2.13 Heat Capacities

Take-home message: Heat capacities are related to changes of entropy with temperature.

See also subsection on Joule-Thomson expansion here.

A heat capacity $C$ is the temperature change per unit heat absorbed by a system during a reversible process: $C {\rm d}T= {}\raise0.44ex\hbox{\bf\symbol{'040}}\llap{d}Q^{\rm rev}$. It is a poor name, since bodies don't contain heat, only energy, but we're stuck with it. (Note the difference between ``heat capacity ($C$)'' and ``specific heat capacity ($c$)''; the latter is the heat capacity per kg or per mole - the units will make clear which.)

The heat capacity is is different for different processes. Useful heat capacities are those at constant volume or constant pressure (for a fluid). Since

$${}\raise0.44ex\hbox{\bf\symbol{'040}}\llap{d}Q^{\rm rev}=T{\rm d}S$$

we have at constant volume $C_V{\rm d}T=T{\rm d}S$, so

$\mbox{\large\colorbox{yellow}{\rule[-3mm]{0mm}{10mm} \ $\displaystyle C_V=T\left({\partial S\over\partial T}\right)_{\!\scriptstyle V}.$ }}$

Similarly

$\mbox{\large\colorbox{yellow}{\rule[-3mm]{0mm}{10mm} \ $\displaystyle C_P=T\left({\partial S\over\partial T}\right)_{\!\scriptstyle P}.$ }}$

Furthermore at constant volume, no work is done on the system and so ${}\raise0.44ex\hbox{\bf\symbol{'040}}\llap{d}Q^{\rm rev}={\rm d}E$; hence

$$C_V=\left({\partial E\over\partial T}\right)_{\!\scriptstyle V}.$$

Also (usefully for chemists), at constant pressure ${}\raise0.44ex\hbox{\bf\symbol{'040}}\llap{d}Q^{\rm rev}={\rm d}H$ where $H$ is the enthalpy, so

$$C_P=\left({\partial H\over\partial T}\right)_{\!\scriptstyle P}.$$

The specific heat capacity is the heat capacity per unit mass (or per mole). Heat capacities are not independent of temperature (or pressure) in general, but over a narrow temperature range they are often treated as such, especially for a solid.

Together with two of Maxwell's relations, we now have expressions for the partial derivatives of the entropy with respect to all easily manipulable variables ($P$, $V$, $T$). These can be used to derive expressions for the entropy change in real processes. (see here for an example.)

We can also derive a relation between $C_P$, $C_V$, and other measurable properties of a substance which can be checked experimentally: if $\alpha$ is the isobaric thermal expansivity and $\kappa_T$ is the isothermal compressibility

$$\alpha={1\over V}\left({\partial V\over\partial T}\right)_{\... ...T}=-{1\over V}\left({\partial V\over\partial P}\right)_{\! T}$$

Then we have the relation

$$C_P-C_V=VT {\alpha^2\over \kappa_{\scriptscriptstyle T}}$$

which is always greater than zero. (The derivation is set as an exercise.) This relation is a firm prediction of thermal physics without any approximations whatsoever. It has to be true! For real gases and compressible liquids and solids it can be checked. For relatively incompressible liquids and solids it is hard to carry out processes at constant volume so $C_V$ may not be well known and this equation can be used to predict it.

For one mole of a van der Waals gas this gives

$$C_P-C_V=R\left(1-{2a(V-b)^2\over R T V^3}\right)^{-1}.$$

In the ideal gas limit $a,b \rightarrow 0$ this reduces to $C_P-C_V=R$ as expected.

References

• Mandl 5.3
• Bowley and Sánchez 2.5-6 & E.3
• Adkins 8.1
• Zemansky 10.6, 10.8