On an alternative formulation of the thermodynamic stability condition

Abstract

In this contribution, we present a new, geometric approach aimed at deriving thermodynamic relations in a formally correct and systematic way. We obtain a number of useful relations, give a formulation of the Gibbs stability condition and describe the limits of its use.

Appendix

Derivation of (10). Consider a function $\varPhi (u,v)=0$, $\varPhi :{\mathbb {R}}^n\times {\mathbb {R}}^m\rightarrow {\mathbb {R}}^n$ s.t. $\frac{\partial \varPhi }{\partial u}\ne 0$ at $(u^*,v^*)$. This implies that one can locally at $(u^*,v^*)$ express $u=f(v)$. Differentiating $\varPhi (u,v)$ w.r.t. v and using the chain rule we get $\frac{\partial \varPhi }{\partial u}\frac{\partial f}{\partial v}+\frac{\partial \varPhi }{\partial v}=0,$ whence $\frac{\partial f}{\partial v}=-\left[ \frac{\partial \varPhi }{\partial u}\right] ^{-1}\frac{\partial \varPhi }{\partial v}.$ Note that

$$\begin{aligned} \left[ \frac{\partial \varPhi }{\partial u}\right] ^{-1}=\frac{1}{\det \left( \frac{\partial \varPhi }{\partial u}\right) } \begin{bmatrix}(-1)^{(1+1)}\det \left( \frac{\partial \varPhi }{\partial u} [1,1]\right) &\dots &(-1)^{(n+1)}\det \left( \frac{\partial \varPhi }{\partial u}[n,1]\right) \\ \vdots && \vdots \\ (-1)^{(n+1)}\det \left( \frac{\partial \varPhi }{\partial u}[1,n]\right) &\dots &(-1)^{(2n)}\det \left( \frac{\partial \varPhi }{\partial u}[n,n]\right) \end{bmatrix}, \end{aligned}$$

where the last matrix is the adjugate, i.e., the transposed cofactor matrix of $\frac{\partial \varPhi }{\partial u}$, [6].

So, for the partial derivative $\frac{\partial f_i}{\partial v_j}$ we have

$$\begin{aligned} \frac{\partial f_i}{\partial v_j}=-\frac{1}{\det \left( \frac{\partial \varPhi }{\partial u}\right) } \begin{bmatrix}(-1)^{(1+i)}\det \left( \frac{\partial \varPhi }{\partial u}[1,i]\right)&\dots&(-1)^{(n+i)}\det \left( \frac{\partial \varPhi }{\partial u}[n,i]\right) \end{bmatrix}\begin{bmatrix}\frac{\partial \varPhi _1}{\partial v_j}\\ \vdots \\ \frac{\partial \varPhi _n}{\partial v_j}\end{bmatrix}, \end{aligned}$$

where the product of the row and the column is the Laplace expansion of $\left[ \frac{\partial \varPhi }{\partial x}\right]$ with respect to the ith column and such that the ith column is composed by the partial derivatives $\frac{\partial \varPhi }{\partial v_j}$. This yields (10). $\square$