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## Homework Statement

The isobaric expansion coefficient and the isothermal compressibility are given by:

$$\alpha_p = (1/V)(\partial V/\partial T)_p \quad \kappa_T = -(1/V)(\partial V / \partial p)_T$$

Suppose they have experimentally been determined to be: $$ \alpha_p = \frac{1}{T} + \frac{3a}{VT^3} \quad \kappa_T = \frac{1}{p}\left(1+\frac{a}{VT^2}\right)$$ with some constant a. Try to determine the equation of state.

## Homework Equations

$$dV(T,p) = \frac{\partial V}{\partial T}dT + \frac{\partial V}{\partial p}dp$$

$$\frac{\partial^2 F}{\partial V^2} = \frac{1}{\kappa_T V}$$ (Not sure if the last one is useful)

## The Attempt at a Solution

I calculated the free energy by plugging in ##\kappa_T## and then solving the equation for ##F## giving: ##F(V,p,T) = pV ln(a+T^2V)+\left(\frac{ap}{T^2}\right)ln(a+T^2V)+c_2V+c_1##

Using the first two equations for ##\alpha_p## and ##\kappa_T## I got $$V(T,p)=\frac{-a~ln~p}{(1+ln~p)T^2}+c,\quad V(T)=\frac{-3a}{2T^2(1-ln~T)}+c$$ by integrating over ##T## for ##\alpha_p## and over ##T## for ##\kappa_T##.

But this approach doesn't seem to make much sense since the two equations for V differ.