E: Interpreting the Fundamental Equations

The fundamental equation for the variation of a state function \(\mathfrak{F}\) expresses the contributions due to changes in its natural independent variables \((x, y, z)\)

\[d \mathfrak{J}= \left(\dfrac{\partial \mathfrak{F} }{\partial x}\right)_{y, z} dx + \left(\dfrac{\partial \mathfrak{F}}{\partial y}\right)_{x, z} dy + \left(\dfrac{\partial \mathfrak{F}}{\partial z}\right)_{x, y} d z \nonumber\]

The following table summarizes the partial derivatives for the most commonly used state functions.

\(\mathfrak{F}\)	fundamental equation	natural independent variables	first derivatives
E	\(d E=T d S-P d V+\Sigma_i \tilde{\mu}_i d n_i\)	\(S, V,\left\{n_{\mathrm{i}}\right\}\)	\(\begin{gathered} (\partial E / \partial S)_{V,\left\{n_i\right\}}=T \\ (\partial E / \partial V)_{S,\left\{n_i\right\}}=-P \\ (\partial E / \partial n i)_{S, V,\left\{n_{j \neq i}\right\}}=\tilde{\mu}_i \end{gathered}\)
H	\(d H=T d S+V d P+\Sigma_i \tilde{\mu}_i d n_i\)	\(S, P,\left\{n_{\mathrm{i}}\right\}\)	\(\begin{gathered} (\partial H / \partial S)_{P,\left\{n_i\right\}}=T \\ (\partial H / \partial P)_{S,\left\{n_i\right\}}=V \\ \left(\partial H / \partial n_i\right)_{S, P,\left\{n_{j \neq i}\right\}}=\tilde{\mu}_i \end{gathered}\)
F	\(d F=-S d T-P d V+\Sigma_i \tilde{\mu}_i d n_i\)	\(T, V,\left\{n_{\mathrm{i}}\right\}\)	\(\begin{gathered} (\partial F / \partial T)_{V,\left\{n_i\right\}}=-S \\ (\partial F / \partial V)_{T,\left\{n_i\right\}}=-P \\ (\partial F / \partial n_i)_{T, V,\left\{n_{j \neq i}\right\}}=\tilde{\mu}_i \end{gathered}\)
G	\(d G=-S d T+V d P+\Sigma_i \tilde{\mu}_i d n_i\)	\(T, P,\left\{n_i\right\}\)	\begin{gathered} (\partial G / \partial T)_{P,\left\{n_i\right\}}=-S \\ (\partial G / \partial P)_{T,\left\{n_i\right\}}=V \\ (\partial G / \partial n_i)_{T, P,\left\{n_{j \neq i}\right\}}=\tilde{\mu}_i \end{gathered}

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