Let’s consider the fundamental equations for the thermodynamic potentials that we have derived in section 8.1:
\[\begin{equation} \begin{aligned} dU(S,V,\{n_i\}) &= \enspace T dS -P dV + \sum_i \mu_i dn_i \\ dH(S,P,\{n_i\}) &= \enspace T dS + V dP + \sum_i \mu_i dn_i \\ dA(T,V,\{n_i\}) &= -S dT -P dV + \sum_i \mu_i dn_i \\ dG(T,P,\{n_i\}) &= -S dT + V dP + \sum_i \mu_i dn_i\;. \end{aligned} \end{equation} \nonumber \]
From the knowledge of the natural variable of each potential, we could reconstruct these formulas by using the total differential formula:
\[\begin{equation} \begin{aligned} dU &= \underbrace{\left(\dfrac{\partial U}{\partial S} \right)_{V,\{n_i\}}}_{T} dS + \underbrace{\left(\dfrac{\partial U}{\partial V} \right)_{S,\{n_i\}}}_{-P} dV + \sum_i \underbrace{\left(\dfrac{\partial U}{\partial n_i} \right)_{S,V,\{n_{j \neq i}\}}}_{\mu_i} dn_i \\ dH &= \underbrace{\left(\dfrac{\partial H}{\partial S} \right)_{P,\{n_i\}}}_{T} dS + \underbrace{\left(\dfrac{\partial H}{\partial P} \right)_{S,\{n_i\}}}_{V} dP + \sum_i \underbrace{\left(\dfrac{\partial H}{\partial n_i} \right)_{S,P,\{n_{j \neq i}\}}}_{\mu_i} dn_i \\ dA &= \underbrace{\left(\dfrac{\partial A}{\partial T} \right)_{V,\{n_i\}}}_{-S} dT + \underbrace{\left(\dfrac{\partial A}{\partial V} \right)_{T,\{n_i\}}}_{-P} dV + \sum_i \underbrace{\left(\dfrac{\partial A}{\partial n_i} \right)_{T,V,\{n_{j \neq i}\}}}_{\mu_i} dn_i \\ dG &= \underbrace{\left(\dfrac{\partial G}{\partial T} \right)_{V,\{n_i\}}}_{-S} dT + \underbrace{\left(\dfrac{\partial G}{\partial P} \right)_{T,\{n_i\}}}_{V} dP + \sum_i \underbrace{\left(\dfrac{\partial G}{\partial n_i} \right)_{T,P,\{n_{j \neq i}\}}}_{\mu_i} dn_i\;, \end{aligned} \end{equation} \nonumber \]
we can derive the following new definitions:
\[\begin{equation} \begin{aligned} T &= \left(\dfrac{\partial U}{\partial S} \right)_{V,\{n_i\}} = \left(\dfrac{\partial H}{\partial S} \right)_{P,\{n_i\}} \\ -P &= \left(\dfrac{\partial U}{\partial V} \right)_{S,\{n_i\}} = \left(\dfrac{\partial A}{\partial V} \right)_{T,\{n_i\}} \\ V &= \left(\dfrac{\partial H}{\partial P} \right)_{S,\{n_i\}} = \left(\dfrac{\partial G}{\partial P} \right)_{T,\{n_i\}} \\ -S &= \left(\dfrac{\partial A}{\partial T} \right)_{V,\{n_i\}} = \left(\dfrac{\partial G}{\partial T} \right)_{V,\{n_i\}} \\ \text{and:} \\ \mu_i &= \left(\dfrac{\partial U}{\partial n_i} \right)_{S,V,\{n_{j \neq i}\}} = \left(\dfrac{\partial H}{\partial n_i} \right)_{S,P,\{n_{j \neq i}\}} \\ &= \left(\dfrac{\partial A}{\partial n_i} \right)_{T,V,\{n_{j \neq i}\}} = \left(\dfrac{\partial G}{\partial n_i} \right)_{T,P,\{n_{j \neq i}\}}\;. \end{aligned} \end{equation} \nonumber \]
Since \(T\), \(P\), \(V\), and \(S\) are now defined as partial first derivatives of a thermodynamic potential, we can now take a second partial derivation with respect to a separate variable, and rely on Schwartz’s theorem to derive the following relations:
\[\begin{equation} \begin{aligned} \dfrac{\partial^2 U }{\partial S \partial V} &=& +\left(\dfrac{\partial T}{\partial V}\right)_{S,\{n_{j \neq i}\}} &=& -\left(\dfrac{\partial P}{\partial S}\right)_{V,\{n_{j \neq i}\}} \\ \dfrac{\partial^2 H }{\partial S \partial P} &=& +\left(\dfrac{\partial T}{\partial P}\right)_{S,\{n_{j \neq i}\}} &=& +\left(\dfrac{\partial V}{\partial S}\right)_{P,\{n_{j \neq i}\}} \\ -\dfrac{\partial^2 A }{\partial T \partial V} &=& +\left(\dfrac{\partial S}{\partial V}\right)_{T,\{n_{j \neq i}\}} &=& +\left(\dfrac{\partial P}{\partial T}\right)_{V,\{n_{j \neq i}\}} \\ \dfrac{\partial^2 G }{\partial T \partial P} &=& -\left(\dfrac{\partial S}{\partial P}\right)_{T,\{n_{j \neq i}\}} &=& +\left(\dfrac{\partial V}{\partial T}\right)_{P,\{n_{j \neq i}\}} \end{aligned} \end{equation}\label{8.4.4} \]
The relations in \ref{8.4.4} are called Maxwell relations, and are useful in experimental settings to relate quantities that are hard to measure with others that are more intuitive.
Exercise \(\PageIndex{1}\)
Derive the last Maxwell relation in Equation \ref{8.4.4}.
- Answer
-
We can start our derivation from the definition of \(V\) and \(S\) as a partial derivative of \(G\):
\[ V = \left(\dfrac{\partial G}{\partial P} \right)_{T,\{n_i\}} \qquad \text{and:} \qquad -S = \left(\dfrac{\partial G}{\partial T} \right)_{V,\{n_i\}}, \nonumber \]
and then take a second partial derivative of each quantity with respect to the second variable:
\[\begin{equation} \begin{aligned} \left(\dfrac{\partial V}{\partial T} \right)_{P,\{n_i\}} &=\dfrac{\partial}{\partial T}\left[ \left(\dfrac{\partial G}{\partial P} \right)_{T,\{n_i\}} \right]_{P,\{n_i\}} \\ \\ -\left(\dfrac{\partial S}{\partial P} \right)_{T,\{n_i\}} &=\dfrac{\partial}{\partial P}\left[ \left(\dfrac{\partial G}{\partial T} \right)_{P,\{n_i\}} \right]_{T,\{n_i\}} \;. \end{aligned} \end{equation} \nonumber \]
These two derivatives are mixed partial second derivatives of \(G\) with respect to \(T\) and \(P\), and therefore, according to Schwartz’s theorem, they are equal to each other:
\[\begin{equation} \begin{aligned} \dfrac{\partial}{\partial T}\left[ \left(\dfrac{\partial G}{\partial P} \right)_{T,\{n_i\}} \right]_{P,\{n_i\}} &= \dfrac{\partial}{\partial P}\left[ \left(\dfrac{\partial G}{\partial T} \right)_{P,\{n_i\}} \right]_{T,\{n_i\}}, \\ \\ \text{hence:} \\ \\ \left(\dfrac{\partial V}{\partial T} \right)_{P,\{n_i\}} &= -\left(\dfrac{\partial S}{\partial P} \right)_{T,\{n_i\}}, \end{aligned} \end{equation} \nonumber \]
which is the last of Maxwell relations, as defined in Equation \ref{8.4.4}. This relation is particularly useful because it connects the quantity \(\left(\dfrac{\partial S}{\partial P} \right)_{T,\{n_i\}}\)—which is impossible to measure in a lab—with the quantity \(\left(\dfrac{\partial V}{\partial T} \right)_{P,\{n_i\}}\)—which is easier to measure from an experiment that determines isobaric volumetric thermal expansion coefficients.
