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A cylinder, thermally isolated from the environment, contains a mole of a mixture of two diatomic ideal gases A and B at temperature \(T\). At one end of the cylinder, there is a membrane permeable to A and not to B, while at the other end, there is a membrane permeable to B and not to A. The membranes are slowly moved to the center, separating the gases in the container. Calculate the work done to move the membranes.

Answer :

Final answer:

According to the first law of thermodynamics, no work is required to separate the two ideal gases in a thermally isolated cylinder as it closely resembles an adiabatic free expansion process.

Explanation:

In the attempt to separate the two diatomic ideal gases A and B in a thermally isolated cylinder using respective permeable membranes, it is important to consider that both gases undergo a kind of adiabatic process. In such a process, there is no heat exchange (Q = 0)

During the process of the gases being separated, each gas will expand into the half of the cylinder previously occupied by the other gas. In this scenario, the expansion of each gas resembles a free expansion in which the gas expands 'against a vacuum'. As such, no work (W) is done by the gas, and the internal energy (Eint) remains unchanged, thus Eint(i) = Eint(f) as per the first law of thermodynamics.

Consequently, considering the stipulated conditions and according to the first law of thermodynamics, the work done to move the membranes and separate the gases is zero, because in a free expansion, both the heat exchange Q and work done W are zero, maintaining the internal energy (Eint) at a constant. Therefore, no work is done to separate the two gases by slowly moving the membranes to the centre of the cylinder.

Learn more about Adiabatic Process here:

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