Problem 2-17
Problem 2-17
In chapter 14 we shall derive an approximate partition function
for dense gas, which is the form
\begin{align} Q(N,V,T)=\frac{1}{N!}(\frac{2\Pi mkT}{h^{2}})^{3N/2} (V-Nb)^{N}e^{aN^{2}/VkT} \end{align}
Calculate the equation of state from this partition function ?
To obtain the equation os state , we apply Eq. 2-23 to obtain the
pressure
\begin{align} \bar{\bar{P}=kT(\frac{\partial\ln Q}{\partial V}})_{N} \end{align}
(3)
\begin{align} =kT(\frac{\partial}{\partial V}\ln[\frac{1}{N!}(\frac{2\pi mkT}{h^{2}})^{3N/2}(V-Nb)^{N}e^{aN^{2}/VkT}])_{N.T} \end{align}
(4)
\begin{align} =kT(\frac{\partial}{\partial V}\{\ln[\frac{1}{N!}\frac{2\pi mkT}{h^{2}})^{3N/2}]+\ln(V-Nb)^{N}+\ln(e^{aN^{2}/VkT})\} \end{align}
(5)
\begin{align} =kT(0+\frac{N}{(V-Nb)}-\frac{aN^{2}}{V^{2}kT}) \end{align}
(6)
\begin{align} =\frac{kTN}{V-Nb}-\frac{aN^{2}}{V^{2}} \end{align}
Which is the Van der Waals equation. The energy of this gas is
(7)\begin{align} \bar{E}=kT^{2}(\frac{\partial\ln Q}{\partial T})_{N.T} \end{align}
(8)
\begin{align} =kT^{2}(\frac{\partial}{\partial T}\ln[\frac{1}{N!}(\frac{2\pi mkT}{h^{2}})^{3N/2}(V-Nb)^{N}e^{aN^{2}/VkT}])_{N,V} \end{align}
(9)
\begin{align} =kT^{2}(\frac{\partial}{\partial T}\{\ln(\frac{1}{N!})+\ln(\frac{2\pi mkT}{h^{2}})^{3N/2}+\ln(V-Nb)^{N}+\ln e^{aN^{2}/VkT})])_{N,V} \end{align}
(10)
\begin{align} =kT^{2}(0+\frac{3}{2}N\frac{h^{2}}{2\pi mkT}\frac{2\pi mk}{h^{2}}+0-\frac{aN^{2}}{VkT^{2}}) \end{align}
(11)
\begin{align} =kT^{2}(\frac{3}{2T}-\frac{aN^{2}}{VkT^{2}}) \end{align}
(12)
\begin{align} =\frac{3}{2}NkT-\frac{aN^{2}}{V} \end{align}
The heat capacity is then
(13)\begin{align} C_{v}=(\frac{\partial E}{\partial T})_{N,V} \end{align}
(14)
\begin{align} =(\frac{\partial}{\partial T}\frac{3}{2}NkT-aN^{2})_{N,V} \end{align}
(15)
\begin{align} =\frac{3}{2}NK \end{align}
Comparing to problem 1-30
(16)\begin{align} (\frac{\partial E}{\partial V})_{T}=(\frac{\partial}{\partial V}\frac{3}{2}NkT-\frac{aN^{2}}{V})_{T} \end{align}
(17)
\begin{align} =\frac{aN^{2}}{V^{2}} \end{align}
which is the same
(18)\begin{align} =T(\frac{\partial E}{\partial V})-p \end{align}
(19)
\begin{align} =T(\frac{\partial}{\partial T}\frac{kTN}{V-Nb}-\frac{aN^{2}}{V^{2}})-\frac{kTN}{V-Nb}+\frac{aN^{2}}{V^{2}} \end{align}
(20)
\begin{align} =\frac{aN^{2}}{V^{2}} \end{align}
