The Journal of
Physical Chemistry
0 Copyright, 1984, by the American Chemical Society
VOLUME 88, NUMBER 12
JUNE 7, 1984
LETTERS A Note on the Molecular Models for Unassoclated Solvent Structure at Polarizable Interfaces Z. Borkowska* and J. Stafiej Institute of Physical Chemistry, Polish Academy of Sciences, Kasprzaka 44/52, 01 -224 Warsaw, Poland (Received: November 29, 1983; In Final Form: January I O , 1984)
The possible degeneracy of the orientational states in the multistate model in general, and the 3-state model in particular, is discussed and introduced into the residual energy term. The importance of this correction in the calculation of the temperature dependence of the inner-layer potential drop and the differential capacity is shown by using experimental data for the Hg/methanol interface published previously.
Introduction Molecular models for the solvent structure at polarizable interfaces have been used with considerable success in describing the dielectric properties of the inner layer at the electrode/electrolyte solution interfere in a variety of solvents.' In these models the inner layer is represented as a monolayer of molecular solvent dipoles, whose orientation depends on the local electrical field. In the case of relatively unassociated solvents such as methanol (MeOH)2 and dimethylformamide (DMF)? the three-state model4 has been particularly successful in describing curves of the inner-layer capacity against charge density, Ci = f(u), for the mercury/electrolyte solution interface. For the mercury/MeOH interface this model reproduces also the qualitative features of the temperature coefficient of the inner-layer capacity, namely, the a( l/Ci)/aT vs. a relationship, and the charge dependence of the temperature coefficient of the inner-layer potential drop dA@oM-2/aT, although substantial quantitative disagreement with ~
~~
~~
( 1 ) Fawcett, W. R. Isr. J . Chem. 1979, 18, 3 . (2) Borkowska, Z . ; Fawcett, W. R. Can. J . Chem. 1981,59, 710. (3) Fawcett, W.R.;Ikada, B.; Sellan, J. B. Can. J. Chem. 1979, 57,2268. (4) Fawcett, W. R. J. Phys. Chem. 82, 1978, 1385.
0022-3654/84/2088-2427$01.50/0
the experimental data is o b ~ e r v e d . ~ It , ~ seems to us that this disagreement originates partially from a simplification in the formalism used in the model.
Theory Let us consider the most general version of the molecular model mentioned above for unassociated solvents, published recently by Fawcett and de Nobriga, in which the solvent dipoles are allowed to assume an unspecified number of orientations6 In this paper the following formula for the mole fractions of the molecules in the ith state of orientation is given (eq 12 or ref 6):
fi = N
~ / N= ~ (l/~)aie-PiIkTeu,~Z/zkt
(1)
whereh is the mole fraction of the molecules in the ith state, Le., with their permanent dipole moment vector, p, forming an angle Oi with an axis normal to the electrode surface. pi is the projection of the permanent dipole moment on this axis. It follows that pi = p cos
f9i
(2)
(5) Borkowska, Z.; Fawcett, W. R. Can. J . Chem. 1982, 60, 1787. (6) Fawcett, W.R.; de Nobriga, R. M. J . Phys. Chem. 1982, 86, 371.
0 1984 American Chemical Society
2428
The Journal of Physical Chemistry, Vol. 88, No. 12, 1984
ai is the polarizability of the molecule in the ith state in the direction normal to the electrode surface. e is the local field which is given by the sum of the field due to the charge on the electrode E and the mean field in the same direction at a given dipole site due to surrounding dipole X . ai is defined as ai = e-Uri/kT (3)
Letters TABLE I: Coordinates of the Capacity Minimum for NaC104 Solutions in Methanol at Various Temperatures together with the Calculated Residual Energies Uriand the Weights w, of a Given Orientational State in the Three-State Modelo
where Uri is the so-called residual energy accounting for nonelectrostatic contributions to the energy of the dipole in the ith orientation state, including the specific interactions between the atoms of the solvent molecules adjacent to the interface and the atoms of the electrode, and q is the partition function:
0 25 54
i
(4)
Equations 1-4 give the basis for the analysis of the dielectric properties of the inner layer,6 such as the potential drop across the surface layer, A@, the differential capacity, C, and the temperature coefficients of these quantities. The expressions for the mole fractions and the partition function (eq 1-4) can be rewritten as follows:
fi = e-ffilkT/q
(5)
q = Ce-ffiIkT
(6)
i
Crnd F m-2
0.05 0.05 0.05
0.0916 0.0950 0.0995
0.05 0.05 0.05
0.0916 0.0950 0.0995
(UZO - ulO)/ kJ mol-’ Na+
(u30
-
ulO)/
kJ mol-‘
-19.98 -19.98 -19.98
-17.95 -18.24 18.54
C104-c 0
q = &;p~/kTeai~~/2kT
C m-2
urnin/
t/’C
25 54
-16.44 -16.44 -16.44
-16.54 -16.79 -17.01
“Parameters used in the calculations are taken from ref 2. p = 1.66 D,(Y = 3.26 X nm3. b d = 0.335 nm, C, = 14.9, wl0= w20 = 1, ~ 3 =0 2.92. ‘d = 0.365 nm, C, = 13.2, wIo = wz0 = 1 , ~ 3 =0 3.76.
I
7 2 -‘ Y I
n
I
where
LL
H, = pit - a,t2/2
+ Urj
“E
0-
(7) c
The Hi given by eq 7 can be interpreted as the energy of interaction of a single dipole of a polarizable molecule with its environment andfi can be interpreted as the statistical-mechanical probability of occupancy of the ith orientation state by this dipole. In the approach of Fawcett and de Nobriga no account is made for the possible degeneracy of the orientational states due to the fact that the energy of a given dipole (eq 7) remains unchanged when rotation around the axis normal to the electrode surface is performed. To account for this degeneracy we introduce the weights wi which can be interpreted as the number of states contributing to the ith orientation state. Equations 5 and 6 under the above consideration become
fi = w,e-ffilkT/q
(8)
q = Cwie-fft/kT
(9)
i
Fawcett and de Nobriga implicitly assume that wi = 1 for all orientations. The weights can be incorporated into the residual energy term in the following way:
Uri = UOri - kT(1n wi)
(10)
It follows that the residual energy term U,, is temperature dependent, a fact that was not previously taken into account in the calculations of the temperature dependence of the inner-layer potential drop and the inner-layer c a p a ~ i t y . ~It, is ~ interesting to note that the configurational entropy, SK,of the monolayer calculated at a given temperature according to the equation
SK= - N T k u . l n f i i
(11)
does not depend on the weights of orientational states providing the latter are incorporated in the residual energies (eq 10). Generally, it should be possible to calculate the weights w ion the basis of quantum mechanical considerations but we do not consider this question further. In the three-state model4we consider three possible orientations of the solvent dipoles in the monolayer, namely, with the positive end toward the metal “up”, with the negative end toward the metal “down”, and with the dipole vector parallel to the metal surface “parallel”. The up and down states are represented by two poles and the parallel state by the equator of a sphere. It follows that there are infinitely more parallel states than up and down ones.
i ’D -\
2-
-2
\I
I
0.1
0
-0.1
Figure 1. The dependence of dc;I/aT on u for a NaC104 solution in methanol. Curve a is calculated according to the three-state model with the parameters for the Na+ ion found in ref 2 and with the weights wol = woz = 1 and ~ 0 =3 2.92. Curve b is calculated as in ref 2, Le., for wol = woz= wo3= 1. Curve c is determined experimentally.
Such a representation is an oversimplification and cannot be true, but one may expect more parallel states than the up and down ones. Application to the Experimental Data According to the three-state model one can evaluate the residual energies from the coordinates of the principal extremum of the inner-layer capacity, Ci,vs. charge density, u, curve provided that the dipole moment p , polarizability a, the effective coordination In the number C,, and the inner-layer thickness d a r e ~pecified.~ previous work,* C,, d , and the residual energy difference of the up and down states, U2,- Ul0,and parallel and down states, U,, - Ulo,have been found for methanolic solutions of various electrolytes at 25 OC by fitting the experimental Ci vs. u curves to the theoretical ones. The values of p and a has been taken as given in the literature for pure methanol. It has been shown that C, and d as well as the residual energies depend on the counterion. That means that the experimentally determined Ci vs. cr curves, unlike the theoretical ones, are not symmetrical around the principal extremum and cannot be described by one set of parameters. In the present work we have calculated the residual energies at three temperatures using two sets of C, and d values as found previously for Na+ and Clod- as counterions and the data for the dependence of C, on u determined for NaC104 solutions at various temperatures.2 The results together with the coordinates of the inner-layer capacity minimum are given in Table I. It is seen from the table that the residual energy difference of the up and down states remains constant with temperature; hence, on the basis
J. Phys. Chem. 1984,88, 2429-2432
I I
t / \
I \
I \I
0.1
0
0.1 G/Cm-2
Figure 2. The dependence of aC;'/aT on u for a NaC104 solution in methanol. Curve a is calculated according to the three-state model with the parameters for the C1Oc ion found in ref 2 and with the weights wol = wo2= 1 and wo3 = 3.76. Curve b is calculated as in ref 2, Le., for wol = woz = wo3 = 1. Curve c is determined experimentally.2
of eq 10 the weights of the up and down states are equal, wl0 = w20 = 1, where subscripts 10 and 20 refer to the up and down states, respectively. The residual energy difference of the parallel and up states changes with temperature. The weight for the parallel state calculated by linear regression according to eq 10 is equal to ~ 3 =0 2.92 f 0.26 for the Na+ ion and ~ 3 =0 3.76 f 0.21 for the C104- ion. Using these values one can calculate the dC;l/aT vs. u curves for both Na' and C104- ions. The resulting curves are presented in Figures 1 and 2 (curves a) together with those calculated for wl0 = w20 = ~ 3 =0 1 (curves b) and those evaluated from experimental data published p r e v i o u ~ l y(curves ~~~ c). The agreement between the experimentally determined curves
2429
and those calculated by using the weights is much better than with those calculated by using the simpler m 0 d e 1 . ~It~ ~is interesting to note that the theoretical curve calculated for the parameters obtained for the Na+ ion2 describe the experimentally determined dependence of dC;'/dT on u more accurately than that for the C104- ion. This can be a reflection of the fact that the calculated values of C,, d, U,,, and w, are based on the coordinates of the capacity minimum, which is situated in the region of negative charge densities where the influence of the cation, Na+, predominates. The temperature coefficient of the inner-layer potential drop at the point of zero charge dA@om-2/dT calculated by us on the basis of the parameters obtained for the Na+ ion is equal to -0.23 mV K-' and agrees much better with the experimental result, -0.166 mV K-',' than that obtained on the basis of the three-state model in its original version (dA@om2/dT= -0.72 mV K-l). The difference still existing between the calculated value of dAaOm2/dT and that estimated experimentally, which is equal to 0.06 mV K-I, may be attributed to the temperature coefficient of the work function of the electron in the meta18~9which has not been taken into account in evaluating the experimental value of dA@om2/dT.7 It is interesting to note that the low value of the weight for the parallel state indicates a high degree of ordering of the parallel dipoles in the surface layer which is consistent with the "headto-tail" arrangement of the dipoles found by Parsons and Reeves'O as a configuration of minimum energy.
Acknowledgment. Thank Prof. W. R. Fawcett for his helpful remarks. The financial support of Research Project M R 1-11 is gratefully acknowledged. (7) Borkowska, Z.; Fawcett, W. R.; Anantawan, S. J . Phys. Chem. 1980, 84, 2169. (8) Trasatti, S.J . Electroanal. Chem. 1971, 33, 351. (9) Trasatti, S. J. Electroanal. Chem. 1977, 82, 391. (10) Parsons, R.; Reeves, R. J . Electroanal. Chem. 1981, 123, 141.
A Model for the Metal-Support Effect Enhancing CO Hydrogenation Rates over Pt-Ti02 Catalysts M. Albert Vannice* and Chakka Sudhakar Department of Chemical Engineering, The Pennsylvania State University, University Park, Pennsylvania 16802 (Received: January 30, 1984; In Final Form: April 3, 1984)
Titanium was deposited on ultrahigh-purity 1-wm Pt particles, then oxidized and reduced. Surface areas determined by BET and H, and CO adsorption measurements did not change significantly as theoretical coverages of 1 and 10 monolayers of Ti were deposited; however, turnover frequencies based on chemisorbed hydrogen increased 4-fold and 40-fold, respectively, and approached the values measured on Ti02-supported Pt. These results argue against morphologicalchanges and alteration of the Pt Fermi level by electron transfer as explanations of this behavior, and they support a model of special active sites existing at the Pt-titania interface.
Introduction ~ i bis one ~ isupport ~ which has been found to have a profound influence on the catalytic activity and selectivity of metals in the co hydrogenation reaction,1-3 and is one metal whose specific activity in this reaction is markedly dependent upon the carrier chosen to disperse it, as turnover frequencies vary more (1) M. A. Vannice and R. L. Garten, J. Cutai., 56, 236 (1979). (2) M. A. Vannice, J . Catai., 74, 199 (1982). (3) "Studies in Surface Science and Catalysis", Vol. 11, "Metal-Support and Metal-Additive Effects in Catalysis", B. Imelik et al., Eds., Elsevier, New York, 1982.
0022-3654/84/2088-2429.$01.50/0
than 100-fold from the least active Pt/Si02 catalysts to the most active Pt/Ti02 catalyst^.^-^ Chemisorption studies on Ti02supported noble metals had shown that adsorption of H2 and C O behaved normally after a low-temperature reduction (LTR) near 473 K but it was markedly suppressed after a high-temperature (HTR) at 773 K, and it was shown not to be a consequence of Pt sintering.' The enhanced activity on titania(4) M. A. Vannice, S . H. Moon, and C . C . Twu, P r e p . Diu. Pet. Chem., Am. Chem. SOC.,23, 303 (1980). (5) M. A. Vannice, C. C . Twu, and S . H. Moon, J . Cutal., 79, 70 (1983). (6) M. A. Vannice and C . C. Twu, J. Cutal., 82, 213 (1983).
0 1984 American Chemical Society
