I've been thinking in circles about the relationship between the key variables in relation to the JT Expansion Effect. I'm particularly talking about a situation where the expansion takes place across a valve. Please help clarify...
I've read almost every source but all that is defined is the JouleThomson coefficient which describes the dT/dP i.e., magnitude of cooling per magnitude of pressure drop. But from my understanding, the actual cooling fundamentally happens due to the expansion, particularly in the form of PV increase.

So shouldn't there be a straightforward relationship between the magnitude of expansion (specifically magnitude of PV increase) and the magnitude of cooling?

Very importantly related to the above question: What actually determines the magnitude of the PV increase?

Also, dT/dp is well known to increase as temperature decreases. Again, what is the effect of temperature on PV increases?
So I guess you can say, with the JT phenomenon, I'm trying to nail down the relationship between T, P and V specifically in terms of magnitude.
Thanks so much in advance! Any help is appreciated!