Kirkwood

Another statistical mechanical approach makes use of the radial distribution function g(r), which gives the probability of finding a molecule at a distance r from a given one. This function may be obtained experimentally from x-ray or neutron scattering on a liquid or from computer simulation or statistical mechanical theories for model potential energies [56]. Kirkwood and Buff [38] showed that for a given potential function, U(r)... 

Slater and Kirkwood s idea [121] of an exponential repulsion plus dispersion needs only one concept, damping fiinctions, see section Al.5.3.3. to lead to a working template for contemporary work. Buckingham and Comer [126] suggested such a potential with an empirical damping fiinction more than 50 years ago ... 

Pekeris C L 1934 The rotation-vibration coupling in diatomic molecules Phys. Rev. 45 98 Slater J C and Kirkwood J G 1931 The van der Waals forces in gases Phys. Rev. 37 682... 

We conclude this section by discussing an expression for the excess chemical potential in temrs of the pair correlation fimction and a parameter X, which couples the interactions of one particle with the rest. The idea of a coupling parameter was mtrodiiced by Onsager [20] and Kirkwood [Hj. The choice of X depends on the system considered. In an electrolyte solution it could be the charge, but in general it is some variable that characterizes the pair potential. The potential energy of the system... 

This is Kirkwood s expression for the chemical potential. To use it, one needs the pair correlation fimction as a fimction of the coupling parameter A as well as its spatial dependence. For instance, if A is the charge on a selected ion in an electrolyte, the excess chemical potential follows from a theory that provides the dependence of g(i 2, A) on the charge and the distance r 2- This method of calculating the chemical potential is known as the Gimtelburg charging process, after Guntelburg who applied it to electrolytes. 

Kirkwood derived an analogous equation that also relates two- and tlnee-particle correlation fiinctions but an approximation is necessary to uncouple them. The superposition approximation mentioned earlier is one such approximation, but unfortunately it is not very accurate. It is equivalent to the assumption that the potential of average force of tlnee or more particles is pairwise additive, which is not the case even if the total potential is pair decomposable. The YBG equation for n = 1, however, is a convenient starting point for perturbation theories of inliomogeneous fluids in an external field. 

Kirkwood J G 1935 Statistical mechanics of fluid mixtures J. Chem. Phys. 3 300 Kirkwood J G 1936 Statistical mechanics of liquid solutions Chem. Rev. 19 275... 

Kirkwood J G and Buff F P 1951 Statistical mechanical theory of solutions I J. Chem. Phys. 19 774... 

The integral under the heat capacity curve is an energy (or enthalpy as the case may be) and is more or less independent of the details of the model. The quasi-chemical treatment improved the heat capacity curve, making it sharper and narrower than the mean-field result, but it still remained finite at the critical point. Further improvements were made by Bethe with a second approximation, and by Kirkwood (1938). Figure A2.5.21 compares the various theoretical calculations [6]. These modifications lead to somewhat lower values of the critical temperature, which could be related to a flattening of the coexistence curve. Moreover, and perhaps more important, they show that a short-range order persists to higher temperatures, as it must because of the preference for unlike pairs the excess heat capacity shows a discontinuity, but it does not drop to zero as mean-field theories predict. Unfortunately these improvements are still analytic and in the vicinity of the critical point still yield a parabolic coexistence curve and a finite heat capacity just as the mean-field treatments do. 

Figure A2.5.21. The heat eapaeity of an order-disorder alloy like p-brass ealeulated from various analytie treatments. Bragg-Williams (mean-field or zeroth approximation) Bethe-1 (first approximation also Guggenheim) Bethe-2 (seeond approximation) Kirkwood. Eaeh approximation makes the heat eapaeity sharper and higher, but still finite. Reprodueed from [6] Nix F C and Shoekley W 1938 Rev. Mod. Phy.s. 10 14, figure 13. Copyright (1938) by the Ameriean Physieal Soeiety.

Kirkwood generalized the Onsager reaction field method to arbitrary charge distributions and, for a spherical cavity, obtained the Gibbs free energy of solvation in tenns of a miiltipole expansion of the electrostatic field generated by the charge distribution [12, 1 3]... 

A version of this material appears in a speoial issue of the Journal of Physical Chemistry ded cated to the Prooeedings of the International Conferenoe on Time-Resolved Vibrational Speotrosoopy (TRVS IX), May 16-22 1999, Tuoson, Arizona. See Kirkwood J C, Ulness D J and Albreoht A C 2000 On the olassifioation of the eleotrio field speotrosoopies applioations to Raman soattering J. Phys. Chem. A 104 4167-73. 

Ulness D J, Stimson M J, Kirkwood J C and Albrecht A C 1997 Interferometric downconversion of high frequency molecular vibrations with time-frequency-resolved coherent Raman scattering using quasi-cw noisy laser light C-H stretching modes of chloroform and benzene J. Rhys. Chem. A 101 4587-91... 

Ulness D J, Kirkwood J C, Stimson M J and Albrecht A C 1997 Theory of coherent Raman scattering with... 

Kose A and Hachisu S 1974 Kirkwood-Alder transition in monodisperse latexes. I. Nonaqueous systems J. Coiioid interface Sc/. 46 460-9... 

Flachisu S and Kobayashi Y 1974 Kirkwood-Alder transition in monodisperse latexes. II. Aqueous latexes of high electrolyte concentration J. Colloid Interface Sol. 46 470-6... 

Edsall, J. T. George Scatchard, John G. Kirkwood, and the electrical interactions of amino acids and proteins. Trends Biochem. Sci. 7 (1982) 414-416. Eigen, M. Proton transfer, acid-base catalysis, and enzymatic hydrolysis. Angew. Chem. Int. Ed. Engl. 3 (1964) 1-19. 

Tanford, C., Kirkwood, J. G. Theory of protein titration curves. I. General equations for impenetrable spheres. J. Am. Chem. Soc. 79 (1957) 5333-5339. 6. Garrett, A. J. M., Poladian, L. Refined derivation, exact solutions, and singular limits of the Poisson-Boltzmann equation. Ann. Phys. 188 (1988) 386-435. Sharp, K. A., Honig, B. Electrostatic interactions in macromolecules. Theory and applications. Ann. Rev. Biophys. Chem. 19 (1990) 301-332. 

J. Wisdom. The origin of the Kirkwood gaps A mapping for asteroidal motion near the 3/1 commensurability. Astr. J., 87 577-593, 1982. 

Wisdom, J. The Origin of the Kirkwood Gaps A Mapping for Asteroidal Motion Near the 3/1 Commensurability. Astron. J. 87 (1982) 577-593 Tuckerman, M., Martyna, G. J., Berne, J. Reversible Multiple Time Scale Molecular Dynamics. J. Chem. Phys. 97 (1992) 1990-2001 Tuckerman, M., Berne, J. Vibrational Relaxation in Simple Fluids Comparison of Theory and Simulation. J. Chem. Phys. 98 (1993) 7301-7318 Humphreys, D. D., Friesner, R. A., Berne, B. J. A Multiple-Time Step Molecular Dynamics Algorithm for Macromolecules. J. Chem. Phys. 98 (1994) 6885-6892... 

This is similar in spirit to the arithmetic-mean rule but with each individual r,) being weighted according to the square of its value. The well depth in this function starts with a formula proposed by Slater and Kirkwood for the Cg coefficient of the dispersion series expansion ... 

How can Equation (11.79) be solved Before computers were available only simple ihapes could be considered. For example, proteins were modelled as spheres or ellipses Tanford-Kirkwood theory) DNA as a uniformly charged cylinder and membranes as planes (Gouy-Chapman theory). With computers, numerical approaches can be used to solve the Poisson-Boltzmann equation. A variety of numerical methods can be employed, including finite element and boundary element methods, but we will restrict our discussion to the finite difference method first introduced for proteins by Warwicker and Watson [Warwicker and Watson 1982]. Several groups have implemented this method here we concentrate on the work of Honig s group, whose DelPhi program has been widely used. 

Kirkwood J G 1934. Theory of Solutions of Molecules Containing Widely Separated Charges witl Special Application to Zwitterions. Journal of Chemical Physics 2 351-361. 

Another way to obtain a relative permitivity is using some simple equations that relate relative permitivity to the molecular dipole moment. These are derived from statistical mechanics. Two of the more well-known equations are the Clausius-Mossotti equation and the Kirkwood equation. These and others are discussed in the review articles referenced at the end of this chapter. The com-... 

Several relations have been devised for the calculation of the parameter C from the molecular properties of two atoms A and B. One of the best known is that of Kirkwood and Miiller "... 

Calculations of the interaction energy in very fine pores are based on one or other of the standard expressions for the pair-wise interaction between atoms, already dealt with in Chapter 1. Anderson and Horlock, for example, used the Kirkwood-Miiller formulation in their calculations for argon adsorbed in slit-shaped pores of active magnesium oxide. They found that maximum enhancement of potential occurred in a pore of width 4-4 A, where its numerical value was 3-2kcalmol , as compared with 1-12, 1-0 and 1-07 kcal mol for positions over a cation, an anion and the centre of a lattice ceil, respectively, on a freely exposed (100) surface of magnesium oxide. 

A detailed hydrodynamic theory has been developed by Kirkwood and Riseman which indeed reduces to the limits predicted above. 

The details of the Kirkwood-Riseman theory are sufficiently involved that we shall not consider the derivation of this theory. We shall, however, examine in somewhat greater detail the cluster of variables we have designated by X as a measure of the permeability of the molecule to the flowing solvent. 

As discussed in connection with Eq. (9.47), the Kirkwood-Riseman theory predicts that a = 1 in the free-draining limit. This limit is expected for small values of n, however, and does not explain a > 0.5 for high molecular weight polymers. 

Random coils. Equation (9.53) gives the Kirkwood-Riseman expression for the friction factor of a random coil. In the free-draining limit, the segmental friction factor can, in turn, be evaluated from f. In the nondraining limit the radius of gyration can be determined. We have already discussed f in Chap. 2 and (rg ) in this chapter and again in Chapter 10, so we shall not examine the information provided by D for the random coil any further. 

C. Craver, ed.. The Coblent Society Desk Book of Infrared Spectra, The Coblent2 Society, Inc., Kirkwood, Mo., 1977. 

C. D. Craver, Infra-Red Spectra of Plasticicyers and Other Additives, 2nd ed.. The Coblentz Society, Kirkwood, Mo., 1980. 

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