Problem3-4
Ebtehal Alrewaily 17 Dec 2013 04:12
State and use Euler's therom to show
(1)\begin{align} \mathit{p=kT(\frac{\partial\ln\Xi}{\partial V}})=kT\frac{\ln\Xi}{V} \end{align}
(2)
\begin{align} 1-\mathit{p=kT}\frac{\ln\Xi}{V} \end{align}
From equation (1-56)
(3)\begin{align} \mathit{dG=-SdT+Vdp+\sum}\mu jdN \end{align}
In The Grand Canonical Ensemble
(4)\begin{align} \mathit{G=\mu N=}E+pV-TS \end{align}
Thus we have
(5)\begin{align} pV=-E+TS-\mu N \end{align}
(6)
\begin{align} \mathit{p=\frac{1}{V}}(-E+TS-\mu N)\rightarrow1 \end{align}
From equation (3-12)
(7)\begin{align} S=\frac{\bar{E}}{T}-\frac{\bar{N}\mu}{T}+\mathit{k}\ln\Xi \end{align}
We can rewrite (1) in term of Entropy in equation (3-12)
(8)\begin{align} p=\frac{1}{V}[-\mathit{E+T}(\frac{\bar{E}}{T}-\frac{\bar{N\mu}}{T}+\mathit{k}\ln\Xi)-\mu N] \mathit{p=\frac{1}{V}}[-E+\bar{\bar{E}-N\mu+}Tk\ln\Xi-\mu N] \end{align}
(9)
\begin{align} p=\frac{1}{v}[\mathit{KT}\ln\Xi] \end{align}
(10)
\begin{align} p=kT\frac{\ln\Xi}{V} \end{align}
Apply Euler's theorem
(11)\begin{align} nF(x_{1,}x_{2,}...,x_{N)=x_{1}}\frac{\partial F}{\partial x_{1}}+.....+\frac{\partial F}{\partial x_{N}} v\ln\Xi(V,T,\mu)=v\frac{\partial\ln\Xi}{\partial v}=v[\frac{\partial}{\partial v}(\frac{pV}{kT})]=v\frac{p}{kT} \end{align}
Therefor
(12)\begin{align} v\ln\Xi(V,T,\mu)=v\frac{p}{kT} \end{align}
We can rewrite it as
(13)\begin{align} p=kT\frac{\ln\Xi}{v} \end{align}
And we can use the same method to state that p=kT(\frac{\partial\ln\Xi}{\partial V})
(14)\begin{align} (\frac{\partial\ln\Xi}{\partial v})_{\mu,T}=\frac{\ln\Xi}{v}=\mathit{\frac{p}{kT}} \end{align}
(15)
\begin{align} p=kT(\frac{\partial\ln\Xi}{\partial v}) \end{align}