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, \(\Delta U\).
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}