5370
J. Phys. Chem. 1991,95, 5370-5371
the same as H with OH-?' The standard free energy change in the heavy water equilibrium corresponding to reaction 1 can therefore be calculated (AGO = -31.7 f 1.0 kJ/mol), and the following thermodynamic cycle can be constructed: ( e - 1 ~ ~+0 H20(1) = (H)H,o + (OH-)H,O AGO = 25.3 f 0.4 kJ/moll ( D ) D+~ (OD-)Dp = (e-)Dp
+ D20(1)
AGO = -31.7 f 1.0 kJ/mol
f/2(H2)g + (OH-)HzO + (cI-)DzO + D20(1) = f/Z(DZ)g + AGO = 4.42 kJ/mo122 (OD-)D@ + (Cl-)H@ + H20(1) ( C l - ) ~ p= (Cl-)~]o AGO = 0.460 kJ/mo12 f/2(DJ8 + (HI, = (D), + y2(H2& AGO = 3.0 k J / m 0 1 ~ ~ AGO = (D), + (H)H,o = (HI, + (D)D,o -0.19 f 0.2 kJ/mol net:
=
AGO = 1.3 f 1.1 kJ/mol
The third line of the proposed cycle comes from coupled light waterlheavy water electrochemical cell experiments.22 The free energy of transfer of CI- from H 2 0 to D 2 0 (fourth line) is based on the standard that AGtmfrfor Na+ is zero.2 The sixth line represents the difference in solvation free energies of D in heavy water vs H in light water. The solvation of these atoms is hydrophobic in nature and presumably much the same as similarsized rare-gas atoms.' Given that AGtmfrfor both He and Ne is -0.19 kJIm01,~' it is safe to use the same number for the atomic hydrogen isotopes.2s (20) Hart, E. J.; Fielden, E. M. J . Phys. Chem. 1968, 72, 577. (21) Han, P.; Bartels, D. M. Chem. Phys. Lerr. 1989, 159, 538. (22) Goldblatt, M.; Jones, W. M. J . Chem. Phys. 1969, 51, 1881. (23) JANAF Thermochemical Tables: Nail. S r a d . Ref Dar. Ser., Narl. Bur. Srand. 1971, No. 37. (24) Abmimov, V. K.; Strakhov, A. N.; Krcstov, G. A. Zh. Srnrkr. Khim. 1976,17, 1027. The dYvalues tabulated in this work arc for raragas transfer from H20 to the deuterium-substituted water, despite the opposite statement in the text which would imply the opposite sign.
The result of this calculation implies that electrons are more stable when solvated in H 2 0 than in D20, again placing them in the "structure-breaking" category. The electron AGmfr is apparently more positive than the halide ions (iodide has the most positive A G e among classical singly charged ions), which suggests it may be the most effective structure breaker in the series, although the error limits are such that we can only be confident of the sign of this quantity.26
Conclusion In conclusion, we infer from both the entropy of hydration and the AGMr(H20--c D20) that electrons are very effective structurebreaking ions in water. The revised interpretation of solvation entropy and the AGwr both appear to be in reasonable qualitative agreement with recent quantum molecular dynamics simulations of the hydrated electron, which indicate a more disordered radial distribution of water molecules around electrons than around classical Acknowledgment. The authors thank Dr. Tom Tuttle for several useful discussions and his critical evaluation of ref 1, which f o r d us to reexamine the basis for the classical entropy equations. This work was performed under the auspices of the Office of Basic Energy Sciences, Division of Chemical Science, US-DOE, under Contract W-3 1-109-ENG-38. (25) H is more polarizable and morc quantum mechanical than He, which introduces some uncertainty into this comparison. However, the *0.2 W/mol uncertainty estimate is ample given the values for heavier (more polarizable) rare gases and the fact that the ,G A must be negative for a hydrophobic spcci~s.'~*'~ Compared to the other uncertainties listed, this one appears negligible. (26) The probable error in the AG,,,,,,is dmppointingly large in the present context, and better precision in the reaction rate of (e-)w with D20is needed to improve the situation. If the activation energy for this reaction were available, one could calculate the , S A and AHd, from the same therm* dynamic cycle, which should provide a much better comparison with the classical ions (AG, is a weaker indicator because of the compensationeffect between TAStmr,and AHHtmt,)*. We hope to address this point in future experiments.
Interactlon Potentlal for Cbo Molecules L.A. Cirifalco Department of Materials Science and Engineering, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6272 (Received: April 5, 1991)
An approximate intermolecular potential was computed for the interaction of two Csomolecules by matching lattice sums to solid-state data for lattice spacing and heat of sublimation. An analytic form was obtained by assuming that individual carbon atoms on different molecules interacted according to a Leonard-Jones potential and that the molecules could be treated as a spherical surface of uniformly distributed carbon atoms. Calculation of solid-state compressibility gave satisfactory agreement with experiment. The potential for two carbon atoms on different molecules was also computed and was found to be similar to that for two carbon atoms on different graphite sheets.
The interaction potential between two Cso molecules is the sum of the van der Waals potentials between the carbon atoms of one molecule with those of the other. This can be approximated by a single-potential function by treating each molecule as if it were a sphere with a surface consisting of a uniform density of smeared-out carbon atoms. The potential energy of interaction between two molecules whose centers are a distance r apart is then
0022-365419112095-5370$02.50/0
where u is the surface density of carbon atoms, given by 60/4.rra2, a being the radius of each sphere, and e is the potential function for the interaction of a carbon atom on one sphere with a carbon atom on the other. The integration is carried out over the surfaces 2, and Z2 of the two spheres. We take the interaction between two carbon atoms a distance x apart to be a 6-12 potential with attractive and repulsive constants A and B, respectively: e(x) = - A / x 6 + B/xI2 (2) 0 1991 American Chemical Society
The Journal of Physical Chemistry, Vol. 95. No. 14. 1991 5371
Letters Integration of (2) according to (1) gives the potential function
between the two spheres as
TABLE I: Coastants for the C,C, P0tClrti.l 10; 7 10.02; 7 10.02; 7.04 IO1%, erg 37.86 38,lO 35.96 1O"B, erg R09
A
10'2~,dyn/cm2
where s = r/2a. The constants
CY
4 =N%/90(2~)~~
a = MA/12(2a)"
TABLE II:
and 4 are given by
E
+
(5)
-CY&', @Si0
where r
1
s4=
--
-1
1
(7) The sums S4and Sloare the interactions of a central molecule with all other molecules in a crystal. These sums are functions only of = A/2u, A being half the edge length of a unit cell. They were evaluated for a number of values of A, including interactions out to intermolecular distances of about 20 sphere diameters. From data on the energy of sublimation, the nearest-neighbor distance, and the sphere diameter, the constants in both the C,-C, and the C-C potentials were obtained by using eq 5 and the condition that the energy must be a minimum at equilibrium. The sublimation energy was taken to be 40.1 kcal/mol.l Several values of the nearest-neighbor distance and the sphere diameter were used, all close to 10 and 7 A, respectively.2 This was done to ascertain the sensitivityof the computed constants to variations in the values of the experimental data. The constants A and B are proportional to the sublimation energy, but their dependence on the nearest-neighbor distance and the sphere diameter is more complex. The results are shown in Tables I and 11. In Table I, &, is the equilibrium distance between two C, molecules, at which the energy is a minimum. In Table 11, Ro is the equilibrium
x
(1) Margrave, J. L.; Hauge, R. H.; Sampaon, M.; Pan, C. J . Phys. Chem. 1991,95,2944. ( 2 ) Kratschmcr, W . ; Lamb, L. D.; Fostiropoulos, K.; Huffman, D. R. Nature 1990, 347, 354-358.
87.61 10.095 8.15
73.97 10.100 6.75
34.45 66.26 10.075 7.99
Coast~Qfor the C-C Potential
(4)
where N is the number of atoms on the sphere (N = 60 for Ca). The constants can be obtained by summing (3) over a FCC lattice and requiring the results to match experimental data for Cm This was done by expressing r in terms of the half-edge of the unit cell with the result that the crystal energy is
85.93 10.024 5.41
10.05; 7.1
1060A,erg cm6
10105E, erg cm12 Ro, A
10; 7 29.69 59.47 3.985
nearest neighbor, A; zu,A 10.02; 7 10.02; 7.04 10.05;7.1 29.88 24.85 29.42 60.63 54.8 1 54.37 3.994 4.050 3.932
distance between two carbon atoms. The intermolecular potential shows a greater degree of sensitivity to small variations in the data used to compute them than does the interatomic potential. For both potentials, however, the equilibrium separation is relatively insensitive to such variations. A calculation of the compressibility can give an independent check on the validity of the potentials. This was done by fitting the lattice sums to a quadratic function of the atomic volume (which reproduced the computed sums to six significant figures) and using the thermodynamic relation between compressibility and energy. The results are shown in the last line of Table I. The computed compressibility varies from 5.41 X to 8.15 X dyn/cm2. These compare well with the experimental value of 7 X lo-'* dyn/cmZm3 The most reasonable potential is probably that for which the nearest-neighbor distance is 10.05 A and the molecular diameter is 7.1 A. This is closest to the data of Fleming et al.,' who performed diffraction experiments on crystals grown from the gas phbse. The carbon-carbon potential between atoms on different C, molecules is similar to that for atoms on two different raphite sheets. For graphite, values of A = 24.3 X 10-l6erg cm and &, = 3.834 A were successfully used to compute the energy of cohesion and compressibility.s These are not too different from the constants in Table 11. The analytic form of the spherical approximation should be useful in the gas phase and in the solid when the temperatures are high enough that the molecules are freely rotating. At low temperatures, the carbon-carbon potential can be used.
f
(3) Fischcr, J. E.; Heincy, P. A.; McGhie, A. R.; Romanow, W. J.; Denenstcin, A. M.; McCaulcy Jr., J. P.; Smith 111, A. E.,private communication; submitted to Science. (4) Fleming, R. M.; Sicgrist, T.; Marsh, P. M.; Hewn, E.;Kortan, A. R.; Murphy, D. W.; Tycko, R.;Dabbagh, G.; Mujw, A. M.; Kaplan, M. L.; Zahurak, S.M. MRT Symp. Proc., in press. (5) Girifalco, L. A.; Lad, R.A. J. Chem. Phys. 1956, 25, 693497.
