group Small Group Activity

30 min.

##### Earthquake waves
Contemporary Challenges 2021 (4 years)

In this activity students use the known speed of earthquake waves to estimate the Young's modulus of the Earth's crust.

assignment Homework

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}