Smith et al. (1970). Equation ( B 2 ) and the first term in Equation ( B 4 ) give the random mixing result. The second and third terms in Equation ( B 4 ) correct the random mixing approximation for clusters of two and three molecules, respectively. The third-and fourth-order terms have been derived, they are of a form similar to Equations (28) and ( 2 9 ) . These are not useful, however, because they require the higherorder distribution functions go( 1234), etc., which are unknown. A good approximation can be made for these terms by writing them in the form of Equation ( B 4 ) and including only the correction for clusters of two molecules. This provides the dominant correction for mixture nonrandomness. The result isThese complex equations allow calculation of mixture properties when the molecules differ in size, shape or intermolecular potential energy.For athermal solutions, the perturbed-hard-chain theory yields the Flory-Huggins combinatorial entropy of mixing for chain molecules, while for mixtures of spherical molecules of the same size, perturbed-hard-chain theory becomes essentially equivalent to Guggenheim's quasichemical approximation. Pcrturbed-hard-chain theory, in effect, interpolates between these two well-established theories of mixtures. This paper shows how two different liquid membranes can selectively separate and concentrate copper or nickel ions. In other words, it demonstrates how copper or nickel can be removed from a dilute solution and concentrated in a more acidic solution. As such, the work is another effort in this laboratory aimed at making mass transfer fast and selective.These membranes contain liquid ion exchangers which function as the mobile carriers so often postulated in biophysics (Stein, 1967). These carriers react with the solutes of interest and diffuse with them across the membrane. The crileria which these carriers must fulfill t o be effective are described elsewhere Schultz et al., 1974).