Maxwell relations

3.26 Maxwell relations

The fundamental equations represent exact differentials, e.g.

\begin{equation*} dG = V dp - S dT \quad \mbox{, i.e.} \quad \left(\frac{\partial^2 G}{\partial p \partial T} \right) = \left(\frac{\partial^2 G}{\partial T \partial p} \right) \quad \mbox{; thus} \quad -\left(\frac{\partial S}{\partial p} \right)_T = \left(\frac{\partial V}{\partial T} \right)_p \label{eq:Maxwell_G} \end{equation*}

(3.62)

Generally

\begin{equation*} df(x,y) = g dx + h dy \quad \mbox{gives} \quad \left(\frac{\partial g}{\partial y} \right)_x = \left(\frac{\partial h}{\partial x} \right)_y \label{eq:Maxwell_general} \end{equation*}

(3.63)

So from the other fundamental equations we get further Maxwell relations:

\begin{equation*} \begin{split} \left(\frac{\partial S}{\partial V} \right)_T = &\;\; \left(\frac{\partial p}{\partial T} \right)_V \\ \left(\frac{\partial T}{\partial V} \right)_S = &\;\; -\left(\frac{\partial p}{\partial S} \right)_V \\ \left(\frac{\partial T}{\partial p} \right)_S = &\;\; \left(\frac{\partial V}{\partial S} \right)_p \\ \left(\frac{\partial V}{\partial T} \right)_p = &\;\; -\left(\frac{\partial S}{\partial p} \right)_T \\ \end{split} \label{eq:Maxwell_additional} \end{equation*}

(3.64)

These Maxwell relations allow for drastic simplification and replacements of difficult-to-measure relations by easy-to-measure relations and thus contribute strongly to the powerful toolbox of thermodynamics as we will see in some subsequent examples.