The internal energy of a fixed quantity of an ideal gas depends only on its temperature. A sample of an ideal gas is allowed to expand at a constant temperature (isothermal expansion). (a) Does the gas do work? (b) Does the gas exchange heat with its surroundings? (c) What happens to the temperature of the gas? (d) What is \(\Delta U\) for the gas?

Short Answer

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(a) The gas does work. (b) The gas does exchange heat with its surroundings. (c) The temperature of the gas remains constant. (d) \(\Delta U = 0\)

Step by step solution

01

Does the gas do work?

In an isothermal expansion, the ideal gas does do work. This is because when the gas expands, it is pushing outward against the pressure of its surroundings, causing it to do work. This is in accordance with the gas law \(W = P \Delta V\), where \(P\) is pressure and \(\Delta V\) is the change in volume. If there's an expansion, there's a positive \(\Delta V\).
02

Does the gas exchange heat with its surroundings?

Yes, in an isothermal expansion, the ideal gas does exchange heat with its surroundings. Because the internal energy of the gas does not change due to the constant temperature, the work done by the gas has to be balanced by an equal amount of heat absorbed from the surroundings. This is in accordance with the first law of thermodynamics, \(\Delta U = Q - W\), if \(\Delta U=0\), then \(Q = W\).
03

What happens to the temperature of the gas?

The temperature of the gas remains constant during an isothermal expansion. This is inherent in the definition of an 'isothermal' process, which states that the process occurs at a constant temperature. Therefore, no matter how much the gas expands or how much work it does, the temperature of the gas remains the same.
04

What is \(\Delta U\) for the gas?

\(\Delta U\), the change in internal energy of the gas, is zero. As mentioned in the question, the internal energy of an ideal gas depends only on its temperature. In an isothermal process, since the temperature of the gas remains constant, there is no change in internal energy: \(\Delta U = 0\).

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