How can we prove the work done by gas expansion in any deformable container?

AI Thread Summary
The discussion focuses on proving the work done by gas expansion in deformable containers, specifically how the formula W=∫p.dV applies beyond simple systems like syringes. The key point is establishing that dV can be expressed as S.h, where S is the surface area and h is a small displacement normal to the surface. The conversation delves into the mathematical complexities involved in generalizing this concept to any deformable container. It suggests that a suitable diffeomorphism can help define the normal to the surface, allowing for the formulation of dW in terms of pressure and volume changes. Overall, the challenge lies in navigating the mathematical intricacies to prove the formula's applicability in more complex scenarios.
raopeng
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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!
 
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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...
 

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