HW 7: Due Friday 11/20

**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!Derive an expression for the

*change*in constant pressure heat capacity, \(\Delta C_p\).Derive an expression for the

*change*in constant volume heat capacity, \(\Delta C_V\).Derive an expression for the internal energy change, \(\Delta U\).

- Derive an expression for the entropy change, \(\Delta S\).

**Isothermal/Adiabatic Compressibility**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}

**Helmholtz Free Energy of a Van Der Waals Gas**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.

**Ideal gas internal energy**In this problem, you will prove that the internal energy of an ideal gas depends only on volume, based soley on the ideal gas equation: \begin{align} pV &= Nk_BT \end{align} and of course your knowledge of thermodynamics. It's a pretty tricky proof, so I'll step you through it.- 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. - Now show that for any material \begin{align} \left(\frac{\partial {S}}{\partial {V}}\right)_{T} &= \left(\frac{\partial {p}}{\partial {T}}\right)_{V}. \end{align}
- Finally, show that for an ideal gas \begin{align} \left(\frac{\partial {U}}{\partial {V}}\right)_{T} &= 0. \end{align}

- 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