Problem 24

plumleyj 16 Jan 2007 19:01

Show that

(1)\begin{align} C_pk_BT^2= \langle H^2 \rangle - \langle H \rangle^2 \end{align}

In the isothermal-isobaric ensemble, the partition function can be written as

(2)\begin{align} \Delta(N,P,T)=\frac{1}{V_o}\int_0^{\infty}dV \sum_i\exp(-\beta(E_i(V)+PV)) \end{align}

From which the average enthalpy can be obtained as

(3)\begin{align} \langle H\rangle =-\left(\frac{\partial\ln\Delta}{\partial\beta}\right)_{N,P} \end{align}

Therefore

(4)\begin{align} \langle H^2 \rangle - \langle H \rangle ^2 &= \frac{1}{\Delta}\left(\frac{\partial^2\Delta}{\partial\beta^2}\right)_{N,P} - \frac{1}{\Delta^2}\left(\frac{\partial\Delta}{\partial\beta}\right)^2_{N,P}\\ &= \frac{\partial}{\partial\beta}\left(\frac{1}{\Delta}\frac{\partial\Delta}{\partial\beta}\right)_{N,P}\\ &=-\frac{\partial}{\partial\beta}\langle H\rangle\\ &=k_BT^2C_P \end{align}

where we have used the definitions $\beta=1/k_BT$ and

(5)\begin{align} C_P=\left(\frac{\partial \langle H\rangle}{\partial T}\right)_{N,P} \end{align}