First Law of Thermodynamics

In an ideal gas, PV = nRT.

Then:

-W = PV = nRT
- dW = nR(dT) = VdP + PdV.

How did you cancel only PdV or VdP, without canceling the other term, by replacement?

It's a bit of a cheat really, because Pressure can be treated as approximately ~constant at an infinitesimal limit, even if we know it isn't really. It leads to dP= 0 since P is a *constant* which leads to dW=-PdV

And what about a real gas:





The equation doesn't cancel out to PV=nRT.

Yes it does. They've taken n=1 there, and already substituted it in, and for an ideal gas (i.e., very dilute) V>>a,b so you can take an approximation that gives PV=RT for 1 mole of gas.

(The actual equation being nRT=(P+an^2/v^2)(V-nb)