The thermodynamic states of pure substances and their phase transitions are often represented in \(PT\) (pressure-temperature) and \(PV\) (pressure-volume) diagrams. While reading the classic paper from Lieb and Yngvason (Lieb and Yngvason 1999a) ( see my commentary here) I learned about a third possible description, in terms of the extensive volume \(V\) and internal energy \(U\).
Beside the important role that the \((U,V)\) parametrization plays in the axiomatic formulation of thermodynamics given in (Lieb and Yngvason 1999a), which makes \(UV\) diagrams interesting per se, there’s a practical advantage in these two variables, in the fact that they uniquely characterize pure substances everywhere in the phase diagram. In particular, a triple “point” becomes a triangle in the UV diagram, as can be seen by the parametrization:
\[ U = m_gu_g+ m_\ell u_\ell+m_su_s,\quad V = m_gv_g+m_\ell v_\ell+m_sv_s, \] where \(m_i\), \(u_i\) and \(v_i\) are the masses, specific internal energy and volumes of the gaseous, liquid and solid phases, respectively (the total mass \(m = m_g + m_\ell + m_s\) is assumed to be fixed). This triangle is projected into a line in the \(PV\) diagram, and into a single point in the \(PT\) diagram.
The subsets of the \(UV\) plane representing the fusion, sublimation and vaporization curves (or any other curve on the \(PT\) diagram representing the coexistence of two distinct phases) are still two dimensional submanifolds, but the parametrization is more involved:
\[ U = m_A u_A(T) + m_B u_B(T),\quad V=m_A v_A(T) + m_B v_B(T), \] where \(A\) and \(B\) are the two coexisting phases, and the specific energies and volumes vary with temperature. These sets are obtained by joining for each value of \(T\) the two corresponding points on the curves \(\gamma _A(T) = m\cdot(u_A(T),v_A(T))\) and \(\gamma _B(T) = m\cdot(u_B(T), v_B(T))\) in the \(UV\) plane.
By pure coincidence, an example of the diagrams I am referring to is shown in (Lieb and Yngvason 1999b), the erratum to the original reference (Lieb and Yngvason 1999a).
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@online{gherardi2024,
author = {Gherardi, Valerio},
title = {The {UV} Diagram for a Pure Substance},
date = {2024-07-01},
url = {https://vgherard.github.io/posts/2024-07-01-the-uv-diagram-for-a-pure-substance/},
langid = {en}
}