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Internal energy of the gas system of a cylinder divided

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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 ?=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

I found that the total internal energy of gases is varies with distance moves compare with the initial total internal energy. So when the pressure on both sides is the same, the sum of energy should be less than 800K? At this time the partition should have maximum kinetic energy. The final temperature in both compartments is actually determined by the sum of internal energy at that moment so I think it will affect final answer.

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The dependence of the internal energy of gas in a cylinder on the size of the cylinder (cnRT for ideal gas) is DIFFERENT from that of a spring (kx^2/2).
In fact, the kinetic ...

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The internal energy of the gas system of a cylinder divided into two compartments are examined.

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See Also This Related BrainMass Solution

Internal Energy of a Gas System: Partition, Conservation

I try to simplify my problem here. This is the place where I may have some misunderstanding in the adiabatic process. Consider a uniform cylinder with insulated wall cylinder that has a very thin and mass less partition at middle divide it into two sections. The left section contains gas molecules and the right side is attached to a mass less spring with vacuum condition.

a) As long as the force exerted by the spring is less than the force exerted by the gas, the partition will move to the right correct?
b) The change in internal energy of the gas will not always be the same as the elastic energy stored by the spring correct?
c) Energy is always conserved so where does the energy goes if it is not the same?
d) Does the process have any differences if the middle partition consists of very thin
and very heavy partition.

All the partition with the contact surface is frictionless.

See the attached file.

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