Assignment:
A cylinder is close at both ends and has insulating walls. It is divided into two compartments by a perfectly insulating partition that is perpendicular to the axis of the cylinder. Each compartment contains 1.00 mol of oxygen, which behaves as an ideal gas with γ=7/5. Initially the two compartments have equals volumes, and their temperatures are 550k and 250k. The partition is then allowed to move slowly until the pressure on its two sides is equal. Find the final temperatures in the two compartments.
a) The author considers (final temperature/internal energy) of gases in both partition when pressure is equal on both side is the same as sum of initial temperature (550+250=800K). Since internal energy is related to temperature. My concern is compare to previous spring mass system or ideal gas system, when the pressure on both side is equal, the kinetic energy of the insulating partition (block) should be maximum? Since force=pressure*area of partition and the area on both side is identical. That is the reason I try to compute the force as a function of distance move.
∫F (X) dx(left partition)=∫F(X) dx(right partition) + 0.5mv^2