# Isothermal compressibility of a gas, how the book calculated it?

I'm currently studying Thermodynamics for real for the first time. So, quick question, the book I'm using (Statistical Physics of Particles, Kardar) defines a force constant and the Isothermal compressibility of gas as:

Force constants measure the (infinitesimal) ratio of displacement to force and are generalizations of the spring constant. Examples include the isothermal compressibility of a gas $$\chi_T = -\frac{1}{V} \left(\frac{\partial V}{\partial P}\right)_T$$

After this, he said that since for an ideal gas the equation of the state is $$PV \propto T$$, $$\chi_T = 1/P$$. I have 2 questions:

1) I don't get why it's $$1/P$$. Using the notation of partial derivatives I learned (from the book of mathematical methods of Mary L. Boas), I'm supposed to write $$V(P,T)$$ and then take the partial derivative in relation to $$P$$. But since $$PV \propto T$$, $$V = cT/P$$ (where c is just a constant of proportionality) and $$\left(\frac{\partial V}{\partial P}\right)_T = -cT/P^2$$ and $$\chi_T = cT/P^2V$$. So, what am I doing it wrong? -- SOLVED --

2) The book says that a force constant measures the ratio of displacement to force. OK, in the ideal gas scenario the generalized displacement is the volume and the generalized force is the pressure, but why in the definition of $$\chi_T$$ it's divided by volume? I don't get why it isn't just the partial derivative.

-- UPDATE --

I just realized that I was right, I just had to replace $$V$$ in the final step and it all cancels out, leaving just $$1/P$$. Anyway, I still don't know the answer for my second question, if anyone can help I would be really grateful!

• Just a suggestion. If you are indeed studying thermodynamics for the first time, you should get a good book on macro thermodynamics before you delve into statistical (micro) thermodynamics. – Bob D Jan 24 at 22:26