5384
J. Phys. Chem. 1993,97, 5384-5391
Hydrogen-Bonding Solutions. 1. Partial Molar Volumes and Optical Susceptibilities for Water, Methanol, and Ethanol in 1,4-Dioxane
William H. Orttung,’ Tzu-Huei Christina Sun, Joseph M. Kim, Jeffrey J. Lehman, and Larissa L. Kalifb Department of Chemistry, University of California, Riverside, California 92521 Received: September 1, I992
Power series or rational polynomials have been fitted to the available density and refractive index data for the three binary solutionsover the full composition range at 25 O C . Excess molar volumes VE and optical susceptibilities YE were evaluated from the fits and were used to evaluate partial molar properties ViE and YiE, where i is 1 for the polar component and 2 for dioxane. The values were generally small and negative, although both VzE and Y2E were sharply negative for dilute dioxane in all three solutions. The YIEvalues appear to be slightly negative at high dilution, but better data are needed to confirm this. The optical susceptibility results were in qualitative agreement with predictions based on the Onsager-B6ttcher model theory, and component contributions to the YiE evaluated from the model theory were found to be large but opposite in sign and nearly canceling. This type of behavior is expected when one component (dioxane) is considerably more polarizable than the other. The available data are not sufficiently precise to resolve the effect of hydrogen bonding on the polarizabilities. The use of thermodynamic methods on molar susceptibilities, as illustrated in this paper, is recommended to both experimentalists and theorists. Analysis of dielectric data is carried out in a companion paper.
Introduction
Theoretical Framework
Guggenheim (e.g., ref 1) may have been the first to suggest the idea of using thermodynamic methods for dielectric analysis of solutions. He proposed that the molar refraction be analyzed like a thermodynamic property. The partial molar quantities combine additively according to the sum rule, and the deviations from the molar refractions of the pure components could be explained by molecular theory. Although the basic idea is a good one, the molar refraction may not be the best function to use for this purpose. The Lorentz local field, which is based on an oversimplified molecular model, is incorporated in its definition and does not always provide a good approximation of the field at the molecule. These thoughts led us to adopt the “molar susceptibility”, as defined in the next section, for the analysis of solution dielectric and optical data by thermodynamic methods. The molar susceptibility depends only on molecular weights, density, and dielectric (or refractive index) data, and it may be analyzed by standard methods for macroscopic variables. The molar susceptibility also has a straightforward interpretation in terms of the moments induced in the moleculesof the solution, as explained in the next section. Many of the traditional molecular model theories utilize the macroscopic dielectric constant in their formulation, and it is tempting to move these “macroscopic factors” into the function involving the experimental quantities. The use of the “molar susceptibility” avoids this temptation. The experimental and theoretical aspects of the problem are clearly separated, and only one function needs to be calculated from the data (rather than a different one for each theory). As a result, the comparison of experimental and theoretical values is more straightforward. The general approach recommended here seems to be implicit in recent computer simulation work on pure liquids (e.g., Sprikz), but it does not seem to have been used in the analysis of solution data.
Molar Susceptibility. The basic electrostatic relationship for an isotropic medium of dielectric constant e (or n*), is3 D=eE=E+kP (1) where D is the field of true (free) charges, E is the average (Maxwell) field in the medium, and P is the polarization (dipole moment per unit volume) induced by E. If the susceptibility x is introduced by the equation P=xE then
X =P -z6 -- 1X
(3)
For a binary mixture: (4)
where N is the total number of molecules per unit volume, Xi is the mole fraction of component i, M i a is the instantaneous dipole moment of a molecule of type i, e, is a unit vector along E, and the average is over all molecular degrees of freedom that affect the result. Equation 4 is sufficiently general for our purposes if the molecular contributions are additive. Interactions such as hydrogen bonding may alter the molecular properties but should not affect the validity of eq 4 if the contributions of the bonding molecules can still be assigned separately. For a binary mixture, N = NA/V, where N A is Avogadro’s number, V = M / p is the molar volume, M = M I X I+ M2Xz is the average molecular weight, and p is the density. (The expression for M is most easily derived by considering the contributions to thedensity.) Equations 3 and 4 maybe combined
0022-3654/93/2091-5384%04.~0/00 1993 American Chemical Society
The Journal of Physical Chemistry, Vol. 97, No. 20, 1993 5385
Hydrogen-Bonding Solutions to give
where Y might be called the 'molar susceptibility". The left side of this equation depends only on experimental data, while the right side can be derived from molecular considerations. The equation therefore allows straightfomard comparison of the data with molecular models of the solution. PvthlMolar Ropertka This topic will bediscussed in terms of the molar property V, but the results should be applicable to any molar property. For nl and n2 moles of components 1 and 2, P = P((P,T,nl,n2), where the asterisk indicates that P is extensive, is., proportional to n = nl + n2. The total differential of P is dP=($)T.cP+(g)p>T+(e)
P.T.n, d n , +
offer our own derivation: Since X I + X2 = 1, the right-hand side of eq 9 can be converted to V I + (V2 - Vl)X2. This expression is in the form of a straight line passing through point V(X2)with slope V2- VIand intercepts VI,V2at X 2 = 0, I , tespcctively. If the straight line is only required to coincide with the curve of V vs. X2 at a single point, then we cannot determine either VIor V2. (Geometrically, the slope of the line is not fixed by one point.) Fortunately, a second relation may be deduced. The slope of the curve of V vs X2 is given by the derivative of eq 9: I
+
d P = V Idn,
+ V, dn, + n , dV, + n, dV,
(7)
Equating this form to eq 6 at constant P and T yields the GibbsDuhem relation: n , dV,
+ n2 dV, = 0
(8)
In its integrated form, the Gibbs-Duhem relation provides a relation between V Iand V I . It is convenient to divide the sum rule by n, the total moles in the system, toobtain a molar property, V=X,V, +X,V2 (9) The Gibbs-Duhem relation may correspondingly be written
XI
av,
av,
ax, + x, ax, = 0
atconstant Pand T. HerethechangesofdVI anddV2areinduced by a change of mole fraction dX2. Equation 10 is often used to evaluate VIfrom VIif V, is a known function of composition. For example, the vapor pressure of solvent over an electrolyte solution can b e d toobtain the partial molar freeenergyoftheelcctrolyte as a function of concentration (relative to its value at infinite dilution). Another well-known use of eq 10 is to show that if one component obeys Raoult's law, then theother must obey Henry's law, and vice versa. A different viewpoint if needed for the prescnt analysis. If the molar property V of a solution is known as a function of composition (e.g., from experiment or theoretical prediction), theneq9and theGibbs-Duhemrelationineq IOallow the partial mdar propertiesofthecomponents tobeevaluated by theintercept method, which is derived below. In this context, we have two equations and two unknowns. This seemingly trivial observation has two important consequences: First, it provides a practical basis for methods of evaluating the V,. Second, it shows that since the partial molar properties can be derived from the molar property and its concentration derivative, they do not contain additional or independent information. On the other hand, the partial molar properties are potentially useful in understanding the interactions that occur in solutions. Intercept Mctk& Many textbooks give derivations of this technique for evaluating the V,from the molar function V. The most satisfying that we have found is by Rowlinson and Swinton.s However. since the method is important for our application, we
P.T
P.T
(11)
The last two terms may be dropped according to the GibbsDuhem relation in eq 10. Thus we see that our straight line is tangent to the V vs X2 curve, so that V and its derivative at X2 provide the information necessary to evaluate V I and V2. In particular, if we sotve eq 1 1 for V2 or VI and substitute in eq 9, we obtain the following explicit equations: V I= V - X
and we define VI= (dP/dn,)p.T,,, as the partial molar property of component i. At constant P and T, eq 6 may be integrated at constant composition (since the V,are then constants) to yield the 'sum rule": P = nlVI n2V2,whose total differential is
( 5 ) +x("i)
(E)
ax2
and V2 = V + X
P.T
' (") ax2
(12) P.T
'Excess" properties are often defined relative to the ideal solution. For some properties (such as molar susceptibility), then is no clear definition of ideal behavior, and we define the excess property to be relative to linear interpolation between the pure liquid properties
v" = v - ( V , O X , + V20X2)= (VI - Vl0)XI + (V, - V20)X2 (13) where the VIoare the pure liquid properties. If we take V,E = V,- VIo,then eq 13 may be written
VE = V,EX, + V,EX,
(14)
The derivation of the intercept method for an excess property YE follows closely that given above for V, and the results are in the same form V I E = V E - X (a @ x2)
P.T
and V : = F + X I( eax)2
P.f
(15)
Thesc equations may be given a slightly different form that is also useful. If we start from the first equality of eq 13, then
and substitution in eqs I S gives VIti= V - VIo - X
(E) 8x2
P.T
V,E =
and
v - V20 + x (") ax2
(17) P.T
Thus the exass partial molar property is obtained from the partial molar property by subtracting V,". The VIEof a p u n component is always zero. When V(X2)or YE(X2)is near X 2 = 0 or I , the extrapolation of the tangent to theother axis may kcome uncertain. particularly if the data show scatter or if the fit oscillatar. This problem is not as important in the evatuation of theoretical models, since the use of smooth functions usually minimizes the difficulty. When needed, the following dilute component method may be useful and can provide an independent analysis of the dilute solution data. Dilute Componnt Method. Let V'be the property of a solution of mole fraction XI in the dilute (solute) component, such that the solution contains 1 mol of (solvent) component i and X ; mol
5386 The Journal of Physical Chemistry, Vol. 97, No.20, 1993
of component j
Orttung et al. modified q s 26
V'= V ( l + X ) ) where Vis the molar property. Then
(18)
Equations 27 and 28 enable us to understand the magnitudes of the yiein more detail. In thermodynamics,it is well-known that the partial molar properties depend on the concentrations of all components of the system. Here we have an example of a moleculartheory predictingthe contributionsof each component to YIEand to Yze. It will be seen below that this information is of considerable interest, even though it cannot be obtained directly from the data.
and
so that eq 18 beuunes
For a set of data, V'and X ; can be calculated for each Vand Xi. We then have V, = (aV'/aXi)p,T,n,, which can be evaluated by standard numerical methods. The Onsager-Btittcher Molecuhr Model. A molecule of type i with polarizability aoiin the vapor is assumed to be in a spherical cavity of radius aiin a liquid of dielectricconstant Q = n2. Equation 5 then takes the form
I--
2n2
+ 1 a;
The derivation of this equation is discussed in a more general context in the following paper.6 For pure component i, we find
If the cavity radii, ai,are estimated from molecular models, then eq 23 provides estimates of aoi for the pure liquids. For polar hydrogen bonding liquids, these estimates are probably more realistic than the actual vapor polarizabilities of the isolated nonhydrogen-bonded molecules. They also provide continuity in the application of the model to solution data. Molar mdPartialMolar Susceptibilitiesfrom Model Theories. Equation 22 may be written more concisely as
Y = X , C , +x,c2 (24) Superficialcomparison with eq 9 suggeststhat Cimight be equal to yi. The correct result is not this simple, as may be seen by the followinganalysis. Equation 24 may be differentiated at constant T and P to give
Substitution in eqs 12 then leads to
Y, = C,- X2( X, aac1~+ X, s ) and
Y 2 = c 2 + x 1 ( x ac2) 1 ~ +(26)x 2 We can now see that Ci = yi only if the Ci satisfy the GibbsDuhem relation as expressed in eq 10, but there is no reason to think that the C,of the Onsager-BOttcher model (or any such model) should obey the Gibbs-Duhem relation. According to eqs 17, eqs 26 can be converted into yiE by subtracting Y," = Crofrom the right side of each equation. The molecular model derivation of the UiE can alsoprovide information about thecomponentcontributions. Thusif yifistheumtribution of component j to yiE,we obtain the following results from the
Compltatid Metbods Analytic expressions are required for the density p, refractive index n, and dielectric constant Q vs composition. Mole fraction seems more appropriate as a composition variable than weight or volume fraction because of eqs 4 and 5. A large size difference between the components can lead to significant differenbetween weight (or volume) fraction and mole fraction. For presentpurposes, component 1 is taken as the polar species (water or alcohol) and component 2 as l,Cdioxane, which is almost nonpolar. We chose to use mole fraction X2. Other variables such as XI- X2are also popular. From the computational point of view, identical results are obtained with either choice in the case of power series (but not in the case of rational polynomials). It has been noted that "Experimental results can be improved or damaged on their way from the laboratory to the practical application..."' Weagreewiththisohation. Thebeetmethod for a given set of data is often not obvious a priori. Before the computer age, even a least-squares fit to a power series was difficult, and more ingenious methods were recommended. (See, for example, Scatchard.8) Power SeriesFits. At the beginning of our study, least-squares fits to power series were attempted in all cases. For property V
Here the value of V at X2 = 0 is PO and the value at X2 = 1 is E&,. Ageneral multiple linear regreasion program was employed. The number of terms was increased until the variance (defined below) stopped decreasing. Severaldifficulties were encountered. Sometimesthe power series was unable to fit the functionalform of the data. More often, the power series oscillated in the gaps between poorly spaced data points. This effect was common with uniformly spaced weight fraction data that was not uniform against mole fraction. Even if oscillation was not obvious by direct plotting, it was sometimes evident in applicationsrequiring derivatives of the fit. R a t i d Polyllomirl Fits. The rational polynomial often provided an attractive alternative to the power series. We used ~the following standard form
+
Po p,x Y(X)
=
1 + Pm,+lX
+ p$ + ...+ p , , P + pm1+2x2+ + Pnppb
(30)
where x is X2 and y(x) is said to be of order [m,,mb]. This bracket notation is common in the theory of Pad6 approximants. The method of numbering parameters in eq 30 was convenient for our programs. The total number of parameters is n, = mt mb 1. To find the best values of the parameters in the least-squares sense, the simplex algorithm was used.9 To avoid false minima
+ +
The Journal of Physical Chemistry, Vol. 97, No. 20, 1993 5387
Hydrogen-Bonding Solutions TABLE I: Density Fit Parameters solno
ref
fit6
n~
lob
aarameters
WD
16 16 21 23 25
RP PS PS
6 6 7 6 6
9.1 5.1 na 11.2
0.9970285,4.223013, 12.77540,0.1084172/3.808397, 12.80399 , 0.9969960,0.4328530, -1.851533,4.071617, -4.992260,3.207838, -0.8375094 0.7865503,0.5602940, -0.6626106,0.6451290, -0.5232023,0.3351769, -0.1412027,0.02786560 0.7864613,0.5552786, -0.7523574, 1.038306, -1.134900,0.7349035, -0.1995117 0.785040,0.379196,4.332873,0.589700,-0).840219,0.637630, -0.190537
MD ED
PS PS
13
WD, MD, and ED refer to dioxane with water, methanol, and ethanol. PS and RP refer to power series and rational polynomial. A slash separates the m, + 1 numerator parameters from the mb denominator parameters in the RP fits. n is n, for a power serics and np for a rational polynomial.
TABLE Ik Refractive Index Fit Parameters soln”
ref
fit
n
104~
WD MD
15 20 23 24 + 26
RP RP
6 5 4 4
2.3 1 .o 1.35 1.9
ED
PS PS
parameters 1.33234,6.43410,9.75367,0.0891640/4.47945,6.92437 1.326698,8.362235, 11.10320/6.150021,7.489020
1.3266550.205 18 16, -0.2141 847,O.14939 18,4.04705430 1.359754,0.08492867,-0.02178330,-0.01429793,0.01161136
See Table I for explanation of the abbreviations.
in the simplex procedure, a first approximation was obtained by generating a rational polynomial from a continued fraction passing through np selected data points. Our implementation followed the references given by Kolling,Io with help from Hildebrand.11 Another treatment of this method was later found. (See ref 12, pp 9 1-4.) The continued fraction derivationlimits the exponents to mb = m,or mb = m,- 1. This limitation was not a serious inconvenience. (The form of the rational polynomial could always be altered before the simplex step, but we have not yet explored this possibility.) Oscillation of the rational polynomial fits between points was not observed. However, in some “hard-to-fit” cases, the coefficients of the denominator terms may be negative and unwanted singularities can occur. With care, this problem can be avoided. It does not occur in the results presented below. Convenient programs for the above procedures were written in the C language.13JZ Excess hoperty Fits. Excess properties (relative to the ideal solution or to linear interpolationbetween pure liquid properties) were often fitted to power series times an X I X Zenvelope function
which guaranteed a zero result at XZ= 0 or 1. This approach seemed adequate for our purposes, but if the p n values are of physical interest, it might be better to fit VE/XIXZ,leaving out the X Z = 0 and 1 end points.14 In many cases, a better representationof the excess property was obtained by calculating it from the fit of the basic data ( p , n, or e). In most cases, we tried both methods. The rational polynomial was not very useful for excess functions, since it had trouble with the two zeros at XZ = 0 and 1. Documentationof Fit Results. Power series are characterized by the maximum power, n,; and rational polynomials by the total number of parameters, np,and by [mt, mb],the maximum powers in the numerator and denominator. In both cases, u is the square root of -~ , i ) ~ / ( n d ,-, np),the variance of the data ye relative to the fit y,. The parameters are then listed consecutively: PO p n , for power series, and PO pm,/pm,+l p n for rational polynomials. Enough significant digits are reportedto reproduce the u of the computer fit, which carried 15 digits. For simplicity (of programming), the same number of digits is reported for each parameter, although some may require fewer than others. The number of significant digits reported should not be taken as a measure of the reliability of the individual parameters. Some authors trim the number of digits to reflect the reliability of the parameters (as estimated from the data), but the resulting “fit” may no longer serve as the best representation of the data. It may not even pass through the data!
-
-
-
Analytic Expressions for the Data The density fit parameters are shown in Table I and the refractive index fits are shown in Table 11. Detailed comments are provided below. Water-Dioxrw. Of the five density data sets considered (HSD,1STK,16Gr,17MS,18KK19) thatofTKhad thebestprecision and was confirmed by MS. Some of the other data (HSD, Gr) disagreed in the dioxane-rich region. Power series and rational polynomial fits of the TK data were obtained. The rational polynomial fit had a larger u but is considered to be more reliable since it did not oscillate at high XZ. Only one set of accurate refractive index data was found (HSDI5). A power series fit of n, = 7 showed small oscillation near X 2 = 1, 80 the rational polynomial fit was used. Methanol-Dioxane. At the beginning of our investigation, the only available density data (LMIC20) had u greater than 10-3. However, excess molar volume data reported at 30 and 35 OC by SVAZ1had simpler functional form than that calculated from the density data and was more precise. These Ve data were fitted to n, = 2 power series times XIXzat each temperature and linearly extrapolated to 25 OC at evenly spaced X2. It was then converted to density using pure component densities of 0.78655 for methanolZ2and 1.02800 for dioxane16 and fitted to an n, = 7 power series. Recently, an additional set of density data has become available (PZz3). We fitted it to an n, = 6 power series in XZ.The u of 1.12 X 10-4 was considerably smaller than the value of 1.39 X 10-4 reported for a similar fit of the reciprocal density.23 (We verified the latter value by repeating their calculation.) The PZ density was about 0.003 below the SVA density at XZ= 0.3. This discrepancyis far beyond the precision of either set of data. Since PZ found linear temperature dependence of the density, it seems unlikely that our linear extrapolation of the SVA data is at fault. Only one set of refractive index data was found initially (LMICZO). The beat power series fit gave u = 8.6 X 10-4 for n, = 3. After deleting points 2 and 3, which deviated from the curve, the best fit was for n, = 5, with u = 2.4 X 1W.Since this fit did not give a very plausible curve for YE, rational polynomial fits were tried. The best fit, for np = 5 and u = 1.0 X 10-4, was obtained when points 2,3,and 8 were deleted. The new PZ data were fitted to a power series of n, = 4 and u = 1.35 X 10-4. It was about 0.0015 higher at Xz= 0.4. As in the case of the density, the discrepancy is beyond the precision of either set of data. Ethanol-Dioxane. Three sets of density data were found (HYL,24 Gr,25 PPKZ6) and power series fits gave 104u= 1.6,1.3, and 16,respectively. We chose to use the Griffths data, from which the HYL data deviated less than 6 X lW. Two sets of
5388 The Journal of Physical Chemistry, Vol. 97, No. 20, 1993 I
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WD(TK) ,O MD(PZ) ,--MD(SVA),O ED(Gr)
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Orttung et al. 0 0
t I
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. 00
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X2 (DIOX MOLE FR) Figure 1. Excess molar volume in cm3 mol-'. Circles are for waterdioxane (TKI6), dashes are for methanoldioxane (SVA2'), squares are for methanol-dioxane (PZZ3),and diamonds are for ethanol4ioxane ( G r 9 . The curves are described in the text.
refractive index data were found (HYL,24PPKZ6), The best fit of the HYL data gave 104u= 1.9 for n, = 4, and the best fit of the PPK data gave 104u = 0.5 for nm= 5 . Plots of the excess functions raised doubts about which data set was more reliable, so the two sets were combined and fitted by n, = 4, with 104u = 1.9. Poor results were usually obtained from attempts to fit combined data sets, but it seemed to be the best choice in this case.
Calculations and Results The molecular weights of water, methanol, ethanol, and dioxane were taken as 18.015,32.04,46.07, and 88.11, respectively. All compositions are X2 (dioxane mole fraction), and all temperatures are 25 OC unless stated otherwise. Excess Partial Molar Volumes. The excess (molar) volume, VE, is given by eq 13. Excess volume is defined relative to the ideal solution, which corresponds to linear interpolation of the 6"of the pure liquids. Thus VE is simply the increment relative to the volume of the unmixed components. The molar volumes ofwater, methanol,ethanol,anddioxaneare 18.069,40.73,58.68, and 85.71 cm3 mol-', respectively. The excess volumes are shown in Figure 1. For water-dioxane, the rational polynomial fit of the TK density datal6 was used to calculate the VE curve on Figure 1. The data points (relative to their own pure liquid values) are also shown. For methanoldioxane, the SVA data21 at 30 and 35 OC were fitted to n, = 2 power series times XlX2 and linearly extrapolated to 25 OC at evenly spaced XZ.The results were then fitted to n, = 2 power series times XlXz and plotted directly. The PZ points23 are considerably less negative, and the curve is calculated from an n, = 6 power series fit of the density. For ethanol-dioxane, the VE data of GriffithsZ5were fitted to an n, = 4 power series times
XIX2. The partial molar volumes were evaluated by the intercept method from evenly spaced VE points (calculatedfrom the analytic expressions) using quadratic (three point) fits for derivatives. The results are shown in Figure 2. The dilute component method gave VZE= -4.9 cm3 mol-l for water-dioxane, in good agreement with the -5.1 from Figure 2 and with Morcom and Smith,18who
:I/ 0
t
'
v
.oo
.40
.80
X2 (DIOX MOLE FR) Figure 2. Excess partial molar volumes by the intercept method from the solid curves of Figure 1: water-dioxane (solid), methanol-dioxane (dashes), and ethanoldioxane (dash-dot-dash). Thecurves starting from zero at X2 = 0 are for the polar component (water, methanol, or ethanol), and the curves starting from zero at X2 = 1 are for dioxane.
reported 4 . 9 . We believe that the qualitative features of Figure 2 are as reliable as the availableinput data and that the situation cannot be improved significantly without better data. Excess PartialMolar OpticalSILpceptibaties. The ex= molar susceptibility, YE, is defined relative to linear interpolation of the pure liquid values according to eqs 5 and 13. The results are shown in Figure 3 for the density and refractive index data fitted in the preceding section. For methanol-dioxane, the SVA density21 and LMIC refractive indexZowere used. The PZ23results for this function werevery similar, perhaps because of cancellation of density and refractive index differences. The solid curves represent the YE values calculated from the fits, and the points were calculated from the refractive index data using the density fits. Since the pure liquid values for the curves differ slightly from the pure liquid data points, the curves may not appear to be the best fit of the YE points, but we believe that the curves provide the most reliable overall representation of YE. There is clearly significant uncertainty in the curves, particularly in the dilute regions near X2 = 0 and 1. The partial molar susceptibilities were evaluated by the intercept method from evenly spaced YE points (calculated from the analytic expressions for the refractive indices). The results are shown in Figure 4. The negative values of Y2 (dioxane) at infinite dilution clearly occur for all three systems. The infinite dilution values of Yl (polar component) are not as clearly determined, but all three are probably slightly negative. Onsager-JXittcher Model Predictions. Cavity radii a, of molecular size were estimated from Courtaulds atomic models,
Hydrogen-Bonding Solutions
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The Journal of Physical Chemistry, Vol. 97, No. 20, 1993 5389 I
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WATER-DIOXANE, 0 HSO (1 936) 0
0
S
0 t I
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METHANOL-DIOXANE 0 LMJC (1985), 0 PZ (1992) VI
I:
FROM EXPERIMENT
0 VI
Y
X
I
. 00
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ETHANOL-DIOXANE 0 HYL (1939), 0 PPK (1987)
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t 0 I
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. 00
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.40 .80 X2 ( D I O X MOLE FR) Figure 4. Excess partial molar optical susceptibilities calculated from the solid curves of Figure 3 by the intercept method: water-dioxane (solid), methanol-dioxane (long dash), ethanol-dioxane (dashdot-dash). The curves starting from zero at X2 = 0 are for the polar component (water, methanol, or ethanol), and the curves starting from zero at X2 = 1 are for dioxane.
.80
40
I
I
. 00
I
I
I
I
I
1 I
.80
.40
X2 ( O I O X MOLE FR) Figure 3. Excess optical molar susceptibilities: water-dioxane (top), methanol-dioxane (middle), and ethanoldioxane (bottom). The solid curves are calculated from the refractive index and density fits. The methanol-dioxane points from PZ2)were added after the solid curve was calculated. Dashed curves are calculated from the Onsager-Bdttcher model, as explained in the text.
and vacuum polarizabilities a,i of the molecules were estimated from the BBttcher formula in eq 23:
ONSAGER-BOTTCHER
'I
I
I
molecule:
water
methanol
ethanol
dioxane
a,/A aoi/A'
1.48 1.35
2.06
2.28 4.77
2.50 7.61
3.09
Equation 24, modified to give YE,was then evaluated to yield the dashed curves on Figure 3. The modified eqs 26 gave the KE curves of Figure 5. Finally, eqs 27 and 28 provided the component contributions to KEin Figures 6 and 7. The near cancellation of component contributions provides a qualitative explanation of the small observed
xE.
Discussion Volume Effects. Although we are not primarily interested in the partial molar volumes, they have some bearing on the optical and dielectric results. The treatment of Terasawa, Itsuki, and Arakawa (TIAZ7)can be applied to the results for water4ioxane solutions. They considered hydrocarbons, ethers (other than dioxane),alcohols, and glycols at high dilution in water. Starting
. 00
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1
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X2 ( D I O X MOLE FR) Figures. Like Figure 4, but for the Onsager-Battcher model, as explained in the text.
from VZE = - 4 . 9 cm3mol-' for infinitely dilute dioxane in water, we obtain V2= V20+ VZE= 85.7 - 4.9 = 80.8 cm3mol-'. From Figure 2 of their paper, the "van der Waals volumen V, of dioxane can then be estimatedas 46.7 cm3 mol-'. This volume is considered to be the actual volume of the molecule. Figure 4 of their paper then gives the "partial molar void volume" Vvoid = 34.3 cm3mol-' as the void volume added to the water per mole of dioxane added. While the latter value is smaller than the void volume in pure liquid dioxane (85.7 - 46.7 = 39.& it is surprisingly large, considering that dioxane is about 4 times larger than water. This analysis is compatible with the TIA treatment of other ether or hydrocarbon systems in water. A similar result is obtained by doing the calculation independently, for example, by using our estimate of the size of the dioxane molecule.
Orttung et al.
5390 The Journal of Physical Chemistry, Vol. 97, No. 20, 1993 0 (u
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X 2 ( D I O X MOLE FR) Figure 7. Like Figure 6, but for Y2E.
The tentativeconclusion of theabove calculation is that dioxane does not fit into the large iceiike cavities of the (pure) water structure. We estimate these cavities to have a diameter of about 3.76 A and a concentration ' / 3 that of the water molecules, corresponding to about 5.6cm3/mol of water. If this happened,
then V2EatX2= 0 would probably be more negative thanobserved. From a different point of view, the large partial molar void volume associated with the dioxane may actually be due in part to an expansion of the water structure upon introductionof the dioxane. This possibility is supported by the dielectric analysis in the following paper: which suggests that the dioxane participates in the hydroen bonding network of the water. The variation of YE with X2 in Figure 1 is also of interest. The marked minimum in YE for water-dioxane nearX2 = ' / 3 is closely related to the curves in Figure 2. The minimum in YE correlates with the equal values and opposite slopes of VI and V2 at this concentration, and it is probably not explainable in terms of a hypothetical DWzdioxanewater complex. Morcomand SmithIs provided data for a range of dioxane-like solutes in water. The minimum remains in the range X2 = 0.3-0.4 for molecules with one, two, or three ether groups, suggesting that the minimum might be explained by a less specific mechanism. The approach of Yoshimura, Nakahara, and co-workers,2swho consdidered differing solute/solvent size ratios and solventpacking fractions, might also be relevant to the present discussion. An excellent survey of important work in this area is included in another recent paper.29 OpticalMolar Susceptibility. The qualitative conclusion from Figure 3 is that the YE values for all three solutions are quite small. On the experimentalside, more accurate data are needed to show significant differences between the different systems. The Onsager-Mttcher model predictsalmost linear interpolation of YE for water-dioxane but not for the other two solutions. The deviation of the experimentalcurves from the Onsager-BBttcher predictions may indicate the effect of hydrogen bonding, but it is probably better to wait for more precise data before attempting this type of analysis. In general, it would be surprising if hydrogen bonding did not alter the polarizabilities of the components as the composition of the solution changed. The partial molar KEshown on Figures 4 and 5 suggest that Y2E is significantly negative near X2 = 0 for all three solutions. Since the Onsager-BBttcher prediction is similar, the explanation probably involves a general effect. There is a resemblance to Figure 2 for the partial molar volumes, but it must be remembered that the volume effects have been taken into considerationin the evaluation of the KE. It is likely that the true YIEvalues near X2= 1 are slightly negative for all three solutions, but this guess can only be verified when more precise data become available. Finally, the component contributions to YIEand Y2Eshown in Figures 6 and 7 are probably the most interesting part of the optical analysis. As pointed out below eq 28, this information can only be obtained from a molecular model calculation. In the present case, the Onsager-BBttcher model was used. The component contributions are quite large but nearly cancel. This fact helps to explain why the YE are so small in Figure 3. From the molecular point of view, the polar component has considerably lower polarizability than dioxane. When the less polarizable component is added to the solution, its own polarizability is raised, but the polarizability of the other component and of the molecules of its own kind already present is lowered. The situation is similar but reversed when the more polarizable component is added. Its own polarizability is lowered, but that of the other component and its own kind already present is raised. Thus a compensating effect is expected on general grounds. For the present systems, the compensation effect leads to surprisingly small YE for these solutions. The information provided by the above analysis is used in the following paper6 to analyze the dielectric data for the same set of solutions. References and Notes ( 1 ) Guggenheim,E. A. Thermodynamics. An Advanced Trearmenifor Chemists and Physicists, 3rd Ed.;North-Holland: Amsterdam, 1957. ( 2 ) Sprik, M.J . Chem. Phys. 1991, 95, 6762.
Hydrogen-Bonding Solutions (3) Brown, W. F.Handb. Phys. 1956, 17, 1. (4) A singleGreeksubscri~dcnotssav~toranda pairofGreeksubscripts denotes a sccond rank tensor. If the same Grtck subscript appcars twice in a symbol or product of symbols, summation over that subscript is implied. This notation is often referred to as the Einstein sum convention. ( 5 ) Rowlinson, J. S.;Swinton, F. L. Liquids and Liquid Mixtures, 3rd ed.;Butterworth: London, 1982. (6) Orttung, W. H.;Sun, T.-H. C.;Kim, J. M.J.Phys. Chem., following paper in this issue. (7) Redlich, 0.;Kister, A. T. Ind. Eng. Chem. 1948.40,345. (8) Scatchard, G. Chem. Rev. 1949,44,7. (9) Caceci, M.S.;Cacheris, W. P. BYTE 1984, May, 340. (10) Kolling, 0.W. A d . Chem. 1985.57. 1721. (1 1) Hildebrand, F. 8. Introduction to Numericul Analysis, 2nd ed.; McGraw-Hill: New York. 1974. (12) Presa, W.H.; Flanncry, B. P.; Teukolsky, S.A,; Vetterling, W. T. Numerical Recipes in C. The Art of Scientific Computing Cambridge University F”w Cambridge, 1988. (13) Harbmn, S.P.; Steele, G. L., Jr. C. A Reference Manual, 3rd. ed.; Prentice-Hall: Englcwood Cliffs, NJ, 1991. (14) Marsh, K. N.; Richards, A. E. Ausr. J. Chem. 1980,33,2121. (1 5 ) Hovorka, F.;Schaefer, R. A.; Dreisbach, D. J. Am. Chem.Soc. 1936, 58, 2264.
The Journal of Physical Chemistry, VoI. 97, No.20, 1993 5391 (16) Tommila, E.; Koivisto, A. Suomen Kemistelehti 1948, 21B, 18. (17) GriMitb, V. S.1.Chun. Soc. 1952,1326. (18) Morcom, K. W.;Smith, R. W. Trans. FaradaySoc. 1918,66,1073. (19) Kbanarian, G.; Kent, L. J. Chsm. Soc., Farday Trans. 2 1981,77, 495. (20) Landauer, 0.; Mateescu, C.; Iulian, 0.; Costcanu. G. Rev. Roum. Chfm. 1985, 30, 651. (21) Singh, P. P.; Verma, D. V.; Arora, P. S . Thermochim.Acta 1976, 15. 267. (22) Gibson, R. E. J. Am. Chem. Soc. 1935,57,1551. (23) Papanastasiou, G. E.;Ziogas, I. 1.1. Chem. Eng. Datu 1992,37,167. (24) Hopkine, R. N.; Yerger, E. S.;Lynch, C. C. 1.Am. Chem. Soc. 1939, 61,2460. (25) Griffiths, V. S.J. Chem. Soc. 1954, 860. (26) Papanastasiou, G. E.; Papoutsis, A. D.; Kokkinidis, G. I. J . Chem. Eng. Data 1987,32, 377. (27) Terasawa, S.;Itsuki, H.;Arakawa, S . J . Phys. Chem. 1975,79.2345. (28) (a) Yoshimura, Y.; Nakahara, M.J . Chem. Phys. 1984,81,4080. (b) Ber. Bunsen-Ges Phys. Chem. 1985,89,426.104. (c) Yoshimura, Y.; Osugi, J.; Nakahara, M. Ber. Bunsen-Ges Phys. Chem. 1985,89,25. (29) Nishimura, N.;Tanaka, T.; Motoyama, T. Can. J . Chem. 1987,65, 2248.
