When I took this class years ago, I simply accepted it as fact. However, now that I'm teaching it, the following bugs me....a lot.
Why do they use Cv to describe the change of internal energy during the carnot cycle?
http://chemwiki.ucdavis.edu/Physical_Chemistry/Thermodynamics/Thermodynamic_Cycles/Carnot_Cycle
There is obviously a volume change. But it is defined that $\Delta U = n*Cv*\Delta T$. How do you justify the use of Cv when there's a volume change?
It is always true that, for an ideal gas, $\Delta U = C_V \Delta T$, regardless of the process. Remeber, we define $C_V=(\delta Q/dT)_V$. Since this is happening at constant volume (aka $\delta W=0$), we have $C_V=(\delta Q/dT)_V=(dU/dT)_V$. Then, since $U$ doesn't depend on volume for an ideal gas, we have that $C_V=dU/dT$ even if volume is changing. So $dU=C_V dT$.
Another way to think about it, just using algebra: for an ideal gas, $U=\alpha NkT$. Thus $\Delta U=\alpha NK\Delta T$. But of course $\alpha NK$ is just $C_V$ for an ideal gas, proving the statement.
In an adiabatic expansion there is not heat exchange, technically the gas cools down because it does work. We are using $C_v$ here only as means to account for the change in internal energy, which happens to be equal than in a constant volume process. So we used $C_v$ here not because there is a constant volume (there is not) but because is equivalent to a process carried out at constant volume where there is the same temperature change.
