Q:

Change in entropy problem (exercise)

This is from the book "Schaum's theory and problems of engineering-thermodynamics".
He says that he does $720 KJ$ of work on the air, so the volume must change, then he says at the end that this a constant volume process and uses the equation of entropy for constant volume process, how does this make any sense? And at first he assumes there's no heat transfer which implies that $Q$ = $0$ then concludes that $-W = mc\Delta T$ ?? I'm lost
What I know that if this is truly an isovolumetric process, then the work done should be zero and then the heat transferred should be $Q=nCv\Delta T$ , would like some help, thank you.

A:

the volume must change, then he says at the end that this a constant volume process and uses the equation of entropy for constant volume process, how does this make any sense?

It is true for gas expansion work that no volume change means no work is done, but the problem is using a different method to perform work on the gas. In the problem a paddle wheel does work on the gas (by pushing on the molecules). In other words it does not always have to be the case that $W=p\Delta V$. This only holds for doing work to compress the gas, or the gas doing work on the surrounding environment while it expands.  

And at first he assumes there's no heat transfer which implies that $Q = 0$ then concludes that $−W=mc\Delta T$ ?? 

The no heat transfer is assumed between the system and the environment. We are still adding energy to the system though, and we determine how much the temperature changes due to this energy addition using the specific heat
$mc=\left(\frac{\partial U}{\partial T}\right)_V$ Usually the constant $V$ means that $\Delta U=Q$, but this is, once again, because you usually just see work as $W=p\Delta V$. Generally the specific heat tells you how much energy is needed to be supplied for a temperature change, but the energy can come from anything.
Additionally, using energy conservation, this change in energy must come from the energy suppied by the wheel, since it can't come from anywhere else.

