8.3: How Enthalpy Depends on Pressure

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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.)