Work done by adiabatic expansion of gas
Part 1. Let's consider first law of thermodynamics.

$\delta Q = dU + \delta W$ (classical sign convention)

or

$\delta Q = dU - \delta W$ (IUPAC sign convention)

In the case of adiabatic process, $\delta Q = 0$.

$0 = dU - \delta W$

$dU = \delta W$

$dU = C_VdT$ (by definition)

$C_VdT = \delta W$

After integration, assuming CV is constant.

$C_V\Delta T =  W$

Part 2. Why not Cp?

By definition, $\delta Q_p = dH = C_pdT$. Obviously, we don't have an isobaric process here.

P. S. Yes, those sign convention differences also make me feel bad.