Heat Capacities

The fact that $dq$ is not a perfect differential is reflected by the observation that the heat capacity depends on path as well.

$$\input{equations/Cv-def}$$ (09-3)

$$\input{equations/Cp-def}$$ (09-4)

$C_V$ is the heat capacity at constant $V$. $C_P$ is the heat capacity at constant $P$.

Question: In materials that expand while heating, they differ considerably. Question: which one should be bigger? Why?

For larger thermal expansion, the difference in heat capacities will be greater.

Gases, which expand considerably with temperature, have a large difference in their heat capacities.

Liquids do not expand as much. For H$_2$O at ${15}^\circ$C and $P=1$ atm, $c_P = 1$   cal$/({}^\circ$   K   gram$)$. (Note use of little $c$ for the derived intensive quantity on a per mass basis), and $c_V$ is only slightly different.

For solids, the difference is very small and usually neglected.