Phase branches and coexistence domains

Fix a constitution $c$ with $C\ge 1$ named species and its equilibrium state domain $D_c\subseteq {\mathbb{R}}^{C+2}$ from Equilibrium state domains. An interior composition is represented by

$$x=(x_1,\dots ,x_{C-1})\in {\mathrm{\Delta }}_{C-1}^{\circ }:=\{x\in {\mathbb{R}}^{C-1}:x_i>0,\sum _{i=1}^{C-1}{x}_i<1\},$$

with $x_C:=1-{\sum }_{i<C}{x}_i>0$; for $C=1$, ${\mathbb{R}}^0=\{0\}$ and $x_1=1$.

A phase branch $\alpha$ is model data consisting of an open chart $W_{\alpha }\subseteq (0,\mathrm{\infty })\times \mathbb{R}\times {\mathrm{\Delta }}_{C-1}^{\circ }\times (0,\mathrm{\infty })$ with coordinates $(T,p,x,m)$ and a $C^1$ diffeomorphism ${\sigma }_{\alpha }:W_{\alpha }\to U_{\alpha }$ onto an open subset $U_{\alpha }\subseteq {\mathbb{R}}^{C+2}$ contained in the equilibrium state domain $D_c$. Here $T$ is in kelvins, $p$ is in pascals, $m$ is total amount in moles, and $C^1$ means continuous first coordinate derivatives as in $C^k$ Euclidean maps and diffeomorphisms. The chart dimension is $C+2$, matching the ambient dimension of $D_c$. The map ${\sigma }_{\alpha }$ takes values in numerical state coordinates; its physical SI state map is explicitly ${\chi }_{\alpha }={\mathrm{\Phi }}_c\circ {\sigma }_{\alpha }$, where ${\mathrm{\Phi }}_c$ is the numerical-to-SI unit map of Entropic temperature, pressure, and molar chemical potentials. Thus the physical amounts at this point are $n_j=m{x}_j$, and the numerical amount coordinates in ${\sigma }_{\alpha }$ are $n_j\mathrm{/}(1\mathrm{m}\mathrm{o}\mathrm{l})$. Derivative-defined physical state functions are evaluated on ${\chi }_{\alpha }(T,p,x,m)$, or equivalently pulled back along ${\mathrm{\Phi }}_c$ to ${\sigma }_{\alpha }(T,p,x,m)$. At ${\sigma }_{\alpha }(T,p,x,m)$ in this precise pullback sense, the derivative-defined state functions of Entropic temperature, pressure, and molar chemical potentials give temperature $T$, pressure $p$, and mole fractions $x_j=n_j\mathrm{/}m$; here $n_{\mathrm{t}\mathrm{o}\mathrm{t}}:={\sum }_{i=1}^C{n}_i=m$ uses the finite-sum convention of Finite sums and finite products, by recursion. An intensive phase branch additionally has each derivative-defined chemical potential ${\mu }_j(T,p,x,m)$ independent of $m$ throughout $W_{\alpha }$; write its value as ${\mu }_j^{\alpha }(T,p,x)$. This scale invariance is an explicit constitutive model property, not a consequence of Additive separated systems and tagged bulk scaling. A valid local response chart and its inverse are as in Thermodynamic response functions on valid local charts and Local thermodynamic Legendre potentials. Existence of $W_{\alpha }$ or ${\sigma }_{\alpha }$ is a model premise, not a consequence of the equations of thermodynamics.

On a branch with a valid natural-variable chart, $G^{\alpha }$ denotes the Gibbs potential of Local thermodynamic Legendre potentials restricted to the branch. For a one-component branch with amount $m_{\alpha }>0$, define $g_{\alpha }:=G^{\alpha }\mathrm{/}{m}_{\alpha }$, $s_{\alpha }:=S_{\alpha }\mathrm{/}{m}_{\alpha }$, and $v_{\alpha }:=V_{\alpha }\mathrm{/}{m}_{\alpha }$, in $\mathrm{J}\mathrm{m}\mathrm{o}{\mathrm{l}}^{-1}$, $\mathrm{J}\mathrm{m}\mathrm{o}{\mathrm{l}}^{-1}{\mathrm{K}}^{-1}$, and ${\mathrm{m}}^3\mathrm{m}\mathrm{o}{\mathrm{l}}^{-1}$, respectively. These quotients need not be independent of amount, and $g_{\alpha }={\mu }^{\alpha }$ is not assumed by this definition. Such an equality requires a separate same-tag homogeneity hypothesis on the branch.

A phase mixture of a finite list of branches is a feasible separated compound whose positive phase amounts $m_{\alpha }$ have fractions ${\lambda }_{\alpha }:=m_{\alpha }\mathrm{/}{\sum }_{\beta }{m}_{\beta }>0$ with ${\sum }_{\alpha }{\lambda }_{\alpha }=1$. Its extensive coordinates and entropy are those supplied by Additive separated systems and tagged bulk scaling and Concave macroscopic fundamental entropy, on the exact feasible set of Isolated compound and feasible equilibrium set. The mixture is in coexistence when it is an attained global entropy-maximizing endpoint under the declared totals and permitted exchanges, as in Stable equilibrium requires an attained global entropy maximum. This definition does not assert that a maximum exists, that all branches can exchange every coordinate, or that metastable states are global equilibria.

The bulk scaling postulate may send a material tag $c$ to a different tag $c^{(\lambda )}$. It therefore does not supply same-tag degree-one homogeneity of $G^{\alpha }$ or $U^{\alpha }$ on a phase branch. Any result using $g_{\alpha }={\mu }^{\alpha }$ must state that additional hypothesis.