2614
J . Phys. Chem. 1989, 93, 2614-2620
Potential excursions in the cathodic direction would shift the equilibrium to the right, favoring the formation of the M, aggregate whose interfacial properties should not be very different from those of the bulk material. The Hz(g) evolution reaction on Pt is a well-known example of the reaction depicted in eq 3. The earlier cited studies of Ag deposition onto an I-coated have shown that codeposition of a full Pt( 1 1 1) single monolayer of Ag results in total place exchange between the Ag and I adatoms. Since the iodine is now completely the outermost layer, formation of direct I-Pt bonds is prohibited. In such a case, one might expect that the resulting 81 vs E plots would be very similar to those for bulk Ag. This is not the case, however, as can be seen from the data in Figure 8. The unique action of monolayer Ag is most evident at pH 0 where removal of the first 20% of the surface-coordinated iodine occurs even sooner than at pure Pt. These results can only mean that the electrochemical properties of the bimetallic interface consisting of one Ag monolayer atop a Pt substrate are distinct from those of either of the pure metals. The nonbulk behavior of monolayer Ag is not unexpected since, as evidenced by the existence of two UPD peaks, the adsorbate-substrate (Ag-Pt) interactions are stronger than the adsorbate-adsorbate (Ag-I and Ag-Ag) interactions. Consequently, the Ag-I binding strength for monolayer Ag is weaker than for bulk deposited material where the innermost Ag-Pt interactions are no longer felt by the outermost Ag-I layer. Closer inspection of the Orvs E data for monolayer Ag in Figure 4 reveals a trace of pH dependence in the reductive desorption of the first 20% of the surface iodine. This result suggests that a small but nonnegligible interaction occurs between the outer I atoms with the inner Pt substrate despite the fact that a gap of one Ag monolayer exists between them. Such through-the-silver interaction is anticipated to vanish when a second layer of Ag atoms is electrodeposited. This expectation is indeed borne out by the data in Figure 6 which compare 8, vs E plots for 2 and 20 layers of Ag. The evidence is indisputable that two layers of codeposited Ag behave just like bulk Ag. From Figure 6, one can
readily extract EoI(,&),the redox potential for the following surface rea~tionl-~ I(ads)
+ e-
-
I-(ads)
(4)
from the potential at which OI = 0.5; this EoI(ads,) value for bulk Ag is -1.1 V. In comparison, Eo1(,d,) for Ir3, Pt , and AuZhave been measured to be -0.32, -0.36, and -0.50 V, respectively. The where the latter term is the redox difference [EoI(ads)potential for the I,(aq)/I-(aq) couple, 0.40 V, is a measure of the relative chemisorption strengths of zerovalent iodine and iodide.'" A(AGo) E AGo1(,k) - AG0I-(,ds) nFIEoI(ads)- EoI(aq)l
- AGod ( 5 )
In eq 5 , AGoI(,dS! and AGoI-(ads)are the respective free energies of adsorption of iodine and iodide, and 2AGod is the energy in2I(aq) dissociation, = 150 kJ/mol.I6 volved in the Iz(aq) Application of data from Figure 6 to eq 5 yields A(AGo) = -200 kJ/mol; that is, the surface-coordination strength of zerovalent iodine is greater than that of iodide by about 220 W/mol. A(AGo) in turn can be used to calculate the ratio of the surface-complex formation constants for zerovalent iodine and iodide, Kf,I/Kf,I:
-
Kf,I/Kf,I- = exp[-A(AGO)/RT]
(6)
The ratio Kf,I/Kf,I-is approximately 3 X lo3*which signifies an overwhelming preference for surface coordination of zerovalent iodine over iodide. Acknowledgment. Acknowledgment is made to the Robert A. Welch Foundation and the Regents of Texas A&M University through the AUF-sponsored Materials Science and Engineering Programs for support of this research. Registry No. I*, 7553-56-2; Ag, 7440-22-4; Pt, 7440-06-4. (16) Cotton, F. A.; Wilkinson, G.Advanced Inorganic Chemistry; Wiley: New York, 1980.
Prediction of Velocity Cross-Correlation Coefficients of Binary Liquid Mixtures Thomas M. Bender7 and R. Pecora* Department of Chemistry, Stanford University, Stanford, California 94305 (Received: February 8, 1988; In Final Form: September 16. 1988)
Velocity cross-correlation coefficients (VCC's) may be used as measures of the coupling between the velocities of different molecules in binary liquid mixtures. The hydrodynamic expression for the VCC's proposed by Friedman and Mills is used to derive expressions for the VCC's in terms of molecular diameters and the Kirkwood-Buff parameters. The main approximation is the use of the Ornstein-Zernike form for the liquid radial distribution functions supplemented by a small separation cutoff. An ideal reference system for the VCC's is proposed in terms of hypothetical ideal Kirkwood-Buff parameters. Experimental VCC's, experimental Kirkwood-Buff parameters, theoretical VCC's using our expression, and reference VCC's are computed from literature data for eight binary liquid systems ranging from nearly thermodynamically ideal to very nonideal. Good agreement is shown between the experimental and predicted VCC's over the entire composition range of these mixtures.
Introduction There are many common binary liquid mixtures for which abundant information is available regarding the system thermodynamic properties and diffusion coefficients. In some cases, it has been possible for specific links to be established between these various static and dynamic properties. In other cases, however, our understanding of the quantitative and qualitative relation between these variables is lacking. Some success has been had in reconciling various of these properties of certain binary liquid 'Present address: Rohm and Haas, Spring House, PA 19477.
0022-3654/89/2093-2614$01.50/0
mixtures by the use of kinetic aggregation models (e.g., clathrate-type models for the tert-butyl alcohol/water system or the 2-butoxyethanol/water system's2). The question must be raised, however, as to whether multistep aggregation, system-specific models are desirable ways for describing binary liquid behavior. In this paper we present a new model relating thermodynamic to diffusion properties of binary liquid mixtures. Through the (1) Ito, N.; Saito, K.; Kato, T.; Fujiyama, T. Bull. Chem. Sac. Jpn. 1981, 54, 991. (2) Bender, T. M.; Pecora, R. J. Phys. Chem. 1986, 90, 1700.
0 1989 American Chemical Society
Velocity Cross-Correlation Coefficients
The Journal of Physical Chemistry, Vol. 93, No. 6, 1989 2615
use of a recent expression for velocity cross correlation coefficients (VCC’s) derived by Friedman and Mills3 using hydrodynamic considerations along with Kirkwood-Buff (KB) parameters4 we demonstrate a mathematical link between solution equilibrium properties and the dynamic properties of the system. This model, although approximate, can be applied to binary systems regardless of the degree of thermodynamic nonideality of the system.
Current Estimates of Reference VCC’s The velocity cross-correlation coefficients (VCC’s) are. defined in terms of the areas under time correlation functions of the velocities of distinct molecules5
f, =
(is, NXj
(cf. eq 1). This failure of the original reference model was then remedied in an ad hoc fashion by HertzIO to give the following reference equations
p,?I = -X,MjDf[ 1 + XjQij]/ M
(6a)
- XjQij]/ M
(6b)
= -XjMjDY[ 1
where Qjj = [MjD;/MjDp- 11. These reference equations (6a, 6b) are the reference equations commonly used by workers in this field. In a similar fashion to the earlier work by Tyrrell et aI.l1 it can be shown12that eq 6a and 6b effectively incorporate the assumption that
(@A)(t).~B)(0))ydt
where X j is the mole fraction of the j t h component and p)(t) is the velocity of a specific molecule (in this case labeled A) of species i at time t and ( )v denotes an ensemble average which is taken in the thermodynamic limit V m with N / V held fixed. The correlation function in eq 1 is always taken between distinct molecules even when the molecules are of the same species (Le,, when i = j ) . It is evident from eq 1 that thefj are measures of the velocity correlation between distinct molecules and are related to the solution diffusion coefficients. An application of the definitions of the Onsager phenomenological coefficients6-8 for a binary liquid mixture along with eq 1 yields -+
hj = -D12MiMJj/(M2a12
xi = D
=
(7) where the lijare the friction coefficients which arise from the Onsager definition of the diffusion coefficients in binary liquid mixturesI3 El2
co
(2a)
~ ~ M ~ Z X -~~p / ( ~ ~ ~ ,(2b) ,
In eq 2a and 2b Dp is defined as the intradiffusion coefficient of component i of the liquid mixture (3) and DL2is the mutual diffusion coefficient of the mixture. Also in eq 2 Mi is the molecular mass of the ith component of the mixture, W = X l M l + X2M2and a 1 2is given by (4)
where y i is the thermodynamic activity coefficient of the ith component of the mixture as defined on a mole fraction basis. Dp is the analogue of the self-diffusion coefficient of a pure fluid. It is useful when interpreting VCC’s for a nonideal system to define a fictitious “ideal” reference system whose VCC’s may be calculated and compared to those for the actual nonideal system. The reference system VCC’s should approximate the measured ones for thermodynamically ideal systems. The original attempt by HertzS to define an ideal reference system for these VCC’s was to invoke a type of microscopic conservation of momentum argument for the calculation. Unfortunately, the resulting reference equations were found to violate the law of microscopic re~ersibility~
ej)
(3) Friedman, H. L.; Mills, R. J. Solution Chem. 1986, 15, 69. (4) Kirkwood, J. G.;Buff, F. P. J. Chem. Phys. 1951, 19, 744. (5) Mills, R.; Hertz, H. G. J . Chem. Phys. 1980, 84, 220. (6) Onsager, L. Phys. Rev. 1931, 37, 405. Onsager, L. Phys. Reo. 1931, 38, 2265. (7) Onsager, L. Ann. N . Y . Sci. 1945, 46, 241. (8) Fitts, D. D. Nonequilibrium Thermodynamics; McGraw-Hill: New York, 1962. (9) Friedman, H. L.; Mills, R. J . Solution Chem. 1981, 10, 395.
(El1522)”2
Dl2 = RTVal2/lI2
+
Dp = RTV/ [XjEii Xjlij]
@a) (8b)
and Vis the molar volume of the liquid mixture. This is unfortunately the only clearly identifiable microscopic assumption which is incorporated in the popular Hertz reference VCC functions. More recently, Friedman and Mills3 have proposed a new hydrodynamic approximation for the VCC’s. They obtained the following expression for the VCC’s using the Oseen tensor for the hydrodynamic interaction between molecules 2kT f.. = X’37v , - j 0 R[hV(R)] dR (9)
+
where hij(R) 1 is the radial distribution function for the species i and j of the binary liquid mixture, 7 is the solution viscosity, and P i s the molar volume of the solution. Note that the solution viscosity now links the VCC’s to solution structural parameters. Their expression was taken as a special case from a more general calculation reported in an earlier workL4on the properties of electrolyte systems consisting of a general electrolyte and a solvent. The general nature of eq 9 can be more fully comprehended if one considers the definition of a Kirkwood-Buff (KB) parameter4 (Gj,) Gjj = Lm[gjj(R)- 1]4aR2 dR
In eq 10 gij(R) is the radial distribution function of species i and + 1, and R is a scalar distance. The KB parameters arise from the general theory of composition fluctuations in the grand canonical ensemble. The KB parameters for a binary liquid mixture can be calculated exactly from the following equations4
j , gjj(R) = hij(R)
Gjj = Gjj = RTK, - V;t5/al2V
(1 l a )
where KT is the solution isothermal compressibility and V, is the partial molar volume of component i. An interesting property of the KB parameters becomes clear on considering the significance of the quantity pjGij. As has been shown by Ben-Naim,I5 the quantity (pjgjj4aR2 dR) measures the average number of j molecules in a spherical shell of width dR at a distance R from the center of an i molecule. The average number of j molecules in the same spherical shell is given by (pj4?rR2dR), where the origin of the sphere has been chosen at random. Therefore the ( I O ) Hertz, H. G.; Leiter, H. Z. Phys. Chem. N.F. 1982, 133, s45-67. (1 1) Harris, K. R.; Tyrrell, H. J. V. J. Chem. SOC.,Faraday Trans. 1 1982, 78, 957. (12) Bender, T. M. Ph.D. Thesis, Stanford University, 1987. (13) Tyrrell, H. J. V.; Harris, K. R. Dvfusion in Liquids: A Theoretical and Experimental Study; Butterworths: London, 1984. (14) Altenberger, A. R.; Friedman, H. L. J . Chem. Phys. 1983, 78,4162. (15) Ben-Naim. A. J. Chem. Phys. 1977, 67, 4884.
2616 The Journal of Physical Chemistry, Vol. 93, No. 6, 1989
quantity (pl[gi,(R) - 1]4aR2dR) is a measure of the excess or deficiency in the average number o f j molecules in the spherical shell at distance R from an i molecule relative to the number obtained by eliminating the condition that an i molecule be placed at the center of the sphere. We then can see that (pici,) is a total average excess or deficiency of j molecules in the entire surroundings of an i molecule. Hence the KB parameters convey information on the local composition around species in the solution. If we rewrite eq 9 as
then we gain insight into the general nature of eq 9. It appears that the coefficient of the mole fraction in eq 9 contains a weighted average of 1/R in which the effect of local structure in the liquid environment is included. In a sense then, the coefficient in eq 9 resembles a modification of the familiar Stokes-Einstein equation for the diffusion of spheres. With eq 9 Friedman and Mills’ were able to calculate the limiting values of X 2 1.0 for thef, by the following approximations
-
hi,(R) = -1
if R
< ufJ
h,(R) = 0
if R
> uIJ
(1 3a)
and
where uv corresponds to the approximate average diameter of a molecule. This set of approximations when inserted into eq 9 gave limiting values of thefj which were in good agreement with the 1.0 limit for 24 binary liquid mixtures. experimentalfj in the X, Another possible approximation to use in obtaining hij(R) was discussed by Friedman and Mills3 but the above approximation proved to give the best agreement with experimental data. In a more recent article, Friedman and MillsI6 introduced a thermodynamic approximation for calculating the hj in the limiting concentration case, but this approximation was found not to be as successful as the hydrodynamic approximation.I6
-
Approximation Using the OZ Form At this point the relation between hi,(R) and fjproposed by Friedman and Mills has only been tested as X 2 1. In the interest of further testing eq 9 and obtaining a set of& which can be effectively used to study binary liquid mixtures we developed the following approximate form for the hij(R). To obtain a useful set of reference functions8 for the VCC’s from eq 9 we must clarify the role of concentration in eq 9. Using Friedman and Mills’ original approximations for hij(R) the only apparent point in eq 9 at which an appreciable concentration dependence of the coefficient of the mole fraction can be obtained is in calculating uij because the dependence of fjon solution viscosity is not sufficient to give the proper concentration dependence. If the ui, calculated by eq 13b is approximately the hard-sphere diameter, as implied by eq 13a, then the concentration dependence of thehjlXj predicted by using eq 9 and 13a,13b will be small. Knowing that thefj/Xj in experimental systems vary significantly with concentration, we then know that we must propose some different set of assumptions other than eq 13a and 13b into eq 9 in order to obtain the desired concentration dependent f j . The most realistic point at which to obtain a nontrivial concentration dependence in eq 9 is by modifying the expression for hi,(R). Unfortunately, accurate expressions for the actual experimental hV(R)are not available for most simple, carbon-based small molecule liquids. A simple approximation which can be made and will lead to the desired concentration-dependent hij(R)
-
(16) Mills, R.; Friedman, H. L. J . Solution Chem. 1987, 16, 927.
Bender and Pecora is the Ornstein-Zernike (OZ) approximation” supplemented with a small separation cutoff
[
gij(R) = Ail R exp - -; j ] In eq 14 the A, and the B , are parameters with the dimension and absolute magnitude of molecular length and R is the scalar distance between particles i and j . The O Z approximation can initially be used to introduce a concentration dependent h,(R) in the following fashion. If one knows the values of the KB parameters for a liquid mixture this will obviously eliminate one parameter from the OZ approximation because of the definition in eq IO. By constructing a reference KB parameters system (to obtain&) which is concentration dependent, or even using the experimental KB parameters (to predict the actual fJ)one adds a significant concentration dependence to eq 9. The OZ h,(R) will still contain one undetermined parameter, E , , after application of the KB definition and the known or assumed KB parameter values. In pure liquids such as water,I8 the decay constants for simple functions fit to experimental radial distribution functions (rdfs) indicate that the decay constant of the exponential in eq 14 is on the order of the molecular diameter. As a first approximation then, one could assume that the decay constant in the OZ approximation is given by the average molecular diameter of the species. The molecular diameters used could be either those that arise from eq 13b or those that result from using standard van der Waals increments.” At this point, with the above assumptions, all the parameters for the OZ approximation of a liquid mixture rdf are available. An examination of the parameters in the OZ rdf approximation reveals at least one fact. If the rdf in eq 14 is integrated to give the G, Gij = -Ja”4aR2 0 dR
+
exp( - :)hR
dR
(1 5 )
it may be clearly seen that either positive to negative values for the KB parameters can be obtained depending on the sign and magnitude of the linear parameter A,. Most common rdf‘s for pure liquids and mixtures which are determined by computer simulations or experiments show a strong peak at a distance equal to the first molecular diameter. This peak corresponds to the presence of a nearest shell of neighbors around the central molecule. Underneath this peak is a large, positive portion of the total area underneath the g,(R) curve. In the case of a liquid mixture, however, there may be an anticorrelation between unlike particles. This anticorrelation will give a strongly negative KB parameter for the system. This implies that the portion of the hv(R) curve from uv to infinity may have an overall negative area. This will result in a diminished or perhaps nonexistent peak in the rdf curve for the anticorrelated molecules. Likewise one would expect that this general lack of solution structure between these pairs gives an rdf with only a few or no maxima in the rdf which have positive ordinate values. By this argument then, one finds whenforcing the OZ hij(R) to apply to a real liquid mixture that no restriction should be placed on the sign of the linear coefficient of the approximation. If the linear coefficient Ail does go negative, however, it should not allow an ordinate value which is more negative than the value in the excluded volume region of the curve (i.e., -1.0).
The Ideal Reference Mixture Model Let us first construct an ideal set of concentration-dependent KB parameters in order to generate a set of&. The ideal binary liquid mixture tht we use is assumed to obey Raoult’s law at all (17) Kruus, P.Liquids and Solutions: Structure and Dynamics; Dekker: New York, 1977. (18) Korsunsky, V. J.; Naberukhin, Yu. I. Acta Crystallogr. 1980, A36, 33. (19) Edwards, J. T. J. Chem. Educ. 1970, 47, 261
Velocity Cross-Correlation Coefficients
The Journal of Physical Chemistry, Vol. 93, No. 6, 1989 2617
composition^.^^ This first assumption implies In y I = In y2 = 0
(16)
Equation 16 combined with eq 4 indicates that a12= 1 at all compositions of this ideal system. It also follows from this assumption that the excess Gibbs free energy of mixing (ACEX) for this reference system is zero, since AGE' = NRTIXl In y1
+ X 2 In y2]
(17)
From the relation
.4
one finds the excess volume of mixing ( A F ) for this ideal system is zero. This in turn implies that
v=e
e
i
1
S
0 0
G, = -(4/3)7ruj?
+ 87rui?Aij exp(-1)
(21)
and 2kT
f j = X.--[-( J37v
1 /2)uij2
+ Aijuijexp(-l)]
(22)
In these expressions the parameter Bij (eq 14) in the OZ approximation has been taken equal to uij and Aij has been allowed either positive values or a restricted range of negative values (cf. previous section on rdfs). If one solves for A , using eq 21 and substitutes it into eq 22 one obtains the following equation for the fj
'"[
f j = x.-( 1 /2)Ui? J37v
+
(
G,
+ (4/3)7rUij3 87ruij
)]
(23)
At this point one may choose either to calculate a reference VCC system using the ideal KB parameters or one may predict the experimentally observed VCC by using the experimental KB parameters. By testing both of these options one can make a reasonable test of the general nature of the Friedman-Mills equation (eq 9) and the present modification of that theory (eq 23).
Experimental KB Parameters The Kirkwood-Buff parameters which we have calculated for the eight binary liquid systems considered in this paper are
0 6
0 6
1
1
U I O
0 0
0 1
0 4
0 6
0 8
I 0
MOLE FRACTION DMF
& =AI
Figure 1. Actual and reference for the systems N,N-dimethylformamide water at 25 "C. Left: Actual and referenceA2andAl (0=A2, W + = reference, A reference). Right: Actual (0)and The units of t h e 4 are IO9 m2 s-]. The reference4 reference were obtained by using the ideal KB parameter model and eq 23.
=A,(.)A2.A2
where pT,i is the isothermal compressibility of neat component i. Because the isothermal compressibility only contributes a small magnitude term to the KB parameters that we require, any errors that arise from making this approximation are quite small. The above equations (16, 19, 20) are the thermodynamic approximations that we will use to calculate the microscopic (KB parameter wise) ideal solution case which corresponds to the macroscopic (bulk thermodynamic property wise) ideal solution case.
New Model The use of (1) the Friedman-Mills expression (eq 9), (2) the OZ approximations discussed in the above sections, and (3) either modeled or experimental KB parameters gives us sufficient information to calculate concentration-dependent VCC's. As the first step in this process one needs to evaluate the expressions for the KB and thefj integrals (eq 10 and 9 ) , respectively, using our version of the OZ approximation. These integrals when evaluated give
0 4
MOLE FRACTION DMF
(19)
where is the molar volume of the neat ith component. The second assumption made to calculate the reference ideal liquid is to assume that the isothermal compressibility of the mixture is given by
0 1
I +
I
A M
I
s0
0
0 1
0 4
0 6
1
n
l
I
-7
0 8
I4 O
40 0
MOLE FRACTION ACETONE
I
L ! 0 1
0 4
0 6
0 8
I O
MOLE FRACTION ACETONE
Figure 2. Actual and reference4 for the system acetone/water at 25 OC. The key and units are the same as in Figure 1. The reference4 were
obtained by using the ideal KB parameter model and eq 23. available as supplementary material (see paragraph at the end of this article). The method of calculationI2 is similar to that of Matteoli and LeporiZ0and the references for the literature data used are also available in the supplementary material. Unfortunately, the few authors who have calculated KB parameters for experimental systems rarely report the numerical values of those parameters. In all cases the present values were in qualitative agreement with the results of other workers, but since no numerical values are presented in the literature, no opportunities existed for checking the quantitative agreement of our results against any previously published KB parameters for the same systems. As a general note, it must be remembered that the error in KB parameters goes as ( 1/a12). This means that the KB parameters calculated for nonideal systems will have the largest errors, and this error will generally reach its largest value where the KB parameters calculated are the largest in value.
Experimental VCC's The experimental velocity cross-correlation coefficients for the two binary liquid systems N,N-dimethylformamide/waterand acetone/water as calculated from eq 2a and 2b are given in Figures 1 and 2. Along with those experimental VCC's are the ideal reference VCC's calculated by using eq 23 along with the ideal KB reference system with thermodynamic parameters appropriate for the respective systems. In these figures the abscissa corresponds to XI. Data sources for these and the other VCC's discussed below (20) Matteoli, E.; Lepori, L. J . Chem. Phys. 1984, 60, 2856.
2618
The Journal of Physical Chemistry, Vol. 93, No. 6, 1989
are given in the supplementary material. In general for the eight binary liquid mixtures studied, the experimental VCC’s are in reasonable quantitative agreement with the literature values. Any quantitative discrepancies are probably due to differences in the calculated a I 2used in the various references. Before comparing the experimental data to the reference VCC functions it is necessary to discuss how the uij parameters for the reference systems were calculated. Several options are available in the calculation of cij. The general opinion in the literature is that the reference functions for the VCC’s should be in good agreement with the experimental VCC’s for thermodynamicakly near ideal binary systems. As a test, therefore, one can check the various models for uii by seeing if the reference VCC’s calculated with the uij from that model are in good agreement with the experimental VCC’s for thermodynamically near-ideal binary systems. The first option for determining cij is to utilize the standard van der Waals incrementsIg for the constituent atoms. Using van der Waals radii and then calculating the molecular volume, one can calculate the radius of a spherical model for the compound. This method gives reasonable agreement with the test systemsf;, but the limiting concentration intercepts for the solvent-solvent VCC cf22)and low-concentration points for that VCC near that intercept quantitatively differ from the experimental data. The second option for obtaining uij is to use the radii which can be calculated from diffusion data by using the Stokes-Einstein equation for spheres. This is the assumption which was used by Friedman and Mills3 in their original paper. This result gives better agreement with the VCC for the thermodynamically near-ideal test systems, particularly in the calculation of the limiting concentration intercept of the solvent-solvent VCC. In VCC’s for the test this approximation the solute-solute systems deivate from the reference model. The agreement of the reference solute-solute VCC with the test systems data can be improved markedly by using the standard relations
Bender and Pecora I, I
I
-2
i i
i -3 1 IO
-5 0 0
0 2
0.4
06
0.8
0 2
00
MOLE FRACTION DMF
0 4
0 6
I O
0 8
MOLE FRACTION DMF
Figure 3. Actual and predicted fjfor the system N,N-dimethylformamide/water at 25 O C . Left: Actual and predictedf,, andf,, (0=fi2, = f l l , + = f i 2 predicted, A = f i l predicted). Right: Actual ( 0 ) and predicted ( m ) f 2 2 . The units of thef,, are lo9 mz s-l. The predictedf, were obtained by using actual KB parameters for the system and eq 23.
vll)
fll
f f l 2 M 2 / M , = -Dl
(244
f22
+ f2IMI/M2 = -0s
(24b)
These relations are a consequence of substituting the expression for D I 2given in eq 2a into the two relations from eq 2b. In this work the solvent-solvent and solvent-solute VCC’s were first calculated by using eq 13b and 23 and then eq 24a was used to calculate the solute-solute VCC. Overall, the best agreement between the thermodynamically near-ideal test systems and the reference VCC’s resulted from using this algorithm. Obviously, other options exist for calculating the VCC. For instance, eq 24a and the sister relation eq 24b could have been used along with the eq 13b and 23 calculated solute-solvent VCC’s to calculate the two remaining VCC’s. However, since the quality of the model for determining uii and the VCC is being judged by the agreement of the predicted VCC’s with the thermodynamically near-ideal test systems, it was decided that the most successful algorithm would be used. This choice of calculating the VCC’s lead to small-magnitude violations of eq 24b but these errors may be in large part a result of the error in the data used, rather than an intrinsic failure on the part of our eq 23. The reference VCC’s presented in Figures 1 and 2 are calculated by using the method described in the above paragraph. In the case of acetone/water a comparison of the present4 and those calculated in the literature show that the two reference systems are quite ~ i m i l a r . ~The positive deviations of the water-water VCC cf22)are indicated by both reference systems. Also, the negative deviation of the water-solute VCC cf12)is indicated by both reference systems. A more thorough test of eq 23 is to now substitute the experimental KB parameters into eq 23 to predict the experimental VCC for these systems. Figures 3-10 show the experimental VCC for eight binary liquid mixtures and the predicted VCC calculated for those systems by use of eq 23 and experimental KB parameters for the respective systems. It becomes apparent by Figure 6 that the use of the experimental KB parameters in eq 23 is leading
d I
-2
i
I
I 00
0 2
0 4
0 6
0 8
3 . 0 0
I O
MOLE FRACTION METHANOL
1 , 0 2
.
, 04
,
. 0 6
,
.
I I O
0 8
MOLE FRACTION METHANOL
Figure 4. Actual and predictedf, for the system methanol/water at 25 OC. The key and units are the same as in Figure 3. The predictedf, were obtained by using actual KB parameters for the system and eq 23. I ,
I I
i
1
-]I
. 0
z
-
e,)
-1
I
3
00
0 2
0 4
0 6
08
MOLE FRACTION ETHANOL
I O
1
00
1 0 2
0 4
06
08
0
MOLE FRACTION ETHANOL
Figure 5. Actual and predictedf,j for the system ethanol/water at 25 OC. The key and units are the same as in Figure 3. The predictedfj were obtained by using actual K B parameters for the system and eq 23.
to qualitatively correct predictions of the VCC’s for the experimental systems. Note in the case of acetone that both the maximum in they2, and the minimum in theyl2 are predicted by the use of the KB parameters. Similar success is had with the system pyridine/water. In the CH,CN/water system one finds that the minimum in the predictedfi2 curve is shifted from the position of the actual f I 2 minimum but that the overall agreement
Velocity Cross-Correlation Coefficients
The Journal of Physical Chemistry, Vol. 93, No. 6,1989 2619
.
a
2-
0 4
0 6
0 8
I O
.OD
MOLE FRACTION ACETONE
0 2
0 4
0 6
0 8
0.00
I O
.
I
+'
0 OS
0.10
0 I5
0 00
MOLE FRACTION 2BE
MOLE FRACTION ACETONE
Figure 6. Actual and predicted J j for the system acetone/water at 25 "C. The key and units are the same as in Figure 3. The predictedf,
.
were obtained by using actual KB parameters for the system and eq 23. 2 5
+
Ei
*I
0 1
+
0
.-
0 0
IT-7
0
os
0 10
0 I5
MOLE FRACTION 2BE
Figure 9. Actual and predicted& for the system 2-butoxyethanol/water at 25 OC. The key and units are the same as in Figure 3. The predicted f J were obtained by using actual KB parameters for the system and eq
23. 10
-1
0 0 0
10
.-
-5.0
I+
I:
:=
-
2-
c
-
c
0
1-
c -6-
0 0
8 -
0.0
0.2
0.4
0.6
0.8
I
0 0
MOLE FRACTION PYRIDINE
0 2
0 4
0 6
0 8
I O
.
6-
c
0 0 2
0 4
0 6
0 1
I O
0 0
0 2
0 4
MOLE FRACTION CCL4
0 4
0 6
0 8
1.0
MOLE FRACTION CCL4
Actual and predicted f, for the system carbon tetrachloride/methanol at 25 OC. The key and units are the same as in Figure 3 . The predictedfi, were obtained by using actual KB parameters for the system and eq 23. Figure 10.
scribed by us,12~21 the BE commonly used by some researchers seems to have a chemical contaminant in it which leads to a phase separation at dilute BE concentrations. Since the presence of such a contaminant has been demonstrated to severely affect diffusion coefficients measured for the system, there is a possibility this diffusion data has a large systemtic error in it. The values of the diffusion coefficients used here are, however, quite reasonable for the size of the molecules in the liquid. The effect of a possible contaminant on the cyl2 value for this system is difficult to predict. Since cylz enters into both calculations for the VCC and the KB parameters it could be a significant error source.
4-
MOLE FRACTION CH3CN
I 0 2
Figure 7. Actual and predicted& for the system pyridine/water at 25 OC. The key and units are the same as in Figure 3. The predictedAj were obtained by using actual KB parameters for the system and eq 23.
S 0 0
-10
MOLE FRACTION PYRIDINE
0 6
0 8
I O
MOLE FRACTION CH3CN
Figure 8. Actual and predicted A, for the system acetonitrile/water at
25 "C. The key and units are the same as in Figure 3. The predicted f Jwere obtained by using actual KB parameters for the system and eq 23.
in qualitative trends in the data is good. Even in the extremely nonideal case of CCl,/MEOH eq 23 gives a prediction which is in good qualitative agreement with the observed VCC. There is, however, one exception to this general satisfactory behavior. In the case of BE/water the agreement of the f12 and f l l values improves with the use of experimental KB parameters but theyzz values suffer. The source of this discrepancy is not known, but there exists at least one possible explanation. As previously de-
Conclusion With the exception of the BE/water system the overall agreement of the current model (eq 23) with experimental VCC's is quite good. The connection of the KB parameters and the VCC's is clearly made by the successful calculation of the qualitative trends in the experimental VCC's by using eq 23 along with the experimental KB parameters. It should be stressed that these tests have been performed on systems which range from near-ideality (DMF/water) to suspected dimerizing nonideal systems (CCl,/MEOH). The use of as simple a hij(R)as that given by Ornstein and Zernike gives surprisingly good results in predicting VCC's for all of these systems. This is pleasing since the use of the O Z function for hi,(R) gives useful predictive equations of very simple mathematical form. Because of the (21) Bender, T. M.; Pecora, R. J . Phys. Chem. 1988, 92, 1675.
J. Phys. Chem. 1989, 93, 2620-2625
2620
success in the reasonably large number of test systems studied here, it would appear that the results obtained are not just fortuitous. Most importantly then, this work indicates that the relation between hij(R)and the VCC seems to be well founded. It would be useful to test eq 12 and 23 using the experimental hij(R) for real chemical systems. Unfortunately this data is currently scarce. One profitable approach might be the use of molecular dynamics computer simulations. Using these techniques one might directly calculate the averge velocity correlation between the different particle pairs. From the equilibrium radial distri-
bution function of the computer-generated binary system one could then also obtain values for the integral in eq 12.
Acknowledgment. This work was supported by NSF grant CHESS-14641 to R.P. Supplementary Material Available: References for the data used in calculating KB parameter and velocity cross-correlation coefficients, and tables of KB parameters for eight binary liquids at 25 "C (13 pages). Ordering information is given on any current masthead page.
X-ray Dtffraction and Raman Studies of Zinc( I I ) Chloride Hydrate Melts, ZnCI,*RH,O ( R = 1.8, 2.5, 3.0, 4.0, and 6.2) Toshio Yamaguchi,* Department of Chemistry, Faculty of Science, Fukuoka University, Nanakuma, Jonan-ku, Fukuoka 814-01, Japan
Shun-ichi Hayashi, and Hitoshi Ohtakit Department of Electronic Chemistry, Tokyo Institute of Technology, Nagatsuta, Midori- ku, Yokohama 227, Japan (Received: August 15, 1988)
X-ray and Raman scattering measurements have been performed at 25 OC on solutions of ZnCl2.RH2Owith R = 1.8, 2.5, 3.0, 4.0, and 6.2. Analysis of the X-ray radial distribution functions and model fittings revealed that tetrahedral species [ZnC1x(OH2)4-x](2-X)+ are predominantly formed in the solutions and that the average number of Zn-C1 interactions changes from -2.4 for the solution of R = 6.2 to -3.4 for the saturated solution of R = 1.8. Both X-ray scattering and Raman data confirmed an aggregate of tetrahedral species linked via C1 atoms at a corner of the tetrahedra in solutions of solute concentrationsabove about 10 mol dm-3;the nearest-neighborZn-Zn distance was -395 pm. Anomalies in the physicochemical data of the ZnCI2-H20 system are discussed from a structural point of view.
Introduction Hydrate melts, which are classified between aqueous solutions and anhydrous molten salts, have recently attracted the attention of people in the fields of both fundamental and applied chemistry; interest includes the basic physicochemical properties of the solutions, crystallization of the salts, supercooling of the solutions, and various technological applications in thermal energy storage.'-5 Among various divalent metal salts, zinc(I1) chloride highly dissolves in water up to a solute to water molar ratio less than 1:2 at ambient temperature (see the phase diagram6 in Figure 1). Various physicochemical data such as activity coefficients,' viscositie~,89~ and transport numbers1° have shown an inflection point at the solute concentration of about 10 mol dm-j, suggesting that some structural change occurs in the solution. Various techniques have been applied to investigate the microscopic structure and dynamical properties of aqueous zinc(I1) chloride solution at various concentrations. Raman spectroscopy has long been used for the system as reviewed in the literature;" their showed the presence of tetrahedral species, [ZnCIx(OH2)s,](2-X)+, together with a hexahydrated Zn2+ion in aqueous solutions less concentrated than 1 0 M. -Beyond 10 M formation of polymeric species similar to Cl--bridged Zn species found in the solid state was suggested. X-ray scattering measurement~'~ were ~ ' ~performed for aqueous ZnCI, solutions over a wide range of concentrations; the results confirmed a tetrahedral coordination of Zn2+ with halide ions and water molecules. However, a detailed analysis of the formation of the polymeric species suggested from the Raman studies was not carried out. 'Author to whom correspondence should be addressed. 'Present address: Coordination Chemistry Laboratories, Institute for Molecular Science, Myodaiji-cho, Okazaki 444, Japan.
0022-3654/89/2093-2620$01.50/0
TABLE I: Composition of ZnC12-RH20Solutions Investigated (mol dm-')" ZNOO ZN18 ZN25 ZN30 ZN40 ZN62
[Zn2*]
[Cl-]
18.7 12.8 11.4 10.4 8.71 6.70
36.9 25.5 22.9 20.7 17.4 13.4
' d is the density measured a t 25 OC.
[H,O] 0
R 0
23.3 28.4 31.1 34.8 41.9
1.8 2.5
O C
3.0 4.0 6.2
d / g cmT3 2.52 2.16 2.05 1.97 1.81 1.67
except that for ZNO024 at 330
Dynamic properties of ions and water molecules in concentrated aqueous solutions of ZnClz have been studied by NMRI9 and ( I ) Marcus, Y. In MoltenSalt Technology; Lovering, D . G., Ed.; Plenum: New York, 1982. (2) Gawron, K.; Schroder, J. Energy Res. 1977, 1 , 351. (3) Pillai, K. K.; Brinkworth, B. J. Appl. Energy 1976, 2, 205. (4) Bajnkzy, G.; Zijld, A. Appl. Energy 1982, 10, 97. (5) Kimura, H.; Kai, J. Sol. Energy 1984, 33, 49. (6) Gmelins Handbuch der Anorganischen Chemie; Verlag Chemie: Weinheim, 1924; Syst. No. 32, Zn, Part D12, p 162. (7) Foxton, F.; Shutt, W. J. Trans. Faraday Soc. 1927, 23, 480. (8) Verner, V.; Khviyuzov, P. Bumarhn. Prom. 1938, 16, 39. (9) Kasbekar, G . S. J . Uniu. Bombay Pt. 3 1940, 9, 55. (10) Harris, A. C.; Parton, H. N. Trans. Faraday Soc. 1940, 36, 1139. (1 1) Irish, D. E. In Ionic Interactions; Petrucci, S.,Ed.; Academic Press: New York, 1971; Vol. 11, pp 187-258. (12) Delwaulle, M. L. C. R. Acad. Sci. 1955, 240, 2132; Bull. SOC.Chim. Fr. 1955, 1294. (13) Irish, D. E.; McCarroll, B.; Young, T. F. J . Chem. Phys. 1963, 39, 3426. (14) Morris, D. F. C.; Short, E. L.; Slater, K. Electrochim. Acta 1963,8, 289.
0 1989 American Chemical Society
