Problem 2-10

Ebtehal Alrewaily 17 Dec 2013 02:22

Problem 2-10

Drive equation (2-31) and (2-32).

(1)\begin{align} \bar{P}=kT(\frac{\partial\ln}{\partial V})_{N,T} \end{align}

Starting from equation 2-15

(2)\begin{align} \bar{P}=-\frac{\sum_{j}(\frac{\partial E_{j}}{\partial V})e^{-\beta Ej}}{\sum_{j}e^{-\beta E_{j}}} \end{align}

(3)
\begin{align} (\frac{\partial Q}{\partial V})_{N,T}=\frac{\partial}{\partial V}\sum_{j}e^{-\beta E_{j}} \end{align}

(4)
\begin{align} =\sum\frac{\partial}{\partial V}e^{-\beta E_{j}} \end{align}

(5)
\begin{align} =\sum-\beta\frac{\partial E_{j}}{\partial V}e^{-\beta E_{j}} \end{align}

Eq. (2.0.1) then can be written in term of Q:

(6)\begin{align} \bar{P}=\frac{1}{\beta}\frac{1}{Q}(\frac{\partial Q}{\partial V})_{N,T} \end{align}

(7)
\begin{align} =kT(\frac{\partial\ln(Q)}{\partial V})_{N,T} \end{align}

(8)
\begin{align} \frac{d\ln Q}{dV}=\frac{1}{Q}\frac{dQ}{dV} \end{align}