If someone ever says "free expansion does no work" all they mean is that it does no work on the vacuum, which is pretty obvious in retrospect. This is because 19th century experimenters and 21st century high schools find it easiest to talk about gas properties in terms of pistons pushing on containers of gas. If the piston is replaced by nothingness, well clearly no work will be extracted from the system.
This doesn't mean the gas doesn't do anything. Think of it this way: First, you have a closed container, sitting in vacuum and containing a gas with some nonzero pressure PP inside. The force on the walls is the same in all directions, no matter the shape of the container, but for simplicity you can picture it as a cube with side length ss. Each wall will have a force Ps2Ps2 pushing on it.
Now remove one wall. There will no longer be any force acting on it (your "free expansion" principle), but until the gas is fully evacuated there will be a force on the opposite wall. So your container has a net force in the opposite direction from the gas expulsion lasting for some time. Momentum is conserved; rockets work.
When you're considering the properties of gases there are often two ways to look at the problem. The first is to use the continuum approximation leading to the usual laws like Boyle's law, Charles' law etc. The second is to treat the gas as many tiny particles (i.e. the gas atoms/molecules) and use Newtonian mechanics. In this case I think the second way is to understand what's going on.
The rocket motor burns a mixture of fuel and oxygen to produce a very hot gas. By very hot we mean that the gas molecules have very high random velocities:
If the fuel were burning in a vaccum the random directions of the atom velocities would mean the ball of atoms expands in a roughly spherical way and the total momentum stays zero. But the fuel is not burning in a vacuum, it's burning inside a combustion chamber:
The reason this matters is that the atoms can't escape to the right or up or down because the walls of the combution chamber are in the way. So they will bounce around until some random collision (with the walls or other atoms) gives them a velocity pointing to the left:
So very quickly all the atoms are going to end up with their velocities pointing in roughly the same direction, because at that point they can escape from the combustion chamber and go flying off into space. Now let's calculate the momentum of all those atoms. If there are NN atoms and the mass of each atom is mm and their average velocity is vv then the total momentum is now NmvNmv (we'll take velocity to the left to be positive). The momentum of the fuel before burning was zero, and after burning it's NmvNmv, so the momentum has changed by NmvNmv. Conservation of momentum means the rocket must have changed its momentum by −Nmv−Nmv so that the total momentum change adds up to zero.
So burning the fuel and allowing it to escape to the left means the rocket must have accelerated to the right. In other words the rocket engine has produced a force on the rocket, and we've calculated this without needing to think of pressures or other macroscopic quantities. In fact we can be more precise about the force. If the rocket produces NsNs particles of exhaust gas per second then the momentum change of the rocket per second is −Nsmv−Nsmv. Momentum change is force times time, so the force on the rocket is simply:
F=Nsmv
F=Nsmv
This force is produced simply because atoms moving to the right bounce off the end of the combustion chamber, and hence push the rocket to the right, but atoms moving to the left don't.
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