CHARLES TANFORD
534s [CONTRIBUTION NO.
1445 FROM
STERLING CHEMISTRY
LABORATORY, YALE
VOl. 79 UNIVERSITY]
The Location of Electrostatic Charges in Kirkwood’s Model of Organic Ions BY CHARLES TANFORD’ RECEIVED MAY8. 1957 The interaction between electrostatic charges on organic ions was treated by Kirkwood in 1934 in terms of a model which formally represents the organic ion as a cavity of low dielectric constant in a solvent continuum. This paper shows that a crucial parameter in determining interaction energies on the basis of this model is the depth, d , within the cavity, a t which charges or dipoles are placed. To account for experimentally observed interaction energies, as reflected in the dissociation consJants of appropriate organic acids (Kirkwood-Westheimer theory), it is necessary that d be assigned a value close to 1.0 A. for discrete charges and about 1.5 A. for dipoles. This result is in agreement with that found earlier in an extension of the Kirkwood model to charge interactions on protein molecules. An empirical procedure, based on this finding, permits the direct calculation of interaction energies, and, hence, of pK differences between related acids differing by the presence of charged or dipolar substituents.
The properties of organic molecules in solution which depend on the interaction between electrostatic charges upon them have been treated with considerable success by the model proposed by Kirkwood2 in 1934. In this model the organic ion is treated as a cavity of low dielectric constant (ordinarily placed equal to 2 ) , embedded in the solvent, which is treated as a continuum with its macroscopic dielectric constant. Using this model, one can show3 that the free energy of interaction between a pair of charges, q1 and q z , separated by a distance R,can be expressed as
where DE is a parameter with the dimensions of a dielectric constant, the value for which depends primarily on the location of the charges with respect to the interface between the cavity and the solvent. Values of DE for spherical and ellipsoidal cavities have been given in tabular and graphical form in a number of p l a ~ e s . ~ - j -4 similar equation may be written3 for the interaction between a charge p and a point dipole of moment p Here R is the distance between the charge and the dipole, and j- the angle between the dipole axis and the line joining the charge to the dipole. Values of DE for use in eq. 2 have been tabulated by Westheimer and c o - ~ o r k e r s . ~ ~ ~ The principal application of the Kirkwood model has been in the procedure used by Kirkwood and Westheimer3 to account quantitatively for differences in the dissociation constants of related acids on the basis of electrostatic interaction between the dissociating proton and a charge or dipole on another part of the molecule. The present calculations result from an attempt to extend this kind of calculation to proteins. It becomes immediately apparent that such a calculation will depend not only on an estimation of the size of a protein mole( I ) Department of Chemistry, State University of Iowa, Iowa City, Iowa. J o h n Simon Guggenheirn hlemorial Fellow, Yale Universitv, 1956-1957. This work also v a s supported by research grant RG-2330 from t h e Sational Institutes of Health, Public Health Service. (2) J. G. Kirkwood, J . C h n . P h r s . , 2, 351 (1934). (3) J. G Kirkwood and F. H Westheimer, i b i d . , 6, 506, 513 ( 1 9 3 8 ) . (1) F. H. R’estheimer. TV, .\ J u n e 5 :ind I
