# First-Order Phase Transformation at Constant Volume: A Continuous Transition?

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## Abstract

**:**

## 1. Introduction

## 2. Recapitulation of a Constant Pressure Transformation

## 3. Constant Volume Transformation

#### 3.1. General Properties

#### 3.2. Calculation of Thermodynamic Quantities

## 4. Application 1: The Liquid–Ice VI Transition in Water

## 5. Application 2: Temperature-Driven Mechanical Actuator

## 6. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Changes of G, U, and H during a Constant Volume Phase Transformation

## Appendix B. Constant Volume Heat Capacity of the Whole System

## References

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**Figure 1.**Qualitative Gibbs energy curves of phases $\beta $ and $\gamma $ as a function of temperature at a constant pressure ${p}_{0}$. Both phases coexist in equilibrium at $T={T}_{eq}$. Solid (dash) lines correspond to stable (metastable) states.

**Figure 2.**Schematic Helmholtz energy curves of phases $\beta $ and $\gamma $ as a function of temperature at a constant volume ${V}_{0}$. At $T={T}_{*}$, the energies of both phases match which, in general, is not associated with a phase transition. See text for details.

**Figure 3.**Volume–Temperature (

**left**) and Pressure–Temperature (

**right**) schematic phase diagrams of a simple system. Two phases, $\beta $ and $\gamma $, transform each other via a first-order transition. The dash-line corresponds to a constant volume process starting at ${V}_{0}({T}_{0},{p}_{0})$.

**Figure 4.**Temperature-pressure phase diagram of water (

**left**) and the molar volume of liquid and solid water along the phase coexistence line (

**right**). Data taken from [6].

**Figure 5.**Temperature–pressure phase diagram of water around the liquid–ice VI transition. The dashed colored line is the trajectory of a process at constant volume ${V}_{0}=$ 13.4 cm${}^{3}$/mol starting in the liquid phase (${T}_{0}$) and ending up in the ice phase (${T}_{3}$). The two phases coexist between ${T}_{1}$ and ${T}_{2}$.

**Figure 6.**The Helmholtz energy change $\Delta F\left(T\right)=F\left(T\right)-{F}_{m}^{liq}\left({T}_{1}\right)$ in a process at constant volume ${V}_{0}=$ 13.4 cm${}^{3}$/mol. The dashed lines are the metastable parts of the single-phase curves.

**Figure 7.**The entropy change $\Delta S\left(T\right)=S\left(T\right)-{S}_{m}^{liq}\left({T}_{1}\right)$ in a process at constant volume ${V}_{0}=$ 13.4 cm${}^{3}$/mol.

**Figure 8.**Heat capacity at constant volume ${V}_{0}=$ 13.4 cm${}^{3}$/mol as a function of temperature. Jumps can be seen when entering and leaving the coexistence region.

**Figure 9.**Cycle for a mechanical actuator using ordinary water as the working substance and the isochoric phase transition between liquid water and ice VI.

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Correa, V.F.; Castro, F.J.
First-Order Phase Transformation at Constant Volume: A Continuous Transition? *Entropy* **2022**, *24*, 31.
https://doi.org/10.3390/e24010031

**AMA Style**

Correa VF, Castro FJ.
First-Order Phase Transformation at Constant Volume: A Continuous Transition? *Entropy*. 2022; 24(1):31.
https://doi.org/10.3390/e24010031

**Chicago/Turabian Style**

Correa, Víctor F., and Facundo J. Castro.
2022. "First-Order Phase Transformation at Constant Volume: A Continuous Transition?" *Entropy* 24, no. 1: 31.
https://doi.org/10.3390/e24010031