# Revisiting the Clausius/Clapeyron Equation and the Cause of Linearity

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## Abstract

**:**

## 1. Introduction

## 2. Clapeyron Equation

#### 2.1. Clausius/Clapeyron Equation

#### 2.2. Updated Clausius/Clapeyron Equation Discussion

## 3. Summary and Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**A Clapeyron plot of NIST reference data for water of the log vapor pressure relative to the vapor pressure at the triple point ($\mathrm{ln}P/{P}^{0}$) versus the inverse absolute temperature times 1000 ($1000/T$) [7,14]. The plot spans from the triple point ($T=273.16$ K, $P=0.0061165$ bar) to approximately the critical point (${T}_{c}=647.096$ K, ${P}_{c}=220.640$ bar).

**Figure 2.**Plots of NIST reference data for water for the temperature (T) dependence of the enthalpy of vaporization relative to the gas constant ($\Delta {H}^{\mathrm{vap}}/R$) and the change in compressibility upon vaporization ($\Delta {Z}^{\mathrm{vap}}={Z}^{V}-{Z}^{L}$) [7,14]. In the third pane we additionally plot $\Delta {H}^{\mathrm{vap}}/R$ versus $\Delta {Z}^{\mathrm{vap}}$. The plot was generated using the MATLAB code accompanying the electronic version of this manuscript. The plots span from the triple point ($T=273.16$ K, $P=0.0061165$ bar) to approximately the critical point (${T}_{c}=647.096$ K, ${P}_{c}=220.640$ bar).

**Figure 3.**Plots of NIST reference data for hexane for the temperature (T) dependence of the enthalpy of vaporization relative to the gas constant ($\Delta {H}^{\mathrm{vap}}/R$) and the change in compressibility upon vaporization ($\Delta {Z}^{\mathrm{vap}}={Z}^{V}-{Z}^{L}$) [7]. In the third pane we additionally plot $\Delta {H}^{\mathrm{vap}}/R$ versus $\Delta {Z}^{\mathrm{vap}}$. The plot was generated using the MATLAB code accompanying the electronic version of this manuscript. The plots span from the triple point ($T=177.83$ K, $P=1.189\times {10}^{-5}$ bar) to approximately the critical point (${T}_{c}=507.82$ K, ${P}_{c}=30.441$ bar).

**Figure 4.**Plots of NIST reference data for water for $\Delta {H}^{\mathrm{vap}}/\left(R\Delta {Z}^{\mathrm{vap}}\right)$ versus T (

**top**pane) and $1000/T$ (

**bottom**pane) [7,14]. The black line corresponds to use of the assumption $\Delta {Z}^{\mathrm{vap}}=1$ while the red line makes use of temperature-dependent values of $\Delta {Z}^{\mathrm{vap}}$. The plots span from the triple point ($T=273.16$ K, $P=0.0061165$ bar) to approximately the critical point (${T}_{c}=647.096$ K, ${P}_{c}=220.640$ bar).

**Figure 5.**Plots of NIST reference data for hexane for $\Delta {H}^{\mathrm{vap}}/\left(R\Delta {Z}^{\mathrm{vap}}\right)$ versus T (

**top**pane) and $1000/T$ (

**bottom**pane) [7]. The black line corresponds to use of the assumption $\Delta {Z}^{\mathrm{vap}}=1$ while the red line makes use of temperature-dependent values of $\Delta {Z}^{\mathrm{vap}}$. The plots span from the triple point ($T=177.83$ K, $P=1.189\times {10}^{-5}$ bar) to approximately the critical point (${T}_{c}=507.82$ K, ${P}_{c}=30.441$ bar).

**Figure 6.**Watson plot of NIST reference data for water for $\mathrm{ln}\Delta {H}^{\mathrm{vap}}/\left(R\Delta {Z}^{\mathrm{vap}}\right)$ versus $\mathrm{ln}\left(1-T/{T}_{c}\right)$, where $T/{T}_{c}={T}_{r}$ is the reduced temperature [7,14]. The black line corresponds to use of the assumption $\Delta {Z}^{\mathrm{vap}}=1$ while the red line makes use of temperature-dependent values of $\Delta {Z}^{\mathrm{vap}}$. The plots span from the triple point ($T=273.16$ K, $P=0.0061165$ bar) to approximately the critical point (${T}_{c}=647.096$ K, ${P}_{c}=220.640$ bar).

**Figure 7.**Watson plot of NIST reference data for hexane for $\mathrm{ln}\Delta {H}^{\mathrm{vap}}/\left(R\Delta {Z}^{\mathrm{vap}}\right)$ versus $\mathrm{ln}\left(1-T/{T}_{c}\right)$, where $T/{T}_{c}={T}_{r}$ is the reduced temperature [7,14]. The black line corresponds to use of the assumption $\Delta {Z}^{\mathrm{vap}}=1$ while the red line makes use of temperature-dependent values of $\Delta {Z}^{\mathrm{vap}}$. The plots span from the triple point ($T=177.83$ K, $P=1.189\times {10}^{-5}$ bar) to approximately the critical point (${T}_{c}=507.82$ K, ${P}_{c}=30.441$ bar).

**Figure 8.**Clapeyron plots of NIST reference data for water of the log vapor pressure relative to the vapor pressure at the triple point ($\mathrm{ln}P/{P}^{0}$) versus the inverse absolute temperature times 1000 ($1000/T$) as compared to predictions made by numerically integrating the Clapeyron equation [7,14]. In the

**top**pane, reference (ref) is compared to predictions made by numerically integrating $\Delta {H}^{\mathrm{vap}}/\left(R\Delta {Z}^{\mathrm{vap}}\right)$ and $\Delta {H}^{\mathrm{vap}}/R$, as indicated, where the latter assumes $\Delta {Z}^{\mathrm{vap}}=1$. In the

**middle**pane, reference (ref) is compared to predictions made by numerically integrating $\Delta {H}^{\mathrm{vap}}/\left(R\Delta {Z}^{\mathrm{vap}}\right)$ which is taken to be constant. We compare the use of the value at the lowest temperature (the triple point, min), the average value (avg) and the average (natural) log value (Lavg), and the value at the highest temperature (0.097 K below the critical point, max). In the

**bottom**pane, reference (ref) is compared to predictions made by numerically integrating $\Delta {H}^{\mathrm{vap}}/R$ which is taken to be constant and assumes $\Delta {Z}^{\mathrm{vap}}=1$. The plots span from the triple point ($T=273.16$ K, $P=0.0061165$ bar) to approximately the critical point (${T}_{c}=647.096$ K, ${P}_{c}=220.640$ bar).

**Figure 9.**Clapeyron plots of NIST reference data for hexane of the log vapor pressure relative to the vapor pressure at the triple point ($\mathrm{ln}P/{P}^{0}$) versus the inverse absolute temperature times 1000 ($1000/T$) as compared to predictions made by numerical integrating the Clapeyron equation [7,14]. In the

**top**pane, reference (ref) is compared to predictions made by numerically integrating $\Delta {H}^{\mathrm{vap}}/\left(R\Delta {Z}^{\mathrm{vap}}\right)$ and $\Delta {H}^{\mathrm{vap}}/R$, as indicated, where the latter assumes $\Delta {Z}^{\mathrm{vap}}=1$. In the

**middle**pane, reference (ref) is compared to predictions made by numerically integrating $\Delta {H}^{\mathrm{vap}}/\left(R\Delta {Z}^{\mathrm{vap}}\right)$ which is taken to be constant. We compare the use of the value at the lowest temperature (the triple point, min), the average value (avg) and the average (natural) log value (Lavg), and the value at the highest temperature (0.82 K below the critical point, max). In the

**bottom**pane, reference (ref) is compared to predictions made by numerically integrating $\Delta {H}^{\mathrm{vap}}/R$ which is taken to be constant and assumes $\Delta {Z}^{\mathrm{vap}}=1$. The plots span from the triple point ($T=177.83$ K, $P=1.189\times {10}^{-5}$ bar) to approximately the critical point (${T}_{c}=507.82$ K, ${P}_{c}=30.441$ bar).

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**MDPI and ACS Style**

Thompson, J.E.; Paluch, A.S.
Revisiting the Clausius/Clapeyron Equation and the Cause of Linearity. *Thermo* **2023**, *3*, 412-423.
https://doi.org/10.3390/thermo3030025

**AMA Style**

Thompson JE, Paluch AS.
Revisiting the Clausius/Clapeyron Equation and the Cause of Linearity. *Thermo*. 2023; 3(3):412-423.
https://doi.org/10.3390/thermo3030025

**Chicago/Turabian Style**

Thompson, Jason E., and Andrew S. Paluch.
2023. "Revisiting the Clausius/Clapeyron Equation and the Cause of Linearity" *Thermo* 3, no. 3: 412-423.
https://doi.org/10.3390/thermo3030025