# 8.3: How Enthalpy Depends on Pressure

$$\newcommand{\vecs}{\overset { \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$$$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

Let us look briefly at the approximations $$\Delta H\left({\mathrm{B}}^{\mathrm{*}}\mathrm{\to }\mathrm{C}\right)\approx 0$$ and $$\Delta H\left({\mathrm{D}}^{\mathrm{*}}\mathrm{\to }\mathrm{A}\right)\approx 0$$ that we used in Section 8.2. In these steps, the pressure changes while the temperature remains constant. In Chapter 10, we find a general relationship for the pressure-dependence of a system’s enthalpy: ${\left(\frac{\partial H}{\partial P}\right)}_T=-T{\left(\frac{\partial V}{\partial T}\right)}_P+V \nonumber$

This evaluates to zero for an ideal gas and to a negligible quantity for many other systems.

For liquids and solids, information on the variation of volume with temperature is collected in tables as the coefficient of thermal expansion, $$\alpha$$, where

$\alpha =\frac{1}{V}{\left(\frac{\partial V}{\partial T}\right)}_P \nonumber$

Consequently, the dependence of enthalpy on pressure is given by ${\left(\frac{\partial H}{\partial P}\right)}_T=V\left(1-\alpha T\right) \nonumber$

For ice, $$\alpha \approx 50\times {10}^{-6}\ {\mathrm{K}}^{-1}$$ and the molar volume near 0 C is $$\mathrm{19.65}\ {\mathrm{cm}}^3\ {\mathrm{mol}}^{-1}$$. The enthalpy change for compressing one mole of ice from the sublimation pressure to 1 atm is $$\Delta H\left({\mathrm{D}}^{\mathrm{*}}\mathrm{\to }\mathrm{A}\right)=2\ \mathrm{J}\mathrm{\ }{\mathrm{mol}}^{-1}$$.

To find the enthalpy change for expanding one mole of water vapor at 100 C from 1 atm to the sublimation pressure, we use the virial equation and tabulated coefficients for water vapor to calculate $${\left({\partial H}/{\partial P}\right)}_{\mathrm{398\ K}}$$. We find $$\Delta H\left({\mathrm{B}}^{\mathrm{*}}\mathrm{\to }\mathrm{C}\right)=220\ \mathrm{J}\ {\mathrm{mol}}^{-1}$$. (See problem 9.)

This page titled 8.3: How Enthalpy Depends on Pressure is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.