) k(Cpm/Vm)/NzVm
(9a)
〈(∆V/〈V〉)(∆S/〈V〉)〉 ) kTRp/〈V〉 ) kTRp/NzVm (10a) The second equalities use the relation, 〈V〉 ) NzVm. On the other hand, we earlier defined q∆ values as1-6
∆ ≡ RT κT/Vm
(8b)
∆ ≡ RCpm/Vm2
(9b)
∆ ≡ RT Rp/Vm
(10b)
V
S
SV
and the partial molar normalized fluctuations of the ith component, q∆i, are defined as1-6
∆i ≡ (1 - xi) (∂ q∆/∂xi)
q
(11)
In comparing two systems A and B, consider the ratios of the left-hand sides of eqs 8a-10a. Using the right-hand sides and eqs 8b-10b
〈(∆V/〈V〉)2〉A/〈(∆V/〈V〉)2〉B ) (V∆A/NzA)/(V∆B/NzB) (8c) 〈(∆S/〈V〉)2〉A/〈(∆S/〈V〉)2〉B ) (S∆A/NzA)/(S∆B/NzB) (9c) 〈(∆V/〈V〉)(∆S/〈V〉)〉A/〈(∆V/〈V〉)(∆S/〈V〉)〉B ) (SV∆A/NzA)/(SV∆B/NzB) (10c) Superscripts A and B denote each system. If we assume that the left-hand sides of eqs 8a-10a are universal and independent of the system in consideration, then the left-hand sides of eqs 8c-10c are an identical unity. If so
NzA > NzB,
if
∆A > q∆B
q
(12)
q ) S, V, or SV and the converse is also true. Namely, the condition, q∆A > q∆B (q ) S, V, or SV), is sufficient and necessary for the conclusion, NzA > NzB. If we relax the above assumption and allow the left-hand sides of eqs 8c-10c to take a positive value, r < 1, dependent on the systems of comparison, then
(q∆A/NzA)/(q∆ B/NzB) ) r < 1
(13)
Under this condition (r < 1), if we find q∆A > q∆B, then it is necessary that NzA > NzB but not sufficient. (If r > 1, the labels A and B can be reversed.) Furthermore, there is another unresolved issue about the relationship between Nz and the extensity of fluctuation, though it could be parallel. Or at least when the extensity of the fluctuation is larger, then the coarse grain size should necessarily be larger. Thus, the usage of q∆ for gaining some qualitative information about the wavelength of fluctuation is based on two assumptions: that the normalized fluctuation functions, the left-hand sides of eqs 8a-10a, are independent of the system in consideration and that the size of the coarse grain is proportional or at least parallel to the extensity or the wavelength of fluctuation. The first assumption could be
relaxed under certain conditions as discussed above, but the statement (12) above is only partially true. We earlier used these fluctuation functions and their temperature and pressure dependences to contrast the nature of H2O from that of n-hexane.13,14 Within the limitation of the usage of q∆ discussed above, we were quite successful in contrasting the peculiar nature of liquid H2O from a typical van der Waals liquid, n-hexane. Briefly, at low temperatures and pressures (less than 50 °C and 200 MPa), the volume fluctuations, Vδ and V∆, show anomalous behaviors consistent with the putative formation/destruction of icelike patches for H2O, which is lacking in n-hexane. Comparison of Vδ and V∆ indicated that the wavelength of fluctuation is much larger for H2O. Sδ and S∆ indicated that not only the amplitude but also the wavelength are larger for H2O than for n-hexane, characteristics of the “bent hydrogen bond model” of H2O. The entropy-volume cross fluctuations, SVδ and SV∆, showed the most conspicuous difference. Due to the putative formation of icelike patches, there is a negative contribution in the cross fluctuation at low temperatures and pressures for H2O. For normal liquids, when a coarse grain fluctuates to give rise to a volume increase, the entropy also increases. For H2O, however, a part of volume increase could be associated with the entropy decrease due to formation of icelike patches. Thus, SVδ and SV∆ are conspicuous measures for the uniqueness of H2O. We thus demonstrated the usefulness of the fluctuation functions introduced above in recovering the general understandings of liquid H2O15-19 and that the “mixture model” and the “bent hydrogen bond model” are only two sides of a coin.20,21 Furthermore, we pointed out the importance of distinguishing the amplitude and the wavelength of fluctuation. Here, we use these fluctuation functions and learn the fluctuation propensities of aqueous urea and NaCl, using the literature data. We did the same for the mono-ols-H2O system earlier and advanced the understandings of hydrophobic hydration.5,6 Both NaCl and urea are non-hydrophobic solutes, but their effects on the molecular organization of H2O are qualitatively different from each other.22,23 It is therefore interesting to see how the fluctuation characteristics differ from each other. Data Analysis for Aqueous NaCl and Urea. The comprehensive compilation of thermodynamic data for NaCl-H2O are available24,25 The data of molar volume, Vm, isothermal compressibility, κΤ, and isobaric expansivity, Rp, are obtained from ref 24, and those of isobaric heat capacity, Cp, from ref 25 together with Vm. Since these data are in a tabular form for a smoothed value of NaCl composition,qδ and q∆ were calculated immediately by eqs 4-6, and eqs 8b-10b for NaCl. For ureaH2O, Cp, and Vm data are available at 15, 25, and 35 °C,26 and Sδ and S∆ were calculated immediately. From the V data at m three temperatures, Rp values were calculated against the mole fraction of urea, xurea. First, the values of Vm were plotted, and smooth curves were drawn through all the data points by means of a flexible ruler. Then at the same values of xurea the values of Vm were read off the smooth curves drawn at three temperatures and evaluated the value of Rp at 25 °C. In order not to lose significant figures in reading a value off a curve, the method described earlier 5 was used. Thus, SV∆ and SVδ were calculated at 25 °C for urea-H2O. Two sets of the adiabatic compressibility data are available in the literature.27,28 To convert them to the isothermal compressibility, κΤ, the required data of Cp, Rp, and Vm were read off from the smooth curves. For comparison, the data of volume compression by raising the pressure from 0.1 to 100 MPa29 were used to evaluate an approximate compressibility at 50 MPa, assuming that the value of Vm at 50 MPa is the average of those at 0.1 and 100 MPa. Sδ
16888 J. Phys. Chem. B, Vol. 109, No. 35, 2005
Figure 1. (a) Amplitude of entropy fluctuation, Sδ, for aqueous urea and NaCl. The filled symbols are for NaCl, and hollow symbols are for urea. (b) The amplitude and the wavelength (qualitative) of entropy fluctuation, S∆, aqueous urea, and NaCl. The filled symbols are for NaCl, and hollow symbols are for urea.
and S∆ were calculated for urea-H2O. All of these results are shown in Figures 1-3. Discussion In our earlier thermodynamic studies using the partial molar enthalpies, and their composition derivatives,30,31 we showed that urea keeps the hydrogen bond connectivity of H2O intact but reduces the degree of fluctuation inherent in H2O, presumably by forming the hydrogen bond with the network of H2O and by breaking the proton donor/acceptor symmetry of H2O.30 Indeed, a simulation study suggested that urea leads to the stiffening of short-time dynamics of both urea and H2O.32 NaCl, on the other hand, hydrates 7.5 molecules of H2O and more importantly leaves the bulk H2O away from the hydration shell
Siu and Koga
Figure 2. (a) Amplitude of volume fluctuation, Vδ, for aqueous urea and NaCl. The filled symbols are for NaCl, and hollow symbols are for urea. The dashed line is an extension of the data by Lo Surdo et al.27 (b) The amplitude and the wavelength (qualitative) of volume fluctuation, V∆, aqueous urea, and NaCl. The filled symbols are for NaCl, and hollow symbols are for urea. The dashed line is an extension of the data by Lo Surdo et al.27
unaffected by the presence of Na+ and Cl-.31 Another recent simulation study for the Na+-H2O system concluded that Na+ contains 5.2 molecules of H2O in its hydration shell and there was no effect of Na+ on the orientational dynamics of H2O beyond the hydration shell.33 Figure 1a indicates that the amplitude of the entropy fluctuation is reduced in a similar rate between urea and NaCl upon increasing the solute composition. A decrease in Sδ by the addition of urea is due to “stiffening” of the dynamics of both urea and H2O as discussed above. For NaCl, since the above discussion indicates that the bulk H2O away from the hydration shell remains unaffected, a decrease in Sδ on addition of NaCl seems odd at first sight. However, this observation may suggest that the hydration shell is rigid and does not participate
Aqueous NaCl and Urea
J. Phys. Chem. B, Vol. 109, No. 35, 2005 16889 does not participate in fluctuation, recalling again that each coarse grain contains the same number of NaCl ion pairs. Figure 2b shows that urea diminishes the wavelength of fluctuation more strongly than NaCl, though qualitatively. Figure 3 provides a similar picture. It is the wavelength of fluctuation that is more strongly affected by urea than NaCl. Thus, the present analysis on the fluctuation functions shows a consistency with our earlier interpretation by thermodynamic studies.30,31 Furthermore, we learn that the hydration shell in aqueous NaCl is rigid and does not participate strongly towards fluctuation. We did not evaluate the partial molar fluctuation functions, eqs 7 and 11, since the data points are not available in small enough increments for graphical differentiation with confidence. Clearly more precise data with smaller increments are required for further investigation. Similar analysis for aqueous solutions of various solutes including amphiphiles, if satisfactory data are available, leads no doubt to a deeper insight into aqueous solutions and is useful toward understanding the function of biopolymers. Acknowledgment. We thank the reviewers assigned to this manuscript for helping us to clarify the nature of the assumptions for the usage of the q∆ values. References and Notes
Figure 3. (a) Amplitude of entropy-volume cross fluctuation, SVδ, for aqueous urea and NaCl. The filled symbols are for NaCl, and hollow symbols are for urea. (b) The amplitude and the wavelength (qualitative) of entropy-volume cross fluctuation, SV∆, aqueous urea, and NaCl. The filled symbols are for NaCl, and hollow symbols are for urea.
in fluctuation, remembering that each coarse grain contains the same number of NaCl. Figure 1b indicates that urea reduces the wavelength of fluctuation more strongly than NaCl. There are two sets of compressibility data for urea-H2O,27,28 out of which Vδ and V∆ were calculated. As shown in Figure 2, these two sets do not seem to match in that there is an initial sharp decrease near xurea ) 0 for the data by Hammes and Schimmel.28 However, in comparison with those at 50 MPa using the data of Moriyoshi and Nakagawa,29 it appears more likely that the results using the compressibility data by Lo Surdo et al.27 and their smooth extension shown as a broken line in the figure are more likely. Accordingly, Vδ drops more sharply for NaCl than for urea. As discussed above, if 7.5 molecules of H2O are hydrated by a pair of Na+ and Cl-, then the number of remaining bulk H2O diminishes rapidly, and at xNaCl ) 0.1, 75% of H2O is in the hydration shell in a rigid manner and
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16890 J. Phys. Chem. B, Vol. 109, No. 35, 2005 (32) Idrissi, A.; Sokolic, F.; Perera, A. J. Chem. Phys. 2000, 112, 9479. (33) White, J. A.; Schwegler, E.; Galli, G.; Gygi, F. J. Chem. Phys. 2000, 113, 4668. (34) It is perhaps more appropriate to normalize with the true volume of a coarse grain rather than its average value. However, the relation of
Siu and Koga 〈(∆V/V)2〉, for example, to derivatives of thermodynamic quantities is not simple. Hence we tentatively use the normalization with 〈V〉. This serves a qualitative convenience in hinting a relative size difference in coarse grain, as discussed below. We thank one of the reviewers for drawing our attention to this point.
