J. Phys. Chem. 1985,89, 1556-1558
1556
A New Method for the Calculation of Excess Electrochemical Free Energies of Mixing of Surface Solutions Marian Karolczak Uniwersytet im. Adama Mickiewicza, Wydzial Chemii, Zaklad Elektrochemii, ul. Grunwaldzka 6, 60-780 Poznan, Poland (Received: September 18, 1984)
A method is described which is free of a priori assumptions regarding the functional dependence of the excas electrochemical free energy. The energy values are extracted directly from the experimental data. It makes calculations easier, faster, and accessible to direct control. A preliminary test has been made of the interfacial behavior of 1-propanol at the potential of maximum adsorption at 293.15 K in aqueous sodium sulfate solutions at constant activity a, = 0.0708. The test revealed that neither the regular solution model nor the Flory-Huggins model of electrosorption can describe some of the thermodynamical implications resulting from the analysis of the experimental data. It has been shown that properties of the surface solution of the investigated system are under entropy control. A discussion of errors associated with the new method of calculations has also been presented.
Introduction The crucial role of the temperature d e p d e n c e of the excess electrochemical free energy of mixing, AGE, in the study of the structure of surface solutions has been recognized some time ago by Mohilner et al.' The significance of the dependence of AGE on xAads in generating the mathematical relationships for electrosorption isotherms, surface pressure equations, and their relations to interfacial models has been presented in an explicit manner icubsequent communications.2d A@ itself represents the net energy associated with all molecular interactions of a system, and its significance in the interpretation of experimental data is also very important. Mohilner et a1.l assumed a polynomial function to describe the dependence of ACE of the surface mole Such an assumption, made at an early fraction of adsorbate, xAndS. stage of analysis of the electrocapillary data, may provide some uncertainty due to the degree of the polynomial used, truncation errors in subsequent calculations, and/or numerical unstability of the formulas and algorithms used. Moreover, it restricts the possibilities of physical interpretation. It was therefore consid_ered worthwhile to search for a method of obtaining values of AGE in a more direct way and to analyze its functional dependence. The present paper develops such a method.
I
In (1 - xAa*)
awads
= (aw/awe) exp[-(yO - y ) / ( n R n m ) I
+ A G E / R T-
From the electrosorption isotherm, eq 2, one can subsequently calculate the activity of the adsorbed species, aAads,divided by a constant value, P' aAads/p = (awads/awads)"aA
AGE = AG, - AG,' AG,'/RT = xAadsIn
+ (1 - xAads)In (1 - xAads)
xAads
and the electrosorption isotherm (2)
derived previously.2 The symbols in these equations were defined in ref 2 and 6. Substitution of the relation between the activity coefficient in the adsorbed state and AGE, Le., eq 5 of ref 2, into eq 1 yields = - n R m , In (awadsuwe/aw)
(7)
is the electrochemical free energy of mixing of an ideal surface solution, and
AG,/RT = xAadsIn aAa*
[aw/(awexwads)]
+ (1 - xAads)In awads
(8)
+ XAads In (aA/[(aw)nXAads]] -
(Xwads
yo - y
(6)
where
Xwads In
(&ads)n
(5)
The excess electrochemical free energy of mixing of the surface solutions is defined as'
A G ~ / R T- xAadsIn p' =
pa),/ (a,)n = a,'*/
(4)
is the electrochemical free energy of mixing of the real surface solution at a specified temperature and electrical state. Equations 4-8 can be used to calculate values of the excess electrochemical free energy of mixing of the surface solutions, diminished by the value of (In P)xAa*for every experimental xAadS. Alternatively, one can use the formula
Results and Discussion The new method uses the equation for surface pressure yo - y = -nRTI',
After rearrangement, this equation allows us to calculate the activity of water in the adsorbed state directly from the experimental interfacial tension data, viz.
(3)
(1) Mohilner, D. M.; Nakadomari, H.; Mohilner, P. R. J . Phys. Chem. 1917, 81, 244. (2) Karolczak, M.; Mohilner, D. M. J. Phys. Chem. 1982, 86, 2840. (3) Karolczak, M.; Mohilner, D. M. J . Phys. Chem. 1982, 86, 2845. (4) Mohilner, D. M.; Karolczak, M. J . Phys. Chem. 1982, 86, 2838. (5) Karolczak, M. Electrochim. Acta, 1984, 29, 51. (6) Karolczak, M. J. Colloid Interface Sci. 1984, 97, 284.
0022-3654/85/2089-1556.$01.50/0
+ nXAads)(yO
- -y)/(nRTT,) ( 9 )
resulting from the substitution of eq 5 and 4 into eq 8 and of eq 8 and 7 into eq 6. The constant value of In P' is related to the standard electrochemical free energy of adsorption, based on the symmetrical choice of standard states for both bulk andd adsorbed species.l It can be calculated from the electrosorption isotherm, via ex= 1, in a manner similar to that for In in.' trapolation to xAadS It can also be obtained from the direct extrapolation of the ex= 1, AGE perimental data calculated from eq 9 since, for xAads = 0, and the extrapolated value must be equal to -In p'. In some instances, the relation derived in ref 6
can be helpful as well. The values of In pe can be determined accurately, as shown in ref 1. Subsequent recalculations via eq 0 1985 American Chemical Society
Excess Gibbs Energy of Mixing for Surface Solution
The Journal of Physical Chemistry, Vol. 89, No. 8, 1985 1557
0.3
0.2 b
-14
'
0.OL
0.1
Xfd" ads
A'
Y 0 0.2 0.L 0.6 0.8 1.o F i i 1. Excess electrochemicalfree energy of mixing AGE as a function of the mole fraction xAaL of adsorbed 1-propanol. The rectangles and bars denote 95% confidence limits. The continuous line represents a nonlinear nonweighted least-squares fit to the van Laar equation, with A , = 2.1787 and BI = 1.0347. The coefficient of determination was 0.9949. Y
V
0
I
I
I
.. I
I
I
I
0.2
5 or 9 at each xAads give the nonbiased value of A G E . We have also examined other methods of calculations of A@, e.g., numerical integration of a slightly modified form of eq 1. However, we have found them to be subject to the same limitation as the above algebraic method. Moreover, such methods do not let us evaluate directly the errors associated with the calculation of A P E .Note that the errors can be calculated in a straightforward manner from eq 9. Below we will discuss this interesting point. The system mercuryll-propanol water sodium sulfate at 293.15 K at the potential of maximum electrosorption, E = 0.0142 V vs. Hg(Pb)lPbS04, N a 2 S 0 4 (a = 0.07087),which we have studied recently,' will be used to illustrate the practical application of the present method. Figure 1 shows AGE vs. xAa*under these conditions. The line drawn represents the theoretical, nonweighted nonlinear least-squares fit to a van Laar isotherm6 which, tentatively, has been introduced at this stage of analysis. The corresponding activities in the adsorbed state are shown in Figure 2.
+
-
0.b
'
OJ2
Figure 3. Excess electrochemicalmixing functions at 278.15 K at the potential of maximum adsorption.
I
0.L 0.6 0.8 1.0 Figure 2. Activity of adsorbed 1-propanol (aA'*) and of adsorbed water (aW&)corresponding to the data of Figure 1.
(7) Karolczak,
'
+
M.; Mohilner, D. M., unpublished investigations.
The temperature dependence of ACE can be used to determine other excess functions of mixing, such as the excess electrochemical enthalpy of mixing, m,and the excesselectrochemical entropy of mixing, hSE. A representative example of the course of electrochemical mixing functions is presented in Figure 3. It is evident from Figure 3 that surface solution properties are under entropy control,' i.e. T p S E I 2 IAPl However, at the time of this writing, this statement can only be made at a moderate confidence level, of about 70%, and further improvement of the accuracy is necessary. It was suggested by Mohilner et a1.l that one should examine whether or not the above unequality is true. The conclusion from Figure 3 provides a partial answer to this question. In light of the discussion in ref 1 it may be concluded that neither the regular solution model nor the so-called Flory-Huggins model8 are consistent with the thermodynamic implications of the experimental data. Thus they must be eliminated from consideration for the investigated system. The Magnitude and Sources of Errors Involved in Calculations of AGE. The errors associated with calculations of ACE should be examined because of their indirect influence on many applications of these data. The total error of ACE has been calculated under the assumptions that the constants R, T, and rm were introduced without errors and that the error in the bulk mole fraction xA can be ignored. Calculations showed that errors in the bulk mole fraction were, indeed, very small. Consideration of other sources of error, using standard statistical procedure? leads to the following estimate for the error in A e E / R T
A ( A c E / R n = I[(n + l)(yo - ~ ) / ( n R m m ) 1 ~ I h A a d S ~
+
+ Ih~~~~ll hl aAl + (AXAadsllh p'I -k IXAadsllA In @'I+ In [XAads(l - XAads)]l + 3 1 h A a d S I
IX~"llAy/(Rm,)l
lhAadsll
(1 1) where IhAa is * thel error associated with the derivation of xAadS from interfacial tension data, and IA In 8'1 is the error associated with the determination of In @'. The error JhAadsl can be evaluated by making use of eq 5 of ref 10, suitably transformed into the new variable, viz. var xAads = (aXAads/ay)T,p,E2 var y (12) where var means "the variance of'. One can obtain the partial derivatives, (axAadS/8y)T,p,w either by numerical differentiation of thexperimental data or by esti(8) Parsons, R. J . Electroanal. Chem. 1964, 8, 93. (9) Dahlquist, C.; BjBrck, A. "Numerical Methods"; Prentice-Hall: Englewood Cliffs, NJ, 1974. (10) Nakadomari, H.; Mohilner, D. M.; Mohilner, P. R. J . Phys. Chem. 1976,80, 1761.
1558 The Journal of Physical Chemistry, Vol. 89, No. 8, 1985
2c
16
1;
e /
J
---
1
ads xA -1
I
I
I
I
0.2
0.4
0.6
2 ,C . -
0.8
I
I
1.0
Figure 4. The total error in the electrochemicalfree energy of mixing, A(AGE/RT), and its principal components as functions of the value of XA'?
mation from eq 23 of ref 2. In the latter case, if (a2AGE/ ~(X,~)~)T3 . ~ 0, . ELe., if the curvature of A@ =AxAa*) is positive or can be ignored, the following estimate can be made: I(ay/axAa*)T,P,El
nRIITm/(l - XAads)
(13)
because, in the common range of experimental concentrations, ~ is possible to introduce this term also (a In ~ J d x , ~ * )=~0.~ (It rigorously into the derivation, but calculations showed that it can be neglected in the range of concentrations used in practice.) Inequality 13 is ensured for systems in which the surface solution exhibits a negative deviation from Raoult's law, or in systems which are far from phase separation. Consequently, substituting inequality 13 into eq 12, one gets
Karolczak lation gives lhyAadslmax = 0.0004 when lAyl = 0.002 pJ cm-2 is used. This value of lAyl has been obtained as the 95% confidence level of the best available interfacial tension data." It was shown" that 1A7I is independent of the electrode potential. It is clear from Figure 1 that the efforts undertaken to improve the precision of electrocapillary measurements have paid off for the present calculations. From eq 14 we expect a decrease of IhxAa*1with an increase in xAa*. Considering a fivefold improvement in precision and the relation of xAa*to 6 or r (ref 2), this result is in agreement with observed behavior of errors, compare Figure 3 of ref 10. The total error of ACE/RTas well as some of its components has been plotted as the function of xAadS.The results are shown in Figure 4. One will note that the major contribution results from the error associated with the determination of the value of In 8'. At the time of this writing, the estimation of In 8' can be achieved with an accuracy of ca. 0.02. If this accuracy could be improved, the most significant source of error could avoided. The total errors are marked as rectangulars or vertical bars in Figure 1. It can be observed from Figures 1 and 4 that the errors are nearly C 0.15. This observations allows constant a t ca. 0.007 for xAads us to use a nonweighted nonlinear least-squares fit over this region, and this considerably simplifies all subsequent calculations.
Conclusions It is possible to calculate values of the electrochemical free energy of mixing of a surface solution from experimental electrmpillary data without having to make a priori assumptions on its functional dependence on surface mole fraction. The errors associated with such calculations can also be evaluated. Neither the regular solution model nor the Flory-Huggins model can describe all features of the electrosorption of 1-propanol at the mercurylaqueous sodium sulfate solution interface. Under the conditions considered here the properties of the investigated electrochemical system are under entropy control. Acknowledgment. I express my gratitude to Prof. R. de Levie of Georgetown University and Prof. D. M. Mohilner of Colorado State University for reading the manuscript as well as for their help and encouragement. Registry No. Mercury, 7439-97-6; 1-propanol,7 1-23-8; sodium sulfate, 1757-82-6.
The maximum value of IAxAadslmcan be estimated if appropriate numbers for IAyI and nRm, are introduced. The calcu-
(1 1) Mohilner, D. M.; Kakiuchi, T. J . Electrochem. SOC.1981, 128, 350.
