J. Phys. Chem. 1984, 88, 3682-3684
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dipole interaction is enhanced in the well-extended *-conjugation system. The effect of the position of the SO3- group is remarkable as seen in the AQaS- ( K = ca. 1000 M-‘ at 25 “C) and AQPS- ( K = 10500 f 1100 M-I) systems. This effect suggests the structure of the complexes. A plausible structure of the TPPS3--AQPScomplex is shown in Figure 2. The planes of the porphyrin and anthraquinone rings may be close to each other and parallel to interact maximally via van der Waals forces. The SO3- group may be exposed to the aqueous phase and the carbonyl groups may be located on the r system of the porphyrin ring. Such a conformation is favorable for dipole-dipole and/or dipole-induced dipole interactions between TPPS3-and AQPS-. In the AQaSsystem, however, the SO3-group of the a-position has to overlap with the hydrophobic porphyrin ring and/or the benzene ring at the meso position if the relative positions of the carbonyl groups do not change. If the SO3-groups of AQaS- are exposed to the aqueous phase, a part of the hydrophobic anthraquinone ring has to be located in the aqueous phase. Thus, AQaS- seems to be a sterically unfavorable complexing agent for TPPS3-. Stacking of aromatic ions having the same charge has been well-known for cationic dyes such as methylene blue, cyanines, proflavin, acridine orange, and thionine.I2 The spontaneous (12) (a) Mukerjee, P.; Ghosh, A. K. J . Phys. Chem. 1963, 67, 193. (b) West, W.; Pearce, S. Ibid. 1965,69, 1894. (c) Robinson, B. H.;Seelig-Loffer, A,; Schwarz, G. J. Chem. SOC.,Faraday Trans. I 1975, 71,815. (d) Dewey, T. G.; Wilson, P. S.; Turner, D. H. J . Am. Chem. SOC.1978, 100,4550. (e) Dewey, T. G . ; Raymond, D. A.; Turner, D. H. Ibid. 1979, 101, 5822.
stacking of these cationic dyes in water is characterized by the large favorable enthalpy and the considerably unfavorable entropy.Izc Robinson et al. have reported the thermodynamic parameters for the stacking of acridine orange as K = 18000 M-’ at 23 “C, AH = -37.4 kJ/mol, and AS = -41.3 J/(mol.K).’2C In thermodynamic studies on self-aggregation, however, no effort has been made to evaluate the effects of the structure of the complexing agent on K , AH, and AS. Although hydrophobic interaction is characterized by large and positive AS, no boundary exists for AS to distinguish the binding force. For example, in spite of the expectation of hydrophobic interaction as a main force for micelle formation, the reported AS values for micelle formation are small and negative or small and ~ 0 s i t i v e . l ~In this thermodynamic study, we used several partners of TPPS3- which can be compared with each other. In conclusion, it was suggested that van der Waals interaction is the predominant force for complexation of TPPS3- with anionic aromatics and hydrophobic interaction assists the stabilization of the complexes in certain cases.
Acknowledgment. We are indebted to Professor Seiji Shinkai for his helpful discussion. Registry No. NB.TPPS3-, 90792-20-4; NBS--TPPS3-, 90792-21-5; AQaS-eTPPS-, 90792-22-6; PyS-.TPPS’-, 90792-24-8; AQDS2-*TPPS3-, 90792-26-0; NphaS-*TPPS’-, 90792-27- 1; NphPS-.TPPS’-, 90792-28-2;
[email protected]’-, 90792-29-3; AnDS2-.TPPS3-,90792-3 1-7. (13) Fendler, J. H.; Fendler, E. J. “Catalysis in Micellar and Macromolecular Systems”; Academic Press: New York, 1975; Chapter 2.
Potential Difference across a Double Layer Containing an Asymmetric Electrolyte in the Mean Spherical Approximation Antoine F. Khater, Instituto de Fisica, Universidade Federal Fluminense, 24000 Niter6, RJ, Brazil, and Departement de Physique, Universiti Lebanaise, Hadath Beirut 61 60, Lebanon
Douglas Henderson,* IBM Research Laboratory, San Jose, California 951 93
Lesser Blum, and L. B. Bhuiyan Department of Physics, University of Puerto Rico, Rio Piedras, Puerto Rico 00931 (Received: November 22, 1983; In Final Form: January 23, 1984)
Analytic expressions, obtained from the mean spherical approximation, for the double-layer potential difference and density profile contact values at the electrode are given for a general electrolyte containing an arbitrary number of components whose diameter and charge are unrestricted. These results are exact, within the mean spherical approximation. A low-concentration expression for the mean spherical approximation density profiles is also given for the general electrolyte. Because of the limitations inherent in the mean spherical approximation, these results are applicable only near the potential of zero charge. However, this is the region where the effects of asymmetries are most apparent.
Introduction A double layer of charge is formed at the interface of an electrolyte and a charged electrode. Most of the theoretical work on the electrical double layer has been concerned with ions which are symmetric in both charge and diameter. Theoretical studies of double layers containing salts with asymmetric charge include those of Grahame’ and Bhuiyan et a].: who used the Poisson-Boltzmann theory, Torrie and V a l l e a ~ , ~ (1) Grahame, D. C. J . Chem. Phys. 1953, 21, 1054.
0022-365418412088-3682$01.50/0
who made a computer simulation using the Monte Carlo method, B l ~ m who , ~ used the mean spherical approximation (MSA), Bhuiyan et al.,5 who used the modified Poisson-Boltzmann approximation, and Lozada-Cassou and Henderson,6 who used the (2) Bhuiyan, L. B.; Blum, L.; Henderson, D. J. Chem. Phys. 1983,78,442. ( 3 ) Torrie, G. M.; Valleau, J. P. J . Phys. Chem. 1982, 86, 3251. (4) Blum, L. J. Phys. Chem. 1977,81, 136. (5) Bhuiyan, L. B.; Outhwaite, C. W.; Levine, S. Mol. Phys. 1981, 42, 1271. Outhwaite, C. W.; Bhuiyan, L. B. J . Chem. SOC.,Faraday Trans. 2
1983, 79, 707.
0 1984 American Chemical Society
Potential Difference across a Double Layer
The Journal of Physical Chemistry, Vol. 88, No. 16, 1984 3683
hypernetted chain approximation. Only the studies of Valleau and Torrie’ and Bhuiyan et aL2 consider salts which are asymmetric in the ionic diameters. Even in these studies, the effect of differences in diameter is only partly included. Both studies use the Poisson-Boltzmann theory and, as a result, the size asymmetry in diameter manifests itself only in the different distances of closest approach of the ions to the electrode. The ions are treated as point ions when the ion-ion interaction is considered. In this note, we apply the MSA to obtain an analytic expression for the density profiles and the potential difference across a double layer in which the ions are completely arbitrary. That is, the salt need not be binary and the ions can differ in either charge or diameter or both. We use the general primitive model for the electrolyte, where the solvent is considered to be a uniform dielectric continuum and the ions are considered to be charged hard spheres whose pair interaction is given by
uij(r) = m r < uij = zizje2/tr r
> ujj
(1)
If the electrode sphere has charge Q and diameter d, eq 3 becomes
where the density of the electrode sphere is 1/ V and is vanishingly small. We have not calculated terms of order l/Varising from c or D because they are independent of Q and so do not contribute to the potential. The term of order 1/ V arising from r in the first two terms in eq 9 is identically zero. Because of this and the fact that the density of the electrode sphere is small, the index in the sums in eq 9 and in the definition of c, r, B, and D can be restricted to the n values corresponding to the bulk ions. Thus, r, c, B, and D have their bulk values. The first two terms in eq 9 give the bulk free energy of the electrode. Thus, the surface energy is
where
=
ajj
(Uii
+ Ujj)/2
(2)
and zie and uii are the charge and diameter of the ions of species i, and c is the dielectric constant of the electrolyte. Because the MSA is a linearized theory, this result is valid only near the potential of zero charge (pzc). This limitation is not too serious since the effects of asymmetry are most apparent for a binary salt, at least, near the pzc. Away from the pzc, the double layer is dominated by the counterions and, for a binary system at least, the properties of the double layer are essentially those of a symmetric salt with the charge and diameter of the counterionS2s3
Theory Blum and collaborators* have obtained the MSA thermodynamic and correlation functions for a completely arbitrary system of charged hard spheres. Our procedure is to treat the electrode as the (n + 1)th species in a system of charged hard spheres. By allowing the (n 1)th ion species to be exceedingly large but present in infinite dilution, these results can be applied to obtain results for an electrical double layer. For example, the MSA expression for the excess Helmholtz free energy (beyond uncharged hard spheres) is
-
where the second term of eq 1 is obtained by expanding in powers of d-l [noting that Q o(&)].The term -@/td is just the energy of an isolated charged sphere. The free energy of the double layer is thus
Q2
2eQ cB
A D L = z - , r ( D )
where we have assumed that higher order terms in h’ are negligible. The potential difference across the double layer is
4 =~ADL/~Q
+
If E / ~ is T the charge density on the electrode sphere, Q = E&/4 and E - 4ecB/D 4= (13)
4w
If all the ions have the same diameter, aii = a, Le., the case of the restricted primitive model, B vanishes because of the bulk neutrality condition
where
czjpj = 0 i
(4)
r2= (flae2/t)CpJriZ i
(5)
xi = (zi - C B U , : / D ) ( ~+ raJ1
(6)
+ I’uii)-’ D = 1 + ccpiuj,yi + ruii)-* B = cpiaijzi(l i
(7)
(8)
I
Also, p i is the mean number density of the ith species, fl = l/kT, k is the Boltzmann constant, and p = The index i in eq 3-8 can take n + 1 values (for the n species in the bulk electrolyte and the (n + 1)th component, the large electrode sphere).
zipi.
(14)
and eq 13 reduces to Blum’s result4
4 = E/[42r)l
(15)
for the restricted primitive model. Because the second term in the numerator of eq 13 is independent of Q (or E ) , the differential capacitance is C D = dQ/dY = (q24r (16) which is the same as Blum’s result4 for the restricted primitive model. If the difference in diameter of the bulk ions is not too large and if the concentration is not too high
( 6 ) Lozada-Cassou, M.; Henderson, D. J . Phys. Chem. 1983,87, 2821. (7) Valleau, J. P.; Torrie, G. M. J . Chem. Phys. 1982, 76, 4623. (8) Blum, L. MoI. Phys. 1975, 30, 1529. Blum, L.; Hmye, J. S.J . Phys. Chem. 1977, 81, 1311. Haye, J. S.; Blum, L. Mol. Phys. 1978, 35, 299. Hiroike, K. Ibid. 1977, 33, 1195.
.
J. Phys. Chem. 1984, 88, 3684-3688
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are not too large, we may write
Substitution of these expressions into eq 13 gives
Pzie(E - Czjpjuii) gSx) where K~ = 4 n B e 2 ( C i ~ , z , 2 ) / c . In a similar manner, the contact values for the density profiles at the electrode can be obtained from the general MSA express i o w 8 The result is
g,(%/2) =
where Em
= (r/6)Cpiuiim
(21)
1
the summation being over the n components of the bulk solution. The first term in eq 20 is the uncharged hard sphere/uncharged hard wall term. The second term is obtained from -2/32XJr,,/[e(u,, 41 by using the same techniques which were used to obtain eq 1 1 . Comparison of eq 13 and 20 shows that the second term in eq 20 vanishes when the potential difference across the double layer is zero. However, the first term does not vanish and is greater for the larger spheres so that the contact values are qualitatively similar to those 0btained~9~ from the Poisson-Boltzmann approximation. If the difference in diameter of the bulk ions is not too large and if the concentration is not too high, CBU,?/Dis small compared to z p Thus
+
- 2ae~z1p,o,) g , ( a , , / 2 )= 1 -
EK
(22)
We do not neglect 2rexJz,p,uJ,compared to E , since E can be small (and is zero at the pzc). If all of the ions have the same diameter, the MSA density profiles are given by
where go@) is the density for hard spheres near a hard wall and if f ( x - u / 2 ) is a complicated function. Explicit results for go(x) and f ( x - u / 2 ) have been given by Henderson and Smith.g In principle, a general expression for the density profiles, valid for ions of different diameter, can be obtained from the general MSA expressions.* Unfortunately, the result is very complex. However, as long as the concentration and the size differences ~~~
(9) Henderson, D.; Smith, W. R. J . Stat. Phys. 1978, 19, 191.
=-
J
e-x(ra/i/2)
tK
x
> uii/2
(24)
Since
f ( x ) N e-Kx (25) at low concentrations, eq 23 is of this form at low concentrations. The mean electrostatic potential +(x) may be obtained by using
as may be verified by differentiating eq 26 twice to obtain Poisson’s equation. Both 4 = $(O) and $ ( x ) are unaffected by a change in the sign of the electrode. This symmetry is a result of the linearization inherent in the MSA.
Comments We have given some MSA results for double layers containing an asymmetric electrolyte. Neither the potential q5 nor the difference between the charged hard sphere/charged hard wall density profile and the hard sphere/hard wall density profile vanishes for an uncharged electrode ( E = 0) unless all the hard spheres have the same diameter. This is true in the PoissonBoltzmann theory als0.~9’ It is conventional to interpret a nonzero potential at zero charge in terms of non-Coulombic forces (Le., specific adsorption). However, it is not necessary to invoke such forces, as differences in diameter can also give rise to a nonzero potential at zero charge. This is not to say that specific adsorption is not an important phenomena in double-layer studies. However, estimates of the importance of specific adsorption, obtained by subtracting equal diameter Poisson-Boltzmann theory results from experimental studies, may well be misleading if there are sizable differences in ionic radii. As long as there are no differences in diameter, the MSA predicts that 4 is symmetric under a change of sign of the electrode charge. This is at variance with other s t ~ d i e s . ’ -However, ~~~~~ it is to be remembered that the MSA is valid only near the pzc and that, in this region, the symmetry is valid. The fact that the MSA results are valid only near the pzc is only a minor inconveniencesince away from the pzc the properties of the double layer are, to an excellent approximation, those of a symmetric electrolyte composed of ions of the charge and diameter of the counterions. Acknowledgment. A.F.K. thanks IBM/Brazil for a travel fellowship which made his visit to IBM/San Jose possible. This work was supported in part by NSF grant no. CHE80-01969.
Enthalpies of Hydration of Alkenes. 2. The n-Heptenes and n-Pentenes Kenneth B. Wiberg,* David J. Wasserrnan, and Eric Martin Department of Chemistry, Yale University, New Haven, Connecticut 0651 I (Received: December 6, 1983) The enthalpies of reaction of the five n-heptenes with trifluoroacetic acid in the presence of a strong acid catalyst have been measured. A combination of these data with the available combustion data for the alkenes allows the enthalpies of formation to be determined with higher precision than previously possible. The differences in enthalpies of formation of the n-pentenes also were determined. The enthalpies of reaction of the three n-heptyl alcohols with trifluoroacetic anhydride were measured, and, when combined with the above data, permits the determination of the enthalpies of formation of the alcohols. Structural effects on the enthalpies of formation are discussed.
We have shown that it is possible to determine the enthalpies of hydration of alkenes via measurements of the enthalpies of 0022-3654/84/2088-3684$01.50/0
reaction of the alkene, the corresponding alcohol, and of water with a reaction medium consisting of 0.25 M trifluoroacetic an@ 1984 American Chemical Society
