2.3. Manipulating partial derivatives

Example: Change in internal energy

We want to compute the change in molar internal energy \(\Delta U\) of a substance as we vary the temperature \(T\) and pressure \(P\) in terms of quantities we can measure. In addition to \(T\) and \(P\), these quantities are the molar volume \(V\), the thermal expansion coefficient \(\alpha_V\), the isothermal compressibility \(\kappa_T\), and the constant-pressure heat capacity \(c_P\):

(2.1)\[\begin{align} \alpha_V &= \frac{1}{V} \td{}{V}{T}{P} \\ \kappa_T &= -\frac{1}{V} \td{}{V}{P}{T} \\ c_P &= \td{}{H}{T}{P} \end{align}\]

where \(H\) is the molar enthalpy. The following total differentials are known from thermodynamics:

(2.2)\[\begin{align} \d{U} &= T \d{S} - P \d{V} \\ \d{H} &= T \d{S} + V \d{P} \\ \d{G} &= -S \d{T} + V \d{P} \end{align}\]

where \(S\) is the molar entropy and \(G\) is the molar Gibbs free energy.

---

First, we express the total differential for \(U\) as a function of \(T\) and \(P\):

(2.3)\[\begin{equation} \d{U} = \td{}{U}{T}{P} \d{T} + \td{}{U}{P}{T} \d{P} \end{equation}\]

Next, we form the derivatives using the given total differential for \(U\):

(2.4)\[\begin{align} \td{}{U}{T}{P} &= T \td{}{S}{T}{P} - P \td{}{V}{T}{P} \\ \td{}{U}{P}{T} &= T \td{}{S}{P}{T} - P \td{}{V}{P}{T} \end{align}\]

Then, we go about replacing what we don’t like because we can’t measure it with things that we can. For \((\partial S/ \partial T)_P\), use the chain rule followed by the inversion rule:

(2.5)\[\begin{align} \td{}{S}{T}{P} &= \td{}{S}{H}{P} \td{}{H}{T}{P} \\ &= \frac{ (\partial H/\partial T)_P }{ (\partial H/\partial S)_P } \\ &= \frac{c_P}{T} \end{align}\]

where the last step used the total differential for \(H\) to replace the derivative in the denominator. Additionally using the definition of \(\alpha_V\) gives

(2.6)\[\begin{align} \td{}{U}{T}{P} &= T \left( \frac{c_P}{T} \right) - P V \alpha_V \\ &= c_P - P V \alpha_V \end{align}\]

For \((\partial S/\partial P)_T\), use the total differential for \(G\) and equate its mixed second derivatives:

(2.7)\[\begin{equation} -\td{}{S}{P}{T} = \td{}{V}{T}{P} = V\alpha_V \end{equation}\]

Additionally using the definition of \(\kappa_T\) gives

(2.8)\[\begin{align} \td{}{U}{P}{T} &= T (-V \alpha_V) - P(-V\kappa_T) \\ &= P V \kappa_T - T V \alpha_V \end{align}\]

Putting it all together:

(2.9)\[\begin{equation} \d{U} = (c_P - PV\alpha_V)\d{T} + V(P\kappa_T - T\alpha_V)\d{P} \end{equation}\]

This total differential is now suitable for integration with respect to \(T\) and \(P\) using only measurable quantities!