Statistical mechanics of polynuclear ring complexes with mixed-valence states by use of the Ising model

Abstract

The correlated-walk theory is useful for estimating reversible current potential curves of a linear polynuclear complex with strongly electronic communication through a conjugated bridge. However, it falls short of a thermodynamic base because of the assumption of the local equilibrium at redox centers. In order to support the theory in the light of the statistical mechanics, the partition function of polynuclear ring complexes was derived by applying the eigenvalue technique of the Ising spin model. It contains the electrode potential, the number of redox centres and interaction energy of the nearest neighbor redox centres. The partition function is essentially equivalent to the generating function of the correlated-walk theory in so far as it is concerned with thermodynamic difference quantities such as the electrode potential and the configurational energy. Thus the microscopically local equilibrium on which the correlated-walk theory relays can reproduce the real equilibrium of the whole system. Expressions for the molar fraction of the oxidized centre, the internal energy and the entropy were obtained analytically as a function of the electrode potential and the nearest neighbour interaction. The internal energy varies with the electrode potential in the mixed-valence potential domain. The entropy is composed of the isomeric configurational contribution and the redox contribution due to different redox states.