Work done by gas expansion

1. Jul 23, 2013

raopeng

In my textbook W=∫p.dV is only proved for a syringe with a piston. This is quite easily done but the book never explains how it extrapolates to the general situation for a gas expanding in any deformable container. It seems the point is to prove dV= S.h where S is the surface area of a given container and h is a small displacement alone the direction of the normal to the surface. I tried in this line of thought but things soon get really mathematical and confusing. How can prove this formula in any deformable containers? Thanks in advance!

2. Jul 23, 2013

raopeng

Well the mathematical idea is that since for a (n-1) sub-manifold a R^n-1 can be introduced, the orthogonal complement to the hyperplane R^n-1 becomes after a suitable diffeomorphism the normal to the surface in the space, Hence it is possible to write dV=S.h for any n-volume. Then dW=F.h=p.S.h=p.dV...