A certain gas is described by the equation of state
What will be an ideal response?
Here, b is a constant and ? is a function of T only. For this gas, determine expressions for the isothermal compressibility ? and the thermal pressure coefficient (?P /?T ) V . These expressions should contain only T, P, ?, d?/dT, and constants.
For this problem, we simply need to use the definitions of the isothermal compressibility and the thermal pressure coefficient, and take the specified partial derivatives. If the equation of state is:
Then we can write V as a function of P and T as
The definition of ? is
So, we just need to take the derivative of V with respect to P at constant T, which is
So,
Similarly, to find the thermal pressure coefficient, we can solve the EOS for P in terms of T and V to get
So, taking the derivative with respect to T at constant V, we have
If we want this only in terms of T, P, ? and d?/dT, we should then substitute into it
Note that we have to do this after taking the derivative, since we need to have V in the expression so that we can hold it constant while taking the derivative. Substituting this in gives
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