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?
Best Answer
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.