Problem 4

plumleyj 04 Dec 2006 19:10

Show that

(1)\begin{align} p=kT\left(\frac{\partial \ln\Xi} {\partial V}\right)_{\mu,T}=kT\frac {\ln \Xi} {V} \end{align}

Utilizing Euler's Theorem;

(2)\begin{align} \ln\Xi\left(\lambda V, T,\mu\right)=\lambda \ln \Xi \left(V,T,\mu\right) \end{align}

Take the derivative of both sides with respect to $\lambda$;

(3)\begin{align} \left(\frac {\partial \ln \Xi} {\partial \lambda V}\right)_{\mu ,T} \frac {\partial \lambda V} {\partial \lambda}= \ln \Xi \end{align}

Set $\lambda$ equal to 1;

(4)\begin{align} \left(\frac {\partial \ln \Xi} {\partial V}\right)_{\mu ,T} V=\ln \Xi \end{align}

Rearrange the equation;

(5)\begin{align} \left(\frac {\partial \ln \Xi} {\partial V}\right)_{\mu ,T}= \frac {\ln \Xi} {V} \end{align}

Substitute into equation 1 to get

(6)\begin{align} p=kT\frac {\ln \Xi} {V} \end{align}