# I often see the expression W=/_\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?

I often see the expression $W=V\mathrm{\Delta }P$ 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?
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arribanti9
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 $dV=S\phantom{\rule{thinmathspace}{0ex}}dx$, 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 $dW=F\cdot dx=p\phantom{\rule{thinmathspace}{0ex}}S\phantom{\rule{thinmathspace}{0ex}}dx=p\phantom{\rule{thinmathspace}{0ex}}dV$. Adding work on small motions yields an integral, $W=\int p\phantom{\rule{thinmathspace}{0ex}}dV$
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.