thermodynamic question

[DatInterviewDood]

A sample of monatomic ideal gas is taken through an adiabatic expansion and is then isothermally compressed until the gas returns to its original pressure. Which of the following is true of this process?

a) The net heat input into the gas is zero.
b) The net work done by the gas is zero.
c) The final state of the gas has a lower total internal energy than the initial state.
d) The final state of the gas has a higher average temperature than the initial state.

My diagram:

Thermo always confuses me. Can anyone correct me on my reasoning?

a) In the first adiabatic step, the heat input is 0. The gas does work in expanding and total energy decreases.

In the second isothermal step, the work is negative since volume decreases. So heat must be added to the system to keep the total energy the same.

b) Work done by the gas only occurs in the first step and is negative. I think the work done in the second step is by the system (on the gas).

c) I think this is the true statement. Energy is lost in the first adiabatic step and conserved in the isothermal step, so the total internal energy should be lower.

d) For an ideal gas, we know that PV = nRT applies. The final pressure is the same, and the final volume is lower, and since n and R are constant, the final temperature must also be lower.

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[aldol16]

A) Whether work is positive or negative depends on the convention you're using. But in general, Q = 0 for adiabatic processes and Q is required for isothermal processes. This is because as you compress the gas, it's losing energy and that energy is going into the environment. Another way to look at it is that you have to do work on the gas in order to compress it. Since you're doing work on the gas, you're adding energy to the gas. For the temperature to remain the same, that same amount of energy must leave the system - in the form of heat. This is actually key to how a refrigerator works. A gas is isothermally compressed as it passes through the container on the exterior of the refrigerator so it deposits heat into the exterior environment. In the inside compartment, it isothermally expands so that it does work on the environment. Since energy must remain constant for temperature to remain constant, it must take on some energy from the environment in the form of heat. These two processes are then connected with adiabatic curves. This is also how a window air conditioner works.

B) Yeah, net work cannot be zero because adiabatic expansion and isothermal compression don't result in 1) returning to the same starting point on the P-V graph and 2) taking the same path to the initial starting point on the P-V curve. Thus, there is a difference in area between these curves and work is just the integral of a P-V graph with the starting and end points being the limits. This integral must be non-zero in this case.

C and D are just reverses of each other, so immediately you should be able to rule out A and B anyway. Higher average temperature means greater energy. The only question then becomes: is energy higher or lower now?

dU = Q + W. In the first adiabatic expansion, Q = 0 by definition and work is done by the gas on the environment (that's negative by convention). Thus, internal energy decreases. This makes sense, because if it's expanding then it's losing energy to the environment by doing work on the latter. Since there's no heat flow, that energy must come solely from the gas's own internal energy. During any isothermal process, dU = 0 because dU = m*C*dT. Therefore, the net internal energy change is negative. The answer is C.