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I am currently studying the Carnot cycle, and am close to understanding it, but something is nagging on my mind. So, during the expansion process of the cycle, work is done on the surroundings as heat energy is transferred from the hot reservoir to the gas piston. During the compression process, work is done by the surroundings as heat energy is transferred from the gas piston to the cold reservoir. So, I know that $dW = pdV$, and therefore less work is required to compress gas at lower pressure (and vice versa, more work released when gas expands at high pressure). This is why there is net work done on the surroundings.

However, during the adiabatic phases, work is done at the expense of the gas's internal energy. To let the gas return back to its original state, shouldn't the work done on the surroundings during the adiabatic expansion be the same as the work done by the surroundings during the adiabatic compression? Since we are trying to regain the internal energy lost, it only makes sense to me this way.

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To let the gas return back to its original state, shouldn't the work done on the surroundings during the adiabatic expansion be the same as the work done by the surroundings during the adiabatic compression?

Yes, you are exactly correct.

The magnitude of the expansion work done by the gas during the reversible adiabatic expansion equals the magnitude of the compression work done on the gas during the reversible adiabatic compression, so that they cancel each other out.

During the expansion there is a decrease in internal energy. Since there is no change in internal energy during the isothermal processes, the adiabatic compression has to increase the internal energy by the same amount in order to return the system to its original internal energy.

Hope this helps

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