Problem 29

costsab 08 Mar 2007 13:24

We evaluate the equation:

(1)\begin{align} \[\left( {\frac{{\partial E}}{{\partial V}}} \right)_T - T\left( {\frac{{\partial p}}{{\partial T}}} \right)_V = - p\] \end{align}

From the thermodynamic potential:

(2)\begin{align} \[dE = TdS - PdV\] \end{align}

for constant N (number of molecules) we can write:

(3)\begin{align} \[\left( {\frac{{\partial E}}{{\partial V}}} \right)_T = T\left( {\frac{{\partial S}}{{\partial V}}} \right)_T - p\left( {\frac{{\partial V}}{{\partial V}}} \right)_T \] \end{align}

And from Maxwell Transformation:

(4)\begin{align} \[\left( {\frac{{\partial E}}{{\partial V}}} \right)_T = T\left( {\frac{{\partial p}}{{\partial T}}} \right)_V - p\] \end{align}

By substituting equation (4) into equation (1) we get:

(5)\begin{align} \[T\left( {\frac{{\partial p}}{{\partial T}}} \right)_V - p - T\left( {\frac{{\partial p}}{{\partial T}}} \right)_V = - p\] \end{align}

Which is true as:

(6)\begin{align} \[ - p = - p\] \end{align}