# Find whether a gas cools or heats up on Joule-Thomson expansion

A gas that follows $P(V - nb)= nRT$ is subjected to Joule-Thomson expansion. Tell whether it cools or heats up.

$$\mu = {\partial T \over \partial P} = {\partial P(V - nb)/nR \over \partial P} = \frac1{nR}\left(V - nb + {\partial V \over \partial P}\right) = \frac1{nR}\left(V - nb - {nRT\over P^2}\right)$$

Now how do I determine whether $\mu >0$ or $\mu < 0$ without knowing anything about temperature or anything else ?

• The equation you gave for $\mu$ is incorrect. That partial derivative is supposed to be at constant enthalpy H. Do you know the mathematical relationship between dH, dT, and dP? Apr 2, 2017 at 1:18
• Yes if you mean $dH = -\mu C_p dP + C_p dT$.
– user31607
Apr 2, 2017 at 1:20
• I set $dH = 0$ then I get $\mu dP = dT$.
– user31607
Apr 2, 2017 at 1:23
• Not that part. The next part. Do you really think that that expression you wrote is equal to the partial derivative of T with respect to P at constant H? Apr 2, 2017 at 1:36
• There are some details in this answer chemistry.stackexchange.com/questions/71543/… . You should find that the coefficient for your gas is $-B/C_p$. Apr 2, 2017 at 9:03

The equation for dH is: $$dH=C_pdT+\left[V-T\left(\frac{\partial V}{\partial T}\right)_P\right]dP$$