Entropy change for reversible and cyclic process

[spaghetti3451]

Homework Statement

An ideal diatomic gas is initially at temperature $T$ and volume $V$. The gas is taken through three reversible processes in the following cycle: adiabatic expansion to the volume ##2V##, constant volume process to the temperature ##T##, isothermal compression to the original volume ##V##.

Which of the following statements about entropy changes in this cycle is true?

(A) The entropy of the gas remains constant during each of the three processes.
(B) The entropy of the surroundings remains constant during each of the three processes.
(C) The combined entropy of the gas and surroundings remains constant during each of the three processes.
(D) For the complete cycle, the combined entropy of the gas and surroundings increases.
(E) For the complete cycle, the entropy of the gas increases.

Homework Equations

The Attempt at a Solution

During a reversible process, the combined entropy of the gas and surroundings remains constant. Therefore, the correct answer is (C).

Here's why the other answers are incorrect:

During the adiabatic expansion, the heat intake is zero, so the entropy of the gas is constant. Therefore, the entropy of the surroundings is also constant.

During the constant volume process, the work done on the gas is zero (as the volume is constant) and the internal energy of the gas increases (as the temperature increases - adiabats are steeper than isotherms, so the adiabatic expansion decreased the temperature initially). So, the heat intake is positive, so the entropy of the gas increases. Therefore, the entropy of the surroundings decreases.

During the isothermal compression, the work done on the gas is positive (as the volume decreases) and the internal energy of the gas is constant (as the temperature is constant). So, the heat intake is negative, so the entropy of the system decreases. Therefore, the entropy of the surroundings increases.

Lastly, the entropy of the gas is a state variable, so the entropy of the gas is constant for the complete cycle. Therefore, the increase in entropy of the gas in the constant volume process is equal to the decrease in entropy of the gas during the isothermal compression.Do you think my analysis is correct? What mistakes have I made?

 

[Chestermiller]

Your analysis is correct if the process is reversible with respect to both the system and the surroundings. However, the problem statement is a little ambiguous. Just because the changes are carried out reversibly for the system does not necessarily guarantee that the changes are also carried out reversibly for the surroundings. For example, if the volume increase is carried out by your hand pressing on the piston and easing off on the piston gradually as the volume increases, this is not a reversible process on your body (which represents the surroundings). On the other hand, if the volume increase is carried out by removing tiny weights from the piston as it rises to different elevations (with the weights stored at these elevations), then the process is reversible for the surroundings.

Chet

 

[spaghetti3451]

Well, this question actually came up in the 1986 Physics GRE test. I wanted to be sure of my understanding of entropy, so I posted the question.

I think what mentioned in your post is an insightful comment given that you are an experienced veteran, but it is perhaps a little too advanced for the typical undergraduate. Hopefully, for the purposes of the Physics GRE test, I won't have to understand your insight.

 

[Chestermiller]

Well, this question actually came up in the 1986 Physics GRE test. I wanted to be sure of my understanding of entropy, so I posted the question.

I think what mentioned in your post is an insightful comment given that you are an experienced veteran, but it is perhaps a little too advanced for the typical undergraduate. Hopefully, for the purposes of the Physics GRE test, I won't have to understand your insight.

Well then, as I said, on that basis, your analysis is correct.

Chet