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During isothermal expansion of an ideal gas, the entropy increases due to an increase in the gas's disorder.

In an isothermal expansion, the temperature of the gas remains constant. This is achieved by adding heat to the system to compensate for the work done by the gas as it expands. The ideal gas law states that PV=nRT, where P is pressure, V is volume, n is the number of moles of gas, R is the gas constant, and T is temperature. If the temperature remains constant, as in an isothermal expansion, then an increase in volume must be accompanied by a decrease in pressure.

Entropy is a measure of the disorder or randomness of a system. In the case of an ideal gas, the entropy is proportional to the natural logarithm of the volume. Therefore, as the volume of the gas increases during the expansion, the entropy of the gas also increases. This is because the gas molecules have more space to move around in, leading to a greater number of possible microstates (ways the gas molecules can be arranged).

The change in entropy can be calculated using the formula ΔS = nRln(Vf/Vi), where ΔS is the change in entropy, n is the number of moles of gas, R is the gas constant, Vf is the final volume, and Vi is the initial volume. This formula shows that the change in entropy is positive when the final volume is greater than the initial volume, as is the case in an expansion.

In summary, the entropy of an ideal gas increases during an isothermal expansion because the gas molecules have more space to move around in, leading to a greater number of possible microstates. This increase in entropy is a measure of the increased disorder or randomness of the gas.

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