Problem 3-15
Ebtehal Alrewaily 17 Dec 2013 04:40
(1)
\begin{align} \Xi(V,T,\mu)=\sum\mathit{Q(N,V,T)}e^{\beta\mu\bar{N}} \end{align}
(2)
\begin{align} \Xi(V,T,\mu)=\sum\mathit{Q(\bar{N},V,T)}e^{\beta\mu\bar{N}} \end{align}
(3)
\begin{align} \Xi(V,T,\mu)=\sum\mathit{Q(\bar{N},V,T)}e^{\beta\mu\bar{N}} \end{align}
(4)
\begin{align} \ln\Xi(V,T,\mu)=\ln\mathit{Q+\beta\mu\bar{N}} \end{align}
(5)
\begin{align} \ln\Xi(V,T,\mu)=\frac{S}{K}-\frac{\bar{E}}{kT}+\beta\mu\bar{N} \end{align}
(6)
\begin{align} \ln\Xi(V,T,\mu)=\frac{1}{kT}\mathit{pV} \end{align}
(7)
\begin{align} \triangle\mathit{(p,T,N)=\sum}\mathit{Q(N,\bar{V},T)}e^{-\beta pV} \end{align}
(8)
\begin{align} \ln\Delta(\mathit{p,T,N)=\ln}\sum\mathit{Q(N,\bar{V,T)}e^{-\mathit{\beta pV}}} \end{align}
(9)
\begin{align} =\ln\mathit{Q-\beta pV} \end{align}
(10)
\begin{align} =\frac{\mathit{S}}{K}-\frac{E}{kT}-\beta pV \end{align}
(11)
\begin{align} =-\mathit{\frac{A}{kT}}-\beta pV \end{align}
(12)
\begin{align} =-\mathit{\frac{G}{kT}} \end{align}
(13)
\begin{align} \phi(V,E,\beta\mu)=\sum\Omega(\mathit{\bar{N,V,E})e}^{\beta\mu\bar{N}} \end{align}
(14)
\begin{align} \phi(V,E,\beta\mu)=\sum\Omega(\mathit{\bar{N,V,E})e}^{\beta\mu\bar{N}} \end{align}
(15)
\begin{align} \mathit{\ln\mathit{\phi=\ln\Omega}}+\beta\mu\bar{N} \end{align}
(16)
\begin{align} \ln\phi=\frac{S}{K}-\frac{\bar{E}}{kT}+\beta\mu\bar{N} \end{align}
(17)
\begin{align} \mathit{\ln\phi=-}\frac{H}{kT} \ln\phi=\frac{S}{K}-\frac{\bar{E}}{kT}+\beta\mu\bar{N} \end{align}
(18)
\begin{align} \ln\psi(V,T,\mu_{1},\mathit{N_{1}})=\sum(N_{1},N_{2},\mathit{T},V)e^{\beta\mu N_{1}} \end{align}
(19)
\begin{align} \ln\psi(V,T,\mu_{1},\mathit{N_{1}})=\sum(\bar{N}_{1},N_{2},\mathit{T},V)e^{\beta\mu\bar{N}_{1}} \end{align}
(20)
\begin{align} \ln\psi=\ln[\mathit{Qe^{\beta\bar{\mu\bar{N_{1}}}}}] \end{align}
(21)
\begin{align} \mathit{\ln\psi=\frac{S}{K}}-\frac{\bar{E}}{kT}+\beta\mu\bar{N_{1}}=\frac{ST-\bar{E}}{kT}-\beta\mu\bar{N_{1}}=\mathit{\frac{A}{kT}}-\beta\mu\bar{N} \end{align}
(22)
\begin{align} \ln\psi=\frac{A}{kT}-\beta\mu\bar{N} \end{align}
(23)
\begin{align} W(p,\gamma,T,N)=\sum\sum Q(N,V,AT)e^{-\beta pV}e^{\beta\gamma A} \end{align}
(24)
\begin{align} W(p,\gamma,T,N)=\sum\sum Q(N,\bar{V},\bar{A,}T)e^{-\beta\bar{P}V}e^{\beta\gamma\bar{A}} \end{align}
(25)
\begin{align} \ln\mathit{W=}\ln[\mathit{Q}e^{-\beta p\bar{V}}e^{\beta\gamma\bar{A}}] \end{align}
(26)
\begin{align} \ln\mathit{W=\frac{S}{K}}-\frac{E}{kT}-\beta\bar{p}V+\beta\gamma A=\frac{ST-E}{kT}-\beta\bar{p}V+\beta\gamma A \end{align}
(27)
\begin{align} \ln W=\frac{-A}{kT}-\beta\bar{p}V+\beta\gamma A=-G \end{align}