Problem 1-30

Ebtehal Alrewaily 17 Dec 2013 02:02

1-30 Drive the equation

(1)\begin{align} dE=[T(\frac{\partial P}{\partial T})-P]dV+C_{v}dT \end{align}

From equation (1-45)

dE=TdS-PV

And equation (1-2)

(2)\begin{align} -P=(\frac{\partial E}{\partial V})-T(\frac{\partial P}{\partial T}) \end{align}

Along with equation (3-18) from An Introduction to Thermal Phyics

book

\begin{align} dS=\frac{C_{v}dT}{T} \end{align}

To drive our equation we will apply equation (1-2) to (3-15) then

to (1-45)

\begin{align} dE=T(\frac{C_{V}dT}{T})-[(\frac{\partial P}{\partial V})_{N,T}-T(\frac{\partial P}{\partial T})]dV \end{align}

(5)
\begin{align} dE=[T(\frac{\partial P}{\partial T})+(\frac{\partial E}{\partial V})]dV+C_{v}dT \end{align}

From equation (1-46)

(6)\begin{align} (\frac{\partial E}{\partial V})=-P \end{align}

We get

(7)\begin{align} dE=[T(\frac{dP}{dT})-P]dV+C_{v}dT \end{align}

Sow that

(8)\begin{align} (\frac{\partial E}{\partial V})_{T}=\frac{a}{V^{2}} \end{align}

Drive equation (1-45) in term of V

(9)\begin{align} (\frac{\partial E}{\partial V})_{T}=-P \end{align}

From Van der waals gas equation (5-49) in the book of

introduction to Thermal Phyics

\begin{align} (P+\frac{aN^{2}}{V^{2}})(V-Nb)=NkT We can write it P=\frac{NkT}{V-Nb}-\frac{aN^{2}}{V^{2}} \end{align}

Drive this equation in term of V at constant T and you will get

(11)\begin{align} P=(-\frac{a}{V^{2}}) \end{align}

We can know apply it to (\frac{\partial E}{\partial V})_{T}

and

have

\begin{align} (\frac{\partial E}{\partial V})_{T}=-(-\frac{a}{V^{2}}) \end{align}

(13)
\begin{align} (\frac{\partial E}{\partial V})_{T}=\frac{a}{V^{2}} \end{align}