Kirkwood-Muller

Table II shows clearly the large differences between various theories for many-electron systems. The Kirkwood-Muller equation always yields somewhat too large coefficients for the atoms which are the only spherical systems but the London equation deviates by a greater amount on the low side. The Slater-Kirkwood equation gives a high value for He but yields coefficients smaller than the empirical ones for all other cases.

The dispersion and repulsion interactions form the Lennard-Jones (Barrer, 1978 Masel, 1996 Razmus and Hall, 1991 Gregg and Sing, 1982 Steele, 1974 Adamson, 1976 Rigby et al., 1986) potential, with an equilibrium distance (r0) where 4 d + 4 r = 0. This distance is taken as the mean of the van der Waals radii of the interacting pair. Once the attractive, dispersion constant, A, is known, B is readily obtained by setting at chp/dr = 0 at r0. Hence, B = Ar /2. The most commonly used expression for calculating A is the Kirkwood-Muller formula ... 

Kirkwood-Muller formula, potential energies for adsorption, 83... 

Kirkwood-Muller formula, 83 Lennard-Jones potential, 83 nonspecific contributions, 82 pillared clays, 120... 

The constant, C, can also be calculated with the help of the Kirkwood-Muller equation, as later explained, or with the Slater-Kirkwood expression. 

In addition, z is the internuclear distance between the adsorbate and adsorbent molecules, (L - ds) is the effective pore width, and Aas is the dispersion constant, which takes into account the adsorbate-adsorbent interaction. The term Aas is calculated with the help of the Kirkwood-Muller formula [8,13,17-19]... 

Finally, Aaa, calculated with the help of the Kirkwood-Muller formula, is the constant characterizing the adsorbate-adsorbate interactions [2,13,17] ... 

By making use of the Kirkwood-Muller equation and substituting the available experimental data for the physical properties of nitrogen and carbon, Horvath and Kawazoe (1983) arrived at the following equation for the adsorption of nitrogen by molecular sieve carbons at 77 K ... 

In equation (4), A is the number density of atoms per unit surface area A is the dispersion constant the subscripts 5 and / refer to the adsorbent and adsorbate, respectively and do = 0.S5 asf is the z-coordinate at which the 10-4 potential for a single planar surface passes through its zero-point value. The 10-4 potential is obtained by integration of the Lennard-Jones 12-6 potential over an infinite planar surface. The dispersion constants A and Aff represent the adsorbate-adsorbent and adsorbate-adsorbate interactions, respectively these coefficients are calculated from the Kirkwood-Muller equations in the original HK paper [6], Combining equations (2-4) yields an equation that relates filling pressure to pore width ... 

In Eq. (33), N is the molecular density (i.e., molecules per unit surface area) A is the dispersion coefficient the subscripts s and f refer to the adsorbent and adsorbate, respectively and d = (2/5) The 10-4 potential is obtained by integrating the I.ennard-Jones potential over an unbounded planar surface [105], The dispersion coefficients are calculated using the Kirkwood-Muller equations (as reported in Ref. 19)... 

Accurate values for the coefficient of the induced-dipole-induced-dipole term, Ce, have been obtained for a number of like and unlike interactions. Kramer and Herschbach used three empirical combination rules and approximate London and Kirkwood-Muller formulas to calculate values of Ce for 153 unlike interactions from the Ce-coefficients of the pure components. They find that a combination rule suggested by Moelwyn-Hughes ... 

The parameters A, and A2 are the Kirkwood-Muller dispersion constants, and are given in eqs. (6.11-8). Here is the collision diameter (can be taken as the arithmetic mean between the diameter of the adsorbate molecule and the surface atom), and and (3 are defined as follows ... 

Somewhat closer agreement between theory and experiment for the rare gases in various forms of zeolite X was obtained by Broier et al. who used the Kirkwood-Muller expression, The calculated values are however quite sensitive to the choice of equilibrium radii, which determine the repulsion constants. The choice of radii is not clearly specified by Broier et al. and it seems likely that the values selected may have been adjusted, within physically reasonable limits, to improve the fit. There is also some uncertainty in the choice of physical parameter values for the estimation of the dispersion constants as well as in the charge distribution. 

---