The isothermal
compressibility is defined as
\begin{equation}
K_{T}=-\frac{1}{V} \left(\frac{\partial V}{\partial p}\right)_{T}
\end{equation}
\(K_T\) is be found by measuring the fractional change in volume when
the the pressure is slightly changed with the temperature held
constant. In contrast, the adiabatic compressibility is defined as
\begin{equation}
K_{S}=-\frac{1}{V} \left(\frac{\partial V}{\partial p}\right)_{S}
\end{equation}
and is measured by making a slight change in pressure without
allowing for any heat transfer. This is the compressibility, for
instance, that would directly affect the speed of sound. Show that
\begin{equation}
\frac{K_{T}}{K_{S}} = \frac{C_{p}}{C_{V}}
\end{equation}
Where the heat capacities at constant pressure and volume are given
by
\begin{align}
C_{p} &= T \left(\frac{\partial S}{\partial T}\right)_{p} \\
C_{V} &= T \left(\frac{\partial S}{\partial T}\right)_{V}
\end{align}

These notes from the fourth week of Thermal and Statistical Physics cover blackbody radiation and the Planck distribution. They include a number of small group activities.