In chapter 11 of his book on thermodynamics, Callen states that Nernst postulate implies the isothermal compressibility (denoted as $\kappa_T$) of any system vanishes as its temperature approaches zero, though this statement is not justified in the text.
However, two related things are demonstrated in the text: one is that the coefficient of thermal expansion vanishes $$\left(\frac{\partial v}{\partial T}\right)_P = \left(\frac{\partial s}{\partial P}\right)_T \rightarrow 0, \quad \Rightarrow \quad \alpha = \frac{1}{v}\left(\frac{\partial v}{\partial T}\right)_P \rightarrow 0$$ and the other is that $$\left(\frac{\partial P}{\partial T}\right)_v = \left(\frac{\partial s}{\partial v}\right)_T \rightarrow 0.$$
Then, using the above equations and identities of partial derivatives, I can write $\kappa_T$ as $$\kappa_T = -\frac{1}{v}\left(\frac{\partial v}{\partial P}\right)_T = \frac{1}{v}\left(\frac{\partial v}{\partial T}\right)_P \bigg/\left(\frac{\partial P}{\partial T}\right)_v \rightarrow \frac{0}{0},$$ which is an indeterminate form.
Is there a way to proceed from here to show that $\kappa_T$ also tends to zero? Or can this be done in a different way?
