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.

The coefficient of thermal expansion α is  

The coefficient of compressibility

Now  considering

From equation (1) we have that

From  ideal equation

So

=>  

=>  

Explanation:

From the question we are told that

   The  coefficient of thermal expansion is

    The coefficient of compressibility is

Generally the ideal gas is  mathematically represented as

=>      

differentiating both side with respect to T at constant P

substituting the equation above into

Recall from ideal gas equation  

So

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

substituting the above  equation into equation of

Recall from ideal gas equation that

So

Now  considering

From equation (1) we have that

From  ideal equation

So

=>  

=>  

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