thermodynamics - Work done in adiabatic process

When we try to establish a relation between the pressure and temperature in adiabatic process we come across a equation..

$dU = dq - PdV$

$dq=0$ (Adiabatic process) and,

$dU=C_v.dT$ (Heat capacity at constant volume)

Therefore, $C_v.dT = -PdV$$\tag1$

In this equation we are using heat capacity defined at constant volume but their still is some work done by the system (i.e, $PdV$ is not $0$ or $dV$ not equals to $0$).

The first part of the equation $(1)$ is implying that the volume is constant but the second part is implying that the volume is not constant (if it was there would be no work done).

Then why there is this contradiction ?

Answer

Even though we call $C_v$ the heat capacity at constant volume, what we really mean by the subscript v is that this is the way we measure $C_v$. At constant volume, we can determine the heat capacity of the material by measuring the heat transferred $dQ=dU=C_vdT$.

But this same heat capacity also applies to all other situations for an ideal gas if we recognize that, for an ideal gas, $U$ depends only on temperature, such that $dU=C_vdT$. It is just that, in these other situations (involving work), dQ is not equal to $C_vdT$.