Maxwell's Thermodynamic Relations

Maxwell's thermodynamic relations are four fundamental equations that establish relationships among temperature (\(T\)), pressure (\(P\)), volume (\(V\)), and entropy (\(S\)). These relations are obtained from the fundamental thermodynamic equations by applying the mathematical properties of exact differentials.

According to the first law of thermodynamics,

\[ dU=\delta Q-\delta W \]

For a reversible process involving only pressure-volume work,

\[ \delta Q_{\mathrm{rev}}=T\,dS \qquad\text{and}\qquad \delta W_{\mathrm{rev}}=P\,dV \]

Therefore,

\[ \boxed{dU=T\,dS-P\,dV} \] (1)

Here, \(U\) is internal energy, \(T\) is absolute temperature, \(S\) is entropy, \(P\) is pressure, and \(V\) is volume.

Let \(U\), \(S\), and \(V\) be differentiable functions of two independent variables \(x\) and \(y\). Their total differentials are

\[ dU= \left(\frac{\partial U}{\partial x}\right)_y dx+ \left(\frac{\partial U}{\partial y}\right)_x dy \]

\[ dS= \left(\frac{\partial S}{\partial x}\right)_y dx+ \left(\frac{\partial S}{\partial y}\right)_x dy \]

\[ dV= \left(\frac{\partial V}{\partial x}\right)_y dx+ \left(\frac{\partial V}{\partial y}\right)_x dy \]

Substituting these expressions into Eq. (1),

\[ \begin{aligned} &\left(\frac{\partial U}{\partial x}\right)_y dx +\left(\frac{\partial U}{\partial y}\right)_x dy\\ &=T\left[ \left(\frac{\partial S}{\partial x}\right)_y dx+ \left(\frac{\partial S}{\partial y}\right)_x dy \right]\\ &\quad-P\left[ \left(\frac{\partial V}{\partial x}\right)_y dx+ \left(\frac{\partial V}{\partial y}\right)_x dy \right]. \end{aligned} \]

Collecting the coefficients of \(dx\) and \(dy\),

\[ \begin{aligned} dU={}& \left[ T\left(\frac{\partial S}{\partial x}\right)_y -P\left(\frac{\partial V}{\partial x}\right)_y \right]dx\\ &+ \left[ T\left(\frac{\partial S}{\partial y}\right)_x -P\left(\frac{\partial V}{\partial y}\right)_x \right]dy. \end{aligned} \]

Comparing the coefficients of \(dx\) and \(dy\) on both sides, we obtain

\[ \boxed{ \left(\frac{\partial U}{\partial x}\right)_y = T\left(\frac{\partial S}{\partial x}\right)_y - P\left(\frac{\partial V}{\partial x}\right)_y } \] (2)

\[ \boxed{ \left(\frac{\partial U}{\partial y}\right)_x = T\left(\frac{\partial S}{\partial y}\right)_x - P\left(\frac{\partial V}{\partial y}\right)_x } \] (3)

Differentiating Eq. (2) partially with respect to \(y\), keeping \(x\) constant,

\[ \begin{aligned} \frac{\partial^2 U}{\partial y\,\partial x} ={}& \left(\frac{\partial T}{\partial y}\right)_x \left(\frac{\partial S}{\partial x}\right)_y\\ &+T\frac{\partial^2 S}{\partial y\,\partial x}\\ &- \left(\frac{\partial P}{\partial y}\right)_x \left(\frac{\partial V}{\partial x}\right)_y\\ &-P\frac{\partial^2 V}{\partial y\,\partial x}. \end{aligned} \]

Similarly, differentiating Eq. (3) partially with respect to \(x\), keeping \(y\) constant,

\[ \begin{aligned} \frac{\partial^2 U}{\partial x\,\partial y} ={}& \left(\frac{\partial T}{\partial x}\right)_y \left(\frac{\partial S}{\partial y}\right)_x\\ &+T\frac{\partial^2 S}{\partial x\,\partial y}\\ &- \left(\frac{\partial P}{\partial x}\right)_y \left(\frac{\partial V}{\partial y}\right)_x\\ &-P\frac{\partial^2 V}{\partial x\,\partial y}. \end{aligned} \]

Since internal energy, entropy, and volume are state functions, their differentials are exact. For continuous second partial derivatives,

\[ \frac{\partial^2 U}{\partial x\,\partial y} = \frac{\partial^2 U}{\partial y\,\partial x} \]

\[ \frac{\partial^2 S}{\partial x\,\partial y} = \frac{\partial^2 S}{\partial y\,\partial x} \]

\[ \frac{\partial^2 V}{\partial x\,\partial y} = \frac{\partial^2 V}{\partial y\,\partial x} \]

Equating the two expressions for the mixed second derivatives of \(U\), and cancelling the corresponding second-derivative terms of \(S\) and \(V\), we obtain

\[ \boxed{ \begin{aligned} & \left(\frac{\partial T}{\partial y}\right)_x \left(\frac{\partial S}{\partial x}\right)_y - \left(\frac{\partial T}{\partial x}\right)_y \left(\frac{\partial S}{\partial y}\right)_x\\[4pt] ={}& \left(\frac{\partial P}{\partial y}\right)_x \left(\frac{\partial V}{\partial x}\right)_y - \left(\frac{\partial P}{\partial x}\right)_y \left(\frac{\partial V}{\partial y}\right)_x \end{aligned} } \tag{4} \]

Equation (4) is the general expression for deriving Maxwell's thermodynamic relations. Four relations are obtained by choosing the independent variable pairs \((S,V)\), \((T,V)\), \((S,P)\), and \((T,P)\).

(i) Maxwell's First Thermodynamic Relation

Let

\[ x=S,\qquad y=V \]

Then,

\[ \begin{aligned} \left(\frac{\partial S}{\partial x}\right)_y&=1, & \left(\frac{\partial S}{\partial y}\right)_x&=0,\\ \left(\frac{\partial V}{\partial x}\right)_y&=0, & \left(\frac{\partial V}{\partial y}\right)_x&=1. \end{aligned} \]

Substituting these values into Eq. (4),

\[ \begin{aligned} & \left(\frac{\partial T}{\partial V}\right)_S(1) - \left(\frac{\partial T}{\partial S}\right)_V(0)\\ ={}& \left(\frac{\partial P}{\partial V}\right)_S(0) - \left(\frac{\partial P}{\partial S}\right)_V(1). \end{aligned} \]

Therefore,

Maxwell's First Relation \[ \boxed{ \left(\frac{\partial T}{\partial V}\right)_S = -\left(\frac{\partial P}{\partial S}\right)_V } \tag{i} \]

This relation is associated with the internal energy \(U(S,V)\), whose fundamental differential is

\[ dU=T\,dS-P\,dV. \]

(ii) Maxwell's Second Thermodynamic Relation

Let

\[ x=T,\qquad y=V. \]

For this choice, it is convenient to introduce the Helmholtz free energy,

\[ F=U-TS. \]

Differentiating,

\[ dF=dU-T\,dS-S\,dT. \]

Using Eq. (1),

\[ dF=T\,dS-P\,dV-T\,dS-S\,dT. \]

Thus,

\[ \boxed{dF=-S\,dT-P\,dV} \]

Since \(F\) is a function of \(T\) and \(V\),

\[ dF= \left(\frac{\partial F}{\partial T}\right)_VdT+ \left(\frac{\partial F}{\partial V}\right)_TdV. \]

Comparing coefficients,

\[ S=-\left(\frac{\partial F}{\partial T}\right)_V \]

\[ P=-\left(\frac{\partial F}{\partial V}\right)_T. \]

Differentiating the first expression with respect to \(V\) at constant \(T\), and the second with respect to \(T\) at constant \(V\),

\[ \left(\frac{\partial S}{\partial V}\right)_T = -\frac{\partial^2 F}{\partial V\,\partial T} \]

\[ \left(\frac{\partial P}{\partial T}\right)_V = -\frac{\partial^2 F}{\partial T\,\partial V}. \]

Since \(F\) is a state function, its mixed second derivatives are equal. Hence,

Maxwell's Second Relation \[ \boxed{ \left(\frac{\partial S}{\partial V}\right)_T = \left(\frac{\partial P}{\partial T}\right)_V } \tag{ii} \]

(iii) Maxwell's Third Thermodynamic Relation

Let

\[ x=S,\qquad y=P. \]

Introducing enthalpy \(H\), defined as

\[ H=U+PV. \]

Differentiating,

\[ dH=dU+P\,dV+V\,dP. \]

Substituting Eq. (1),

\[ dH=T\,dS-P\,dV+P\,dV+V\,dP. \]

Therefore,

\[ \boxed{dH=T\,dS+V\,dP} \]

Since \(H\) is a function of \(S\) and \(P\),

\[ dH= \left(\frac{\partial H}{\partial S}\right)_PdS+ \left(\frac{\partial H}{\partial P}\right)_SdP. \]

Comparing coefficients,

\[ T=\left(\frac{\partial H}{\partial S}\right)_P \]

\[ V=\left(\frac{\partial H}{\partial P}\right)_S. \]

Taking mixed partial derivatives,

\[ \left(\frac{\partial T}{\partial P}\right)_S = \frac{\partial^2 H}{\partial P\,\partial S} \]

\[ \left(\frac{\partial V}{\partial S}\right)_P = \frac{\partial^2 H}{\partial S\,\partial P}. \]

As \(H\) is a state function, the mixed second derivatives are equal. Hence,

Maxwell's Third Relation \[ \boxed{ \left(\frac{\partial T}{\partial P}\right)_S = \left(\frac{\partial V}{\partial S}\right)_P } \tag{iii} \]

(iv) Maxwell's Fourth Thermodynamic Relation

Let

\[ x=T,\qquad y=P. \]

Introducing Gibbs free energy \(G\), defined as

\[ G=U+PV-TS. \]

Differentiating,

\[ dG=dU+P\,dV+V\,dP-T\,dS-S\,dT. \]

Substituting Eq. (1),

\[ \begin{aligned} dG={}&T\,dS-P\,dV+P\,dV\\ &+V\,dP-T\,dS-S\,dT. \end{aligned} \]

Therefore,

\[ \boxed{dG=-S\,dT+V\,dP} \]

Since \(G\) is a function of \(T\) and \(P\),

\[ dG= \left(\frac{\partial G}{\partial T}\right)_PdT+ \left(\frac{\partial G}{\partial P}\right)_TdP. \]

Comparing coefficients,

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

\[ V=\left(\frac{\partial G}{\partial P}\right)_T. \]

Taking mixed partial derivatives,

\[ \left(\frac{\partial S}{\partial P}\right)_T = -\frac{\partial^2 G}{\partial P\,\partial T} \]

\[ \left(\frac{\partial V}{\partial T}\right)_P = \frac{\partial^2 G}{\partial T\,\partial P}. \]

Since the mixed second derivatives of \(G\) are equal,

Maxwell's Fourth Relation \[ \boxed{ \left(\frac{\partial S}{\partial P}\right)_T = -\left(\frac{\partial V}{\partial T}\right)_P } \tag{iv} \]

Summary of Maxwell's Relations

The four Maxwell relations and their corresponding thermodynamic potentials are summarized below.

Relation	Thermodynamic Potential	Maxwell Relation
First	\(U(S,V)\)	\(\displaystyle \left(\frac{\partial T}{\partial V}\right)_S = -\left(\frac{\partial P}{\partial S}\right)_V \)
Second	\(F(T,V)\)	\(\displaystyle \left(\frac{\partial S}{\partial V}\right)_T = \left(\frac{\partial P}{\partial T}\right)_V \)
Third	\(H(S,P)\)	\(\displaystyle \left(\frac{\partial T}{\partial P}\right)_S = \left(\frac{\partial V}{\partial S}\right)_P \)
Fourth	\(G(T,P)\)	\(\displaystyle \left(\frac{\partial S}{\partial P}\right)_T = -\left(\frac{\partial V}{\partial T}\right)_P \)