Joule expansion

Problem Statement:

A container with adiabatic walls is divided into two equal compartments. The left compartment is filled with n moles of an ideal gas, while the right compartment is evacuated to create a void. If the partition between the two compartments is opened, and the gas fills both sides of the container, determine the entropy change of the gas and that of the universe.

Givens: R = 8.31 J/mol K

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Solution:

The process undergone by the ideal gas between states A and B is known as Joule expansion (or free expansion). It is an irreversible process, because the gas expands into a void, and it cannot return from state B to A.

The entropy change is defined:

Where δQR is the heat exchanged by the system when it undergoes a reversible process.

When the gas expands from state A to B, there is no heat transfer with the surroundings because the gas is enclosed in a container with adiabatic walls. Therefore, for process AB we have:

Where the superscript I is used to indicate that the heat transfer is irreversible and thus it cannot be used to calculate the entropy change between states A and B.

Nonetheless, as entropy is a state function, the entropy changes by the same amount whether the path AB is followed by a reversible or irreversible process. We can consider any hypothetical reversible process from state A to state B to calculate the entropy change as long as the initial and final states are A and B.

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There is no work done (W = 0) when the gas expands freely from state A to B since it expands against zero pressure. Furthermore, we have seen that it does not transfer heat. So using the first Law of Thermodynamics we have:

That means that the internal energy of the ideal gas is the same in states A and B. But the internal energy of an ideal gas depends only of its temperature:

Since the temperature of the ideal gas is the same in states A and B, we can connect both through an isothermal process in order to calculate the entropy change between A and B. Said process is represented (in green) in the following PV diagram:

The entropy change of an isothermal process between states A and B is:

The two container compartments have the same volume, therefore the volume occupied by the gas in state B is twice the volume occupied in state A. After substituting we have:

The entropy change of the universe is the sum of the entropy change of the gas and its surroundings. But the latter is zero because the gas is enclosed in a container with adiabatic walls and therefore does not transfer heat with the surroundings.

Which is positive because the gas has undergone an irreversible process and in this case the entropy of the universe always increases.

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