Confusion with usage of heat capacity in Adiabatic expansion

My book says, that if a system is expanding adiabatically then $dq=0$ and $U= q+w$, so $U=w$. 


It also states that $$\delta U=(C\_v)\cdot\delta T$$


Here $C\_v$, is the molar heat capacity at constant volume. How can they just $C\_v$, which is defined for a specific volume, in a system that is expanding, i.e. its volume is changing?




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.


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