On the relationship between principal fluctuations and stability coefficients in multicomponent systems

There is a univocal correspondence between each of the (n+2)! stability coefficients of an n-component system and mean squared fluctuations of Gibbs-space variables (entropy, volume, molecule number) in which the subsystem under study is defined through a single extensive Gibbs-space variable (principal fluctuations). The relationship between principal fluctuations and stability coefficients [Eq. (53)] is derived in this paper starting from the usual expressions involving explicit calculation of the inverse of an (n+1)×(n+1) matrix. A general equation between higher order energy variations in fluctuating subsystems [Eq. (22)] is also derived. The relationship between principal fluctuations and stability coefficients yields, as particular cases, expressions which had been used previously for the limiting case of single-component systems, or extended to solutions through heuristic arguments.