Work for constant volume compression by adding moles of gas

Question

I often see the expression $W = V \Delta P$ for the work of a constant volume compression where there are a fixed number of moles and the compression is caused by heating. Is this the work equation for a constant volume, isothermal process where the pressure is increased by adding moles of a gas?

Answer 1

In classical gas dynamics, in order to have work, you need a volume change $dV$. If volume $V$ is constant, no work is performed.

The traditional picture is to think of a piston closing a cylinder containing the gas. The gas inside the cylinder will be characterized by pressure $p$, temperature $T$ and volume $V$. To actually perform work, the piston needs to be moved, which implies a change of volume $dV=S\, dx$, where $S$ is the moving surface of the piston and $dx$ the distance it moved. Work is then performed by the force pushing the piston, related to pressure by $F=p\cdot S$, and during a slight motion $dx$ of the piston, work performed is $dW=F\cdot dx=p\,S\,dx=p\, dV$. Adding work on small motions yields an integral, $W=\int p\, dV$.

If volume doesn't change, no work is performed. Energy may still be transfered into or out of the "cylinder" (whatever the gas' container is), but in a disorderly, non-directional fashion, that is as Heat, not Work.