Question

The coefficient of thermal expansion α = (1/V)(∂V/∂T)p. Using the equation of state, compute the value of α for an ideal gas. The coefficient of compressibility β is define by β = -(1/V)(∂V/∂p)T. Compute the value of β for an ideal gas. For an ideal gas, express the derivative (∂p/∂T)v in terms of α and β. Do the same derivative for van der Waals gas.

Answer 1

Answer:

The coefficient of thermal expansion α is  

      $\alpha = \frac{1}{T}$

The coefficient of compressibility

      $\beta = \frac{1}{P}$

Now  considering $(\frac{ \delta P }{\delta T} )V$

From equation (1) we have that

       $\frac{ \delta P}{\delta T} = \frac{n R }{V}$

From  ideal equation

         $nR = \frac{PV}{T}$

So

     $\frac{\delta P}{\delta T} = \frac{PV}{TV}$

=>  $\frac{\delta P}{\delta T} = \frac{P}{T}$

=>   $\frac{\delta P}{\delta T} = \frac{\alpha }{\beta}$

Explanation:

From the question we are told that

   The  coefficient of thermal expansion is $\alpha = \frac{1}{V} * (\frac{\delta V}{ \delta P}) P$

    The coefficient of compressibility is $\beta = - (\frac{1}{V} ) * (\frac{\delta V}{ \delta P} ) T$

Generally the ideal gas is  mathematically represented as

        $PV = nRT$

=>      $V = \frac{nRT}{P} --- (1)$

differentiating both side with respect to T at constant P

       $\frac{\delta V}{\delta T } = \frac{ n R }{P}$

substituting the equation above into $\alpha$

       $\alpha = \frac{1}{V} * ( \frac{ n R }{P}) P$

        $\alpha = \frac{nR}{PV}$

Recall from ideal gas equation  $T = \frac{PV}{nR}$

So

          $\alpha = \frac{1}{T}$

Now differentiate equation (1) above with respect to  P  at constant T

          $\frac{\delta V}{ \delta P} = -\frac{nRT}{P^2}$

substituting the above  equation into equation of $\beta$

        $\beta = - (\frac{1}{V} ) * (-\frac{nRT}{P^2} ) T$

        $\beta =\frac{ (\frac{n RT}{PV} )}{P}$

Recall from ideal gas equation that

       $\frac{PV}{nRT} = 1$

So

       $\beta = \frac{1}{P}$

Now  considering $(\frac{ \delta P }{\delta T} )V$

From equation (1) we have that

       $\frac{ \delta P}{\delta T} = \frac{n R }{V}$

From  ideal equation

         $nR = \frac{PV}{T}$

So

     $\frac{\delta P}{\delta T} = \frac{PV}{TV}$

=>  $\frac{\delta P}{\delta T} = \frac{P}{T}$

=>   $\frac{\delta P}{\delta T} = \frac{\alpha }{\beta}$