Theoretical Basis for Coomber's Relationship

Abstract

THE almost linear relation found empirically by Coomber^1,2 between the dielectric constant ε and internal pressure p_i for several non-polar liquids appears so far to have received no theoretical justification. By making some assumptions it is possible to show that a simple explanation follows from the nature of intermolecular forces. The internal pressure of a liquid is defined as where E is the potential energy of N molecules occupying a liquid volume V at absolute temperature T. Hildebrand and Scott^3,4 have shown that E may be written in the form −a(T).V⁻ⁿ where n is approximately unity for many organic dielectric liquids⁵, a(T) being slightly dependent on temperature, but not volume, so that p_i = n.a(T).V⁻ⁿ⁻¹. If molecular repulsion is assumed⁵ to contribute negligibly to E, then for non-polar liquids E is due only to the attractive dispersion forces of London⁵. According to London, the dispersion energy of a molecular pair having random mutual orientation and separation r may be written −3/4.I.α².f(r), where f(r) = r⁻⁶ for monatomic atoms and is a more complicated function for polyatomic molecules; α and I are the atomic or molecular polarizability and ionization potential respectively. Noting that N molecules consist of 1/2N² molecular pairs and making the usual assumption that E may be calculated from a pair interaction summation, then a(T) may be factorized into a temperature independent part equal to IN²α² and a temperature dependent part denoted by b(T) which includes numerical constants of the pair summation arising from the interparticle geometry. Thus Assuming the Clausius–Mossoti equation to be valid for non-polar dielectrics, then Thus For the range 2 < ε < 3, in which most non-polar dielectric liquids lie, the function (ε−1/ε+2)² can be represented by 0.0975ε − 0.1325 with a maximum error of ±0.2 per cent. Because n differs from unity by typically 0.1 and b varies slowly⁴ with T, p_i is seen to be a linear function of ε, as found by Coomber. Values of I differ from molecule to molecule and so the linear relation found by Coomber between p_i and ε for different liquids cannot be explained by the theory above. The theory does, however, predict that dielectric constants when extrapolated to zero p_i have a value 1.36 for all liquids for which n = 1. Values of n have been given as 1.09, 1.09, 1.05 and 0.89 for hexane⁴, carbon tetrachloride⁴, benzene⁵ and carbon disulphide⁵ respectively. Coomber¹ (his Fig. 2) has plotted ε against p_i for the liquids hexane, carbon tetrachloride, cyclohexane, benzene and carbon disulphide, and finds each graph to be linear as demanded by the above theory. Moreover, these lines intersect the p_i = 0 axis at values of 1.35, 1.37, 1.35, 1.40 and 1.54 respectively, compared with the theoretical intercept of 1.36. The large discrepancy in the case of carbon disulphide may be due to neglect of molecular repulsion forces in the theory. Hildebrand⁴ has shown that its potential energy does not follow the V⁻¹ law in the density range considered by Coomber.