Heat capacity

Heat capacity $C$ relates heat to change in temperature. Generally

C = \frac{\db Q}{\d T}.

However, the precise value of the heat capacity depends on the circumstances, for example on the work done on the system, as well as it’s macrostate.

We define special variables for several common cases.

When no work is done on the system, i.e. when it’s volume is constant, we define

C_V = \left. \frac{\db Q}{\d T} \right|_V.

From the first law we can write

\begin{align*} \d U &= \db Q + \db W \\ &= C_v \d T + P \d V \\ C_v &= \left.\frac{\partial U}{\partial T}\right|_V. \end{align*}

At constant pressure, we say

\begin{align*} C_P &= \left.\frac{\db Q}{\d T}\right|_P \\ &= \frac{\d U - \db W}{\d T} \\ &= \left.\frac{\partial U}{\partial T}\right|_P - P \left.\frac{\partial V}{\partial T}\right|_P. \end{align*}

Since both of these heat capacities depend only on exact derivatives, we conclude that heat capacity is a state function.

For an ideal gas, the heat capacities are related to the number of active degrees of freedom $f$ .

\begin{align*} C_p &= \left (\frac f2 + 1 \right) N k_B \\ C_v &= \frac f2 N k_B. \end{align*}