Confusion with usage of heat capacity in Adiabatic expansion
For an ideal gas, U is a function only of T (and not V).  So the derivative of U with respect to T is an ordinary derivative, and is independent of V.  So why do we call $C_v$ the heat capacity at constant volume, if, for an ideal gas, it doesn't even depend on volume?  It has to do with how we measure $C_v$ experimentally. If the volume is held constant, then the amount of work is zero and $$Q=\Delta U=C_v\Delta T$$  So, in the case of constant volume, we can measure $C_v$ by measuring the amount of heat added Q.  If the volume is not held constant, so that some work is done, we can't determine $C_v$ from the amount of heat added.