Why does a system expand when doing work?

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Is it because for there to be work there must be heat entering the system, which is what is really causing the expansion of the system? Or is it something else? Also vice versa, why does a system compress when work is done on it?

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I think you're mixing up cause and effect with your post. A system must do work against its surroundings in order to expand, and the surroundings must do work on the system in order to compress it.
 
A spring under tension does work on its surroundings when it contracts and must have work done on it in order to expand.
 
For a system of particles to do work to the environment, it must expand due to that while particles keeps exerting force on the boundaries(a wall or piston), work must be done when there is displacement. If it does not then there is merely energy interchanging in the system.
 
Seems if the boundary does not move, it would imply, in the equations of conservation of momentum of an elastic collision, the mass of the particle is negligible comparing to the boundary. So the particle would have a final velocity with opposite direction but the same magnitude. So no energy is lost to the environment.