Energy and Entropy: Fall-2020
HW 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, \(\Delta S\).

  2. 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}

  3. 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.

  4. 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.
    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.
    2. 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}
    3. Finally, show that for an ideal gas \begin{align} \left(\frac{\partial {U}}{\partial {V}}\right)_{T} &= 0. \end{align}