The statistical thermodynamics of a one-dimensional multicomponent assembly

Summary

It is shown that a one-dimension multicomponent mixture with nearest neighbour interactions u_ij,(r) obeys the implicit equation of state

$$\Delta (\lambda _A ,\lambda _B ,...,T,P) \equiv \left| {\lambda _i \eta _{ij} - \delta _{ij} } \right| = 0$$

where λ_i is the absolute activity of speciesi (divided by a kinetic factor) and

$$\eta _{ij} (T,P) = \int\limits_0^\infty {\exp [ - \{ u_{ij} (r)} + P_r \} /kT]dr$$

The concentration of speciesi is thence determined as

$$ - \lambda _i \frac{{\partial \Delta }}{{\partial \lambda _i }}/kT\frac{{\partial \Delta }}{{\partial P}}$$

A one-dimensional solution is ideal if, and only if, the matrix [η_ij] is of rank unity. The excess free energy of a nearly ideal solution is found to be

$$(\Delta ^ * G)_{T,P} = \mathop {\Sigma \Sigma }\limits_{i< j} x_i x_j (G_{ij} + G_{ji} - G_{ii} - G_{jj} )$$

where G_ij (T, P) is the (conflgurational) molar Gibbs free energy of a «hybrid » species with interaction potential u_ij(r).