## Energy and Entropy: Fall-2020HW 7: Due Friday 11/20

1. Temperature Change of an Ideal Gas

The enthalpy $H\equiv U+pV$ of a certain gas was determined over a range in temperature while pressure was kept constant at the value $p_0$ and the results are summarized in the expression: \begin{equation*} H=a T+b T^{3} \end{equation*} where $a$ and $b$ are experimentally determined constants. Consider the process when the gas is heated from temperature $T_1$ to temperature $T_2$, both values within the range of validity of the expression above, while the pressure was kept constant at the value $p_0$ used in the experiments described above. You may assume that the equation of state for this gas is given by the ideal gas law $pV=Nk_BT$, and that the internal energy of the gas is a function of $T$ only, i.e. \begin{align} \left(\frac{\partial {U}}{\partial {V}}\right)_{T} &= 0. \end{align} Note that this property is only true for an ideal gas!

1. Derive an expression for the change in constant pressure heat capacity, $\Delta C_p$.

2. Derive an expression for the change in constant volume heat capacity, $\Delta C_V$.

3. Derive an expression for the internal energy change, $\Delta U$.

4. Derive an expression for the entropy change, $\Delta S$.

The isothermal compressibility is defined as $$K_{T}=-\frac{1}{V} \left(\frac{\partial V}{\partial p}\right)_{T}$$ $K_T$ 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 $$K_{S}=-\frac{1}{V} \left(\frac{\partial V}{\partial p}\right)_{S}$$ 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 $$\frac{K_{T}}{K_{S}} = \frac{C_{p}}{C_{V}}$$ 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}
The Helmholtz free energy of a van der Waals gas can be written as: \begin{equation*} F=-N k T\left\{1+\ln \left[\frac{(V-N b) T^{\frac{3}{2}}}{N}\right]\right\}-\frac{a N^{2}}{V} \end{equation*} Where $a$ and $b$ are constants. Derive the equation of state (relationship between $p$, $T$, and $V$) for this Helmholtz free energy and sketch (or plot) the pressure as a function of volume at fixed temperature.
1. To begin with, use the Helmholtz free energy $F=U-TS$ to show that \begin{align} \left(\frac{\partial {U}}{\partial {V}}\right)_{T} &= -p + T\left(\frac{\partial {S}}{\partial {V}}\right)_{T} \end{align} for any material.