# The UV diagram for a pure substance

Describing a pure substance at a phase transition in terms of internal energy and volume removes all degeneracies.

Valerio Gherardi https://vgherard.github.io
2024-07-01

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 ( 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 , 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 , the erratum to the original reference .

Lieb, Elliott H., and Jakob Yngvason. 1999a. “The Physics and Mathematics of the Second Law of Thermodynamics.” Physics Reports 310 (1): 1–96. https://doi.org/https://doi.org/10.1016/S0370-1573(98)00082-9.
———. 1999b. “The Physics and Mathematics of the Second Law of Thermodynamics (Physics Reports 310 (1999) 1–96).” Physics Reports 314 (6): 669. https://doi.org/https://doi.org/10.1016/S0370-1573(99)00029-0.

### Corrections

If you see mistakes or want to suggest changes, please create an issue on the source repository.

### Reuse

Text and figures are licensed under Creative Commons Attribution CC BY-SA 4.0. Source code is available at https://github.com/vgherard/vgherard.github.io/, unless otherwise noted. The figures that have been reused from other sources don't fall under this license and can be recognized by a note in their caption: "Figure from ...".

### Citation

Gherardi (2024, July 1). vgherard: The UV diagram for a pure substance. Retrieved from https://vgherard.github.io/posts/2024-07-01-the-uv-diagram-for-a-pure-substance/
@misc{gherardi2024the,
}