# How is linear momentum conserved in a cooling gas?

Consider a closed cylindrical tank with no external forces acting on it which is filled with gas which is cooling due to radiative heat loss to the surroundings. Consider a smaller tank at a higher pressure fixed within this tank, which releases a blast of gas inside the larger tank from one end of the cylinder towards the other. The impulse from the blast imparts momentum to the tanks. Conservation of momentum tells me that the total momentum getting transferred molecule to molecule by the resulting pressure wave as it travels the length of the cylinder from one end of the cylinder to the other must be independent of gas temperature and pressure, such that the axial momentum of the tanks is exactly cancelled when the wave is done bouncing around inside (with the velocity of the CG of the tank-gas system remaining constant).

During the time it takes the wave to travel the length of the tank the average velocities of all the gas molecules have decreased, due to decrease in temperature. I assume that the localized density at the wave front must compensate for the loss of molecular velocity such that exactly 100% of the axial component of momentum is transferred back into the tank, but this does not seem obvious given that all the molecular velocities are lower than at the start, and that the degree to which they are lower is not tied to the energy or momentum of of the initial blast since it is just a function of the rate of heat transfer to the surroundings. How is the axial component of momentum conserved when the molecular velocities have all decreased since the initial impulse?

Edited for brevity and to add basic conceptual diagram:

• Welcome to Physics! A diagram would really help elucidate what you're talking about. I've read your setup paragraph a couple times now and I'm still not sure that I've got it right. Nov 15 '21 at 18:12
• Hello! Please only ask one question per post – otherwise it might get closed due to lack of focus. You can always edit your question or ask a new one. Thanks! Nov 15 '21 at 19:30
• As it stands, this question asks multiple questions and may be closed for "needing more focus". The first question is directly about your physical setup, but the second and third questions are more general. Your third question in particular will likely get more attention as a stand alone question. Nov 15 '21 at 19:32
• Most introductory thermodynamics focuses on equilibrium systems. You may want to read about the Boltzmann equation, to get an idea of the scope of the calculation you are asking about. Nov 15 '21 at 19:40
• Thanks for the pointers. I'm hoping for a qualitative explanation of how momentum is conserved in this scenario. I think I can almost answer my own question, but I am out of my depth. Nov 15 '21 at 20:09