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9.4: The Enthalpy of an Ideal Gas

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    How does pressure affect enthalpy \(H\)? As we showed above we have the following relations of first and second order for \(G\)

    \[\left( \dfrac{\partial G}{\partial T} \right)_P = -S \nonumber \]

    \[ \left( \dfrac{\partial G}{\partial P} \right)_T = -V \nonumber \]

    \[ -\left (\dfrac{\partial S}{\partial P }\right)_T = \left (\dfrac{\partial V}{\partial T} \right)_P \nonumber \]

    We also know that by definition:

    \[G = H - TS \label{def} \]

    Consider an isothermal change in pressure, so taking the partial derivative of each side of Equation \(\ref{def}\), we get:

    \[ \left( \dfrac{\partial G}{\partial P}\right)_T = \left( \dfrac{\partial H}{ \partial P}\right)_T -T \left( \dfrac{\partial S}{\partial P}\right)_T \nonumber \]

    \[ \left( \dfrac{\partial H}{\partial P}\right)_T = V -T \left( \dfrac{\partial V}{\partial T}\right)_P \label{Eq12} \]

    For an ideal gas

    \[\dfrac{\partial V}{\partial T} = \dfrac{nR}{P} \nonumber \]

    so Equation \(\ref{Eq12}\) becomes

    \[ \left( \dfrac{\partial H}{\partial P}\right)_T = V - T \left( \dfrac{nR}{P}\right) = 0 \nonumber \]

    As we can see for an ideal gas, there is no dependence of \(H\) on \(P\).

    9.4: The Enthalpy of an Ideal Gas is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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