Is my interpretation of how a Carnot Cycle works correct?

Question

I'm trying to understand how the Carnot cycle works in my learning of it. Here's what I so far believe is what is happening:

1. (A $\to$ B) A cylinder of gas is connected to a frictionless piston, and connected to a hot thermal reservoir. I have a bit of trouble with this bit, but there are two logical things that happen here: a) the gas is heated by the thermal reservoir and expands, but since it stays in thermal contact with a reservoir that won't lose any temperature, it stays at that new temperature throughout the expansion, or (b) doesn't go any temperature change but merely expands, reducing its pressure after the volume change takes place, both doing work, pushing up a piston. The problem with this is I have trouble understanding how a gas can expand if its temperature doesn't increase, so I'm more inclined to believe in (a) though pedantically I believe (b) is correct.

2. (B $\to$ C) The cylinder now is no longer in thermal contact with a heat reservoir, but is now perfectly insulated. Since the gas is hot, it is continuing to push the piston up, or the piston starts expanding the gas itself, not sure which, but its temperature is no longer able to remain constant, and thus the particles that put pressure on the piston are no longer able to regain the energy they lost pushing on it, so the temperature of the gas reduces. This is an adiabatic expansion, where the volume increased while the pressure and the temperature decreased, so the pressure decreased more than it would in an isothermal expansion.

3. (C $\to$ D) We now need to have this process go back to A for this to be a meaningful way to produce work cyclically for us. The way to do this in the Carnot cycle is to bring the gas in contact with a cold reservoir now which is ideally a thing that drains the energy out of things, which cools the gas to the reservoir's temperature and then causes it to compress, as either (c) the piston is now net pushing down on the gas harder than the gas can push up on the piston or (d) some other reason I can't explain other than gases tend to contract when cooled. Since this is isothermal, the temperature in the volume change stays constant, so once the gas is reduced in temperature it doesn't have any other change in internal energy.
4. (D $\to$ A) The cold reservoir is now removed, and the gas is now completely thermally insulated. This causes it to continue contracting, as.. it was already so it shouldn't all of a sudden stop? Perhaps the piston itself is just compressing it. I can't explain this well. But in doing so, it gains an increased temperature as the pressure is increased as the gas contracts, as the gas has work done it, so pressure increased by virtue of the fact that the gas was contracted and the temperature rose.

I don't think the piston actually does anything as the point of the Carnot cycle is to not need to power something like a piston, I thought, and instead gain work by doing actions on a gas. Anyway, please let me know which parts of my explanations of each process are wrong.

Answer 1

1. At point A the gas is already at a temperature just less than that of the hot reservoir. It is allowed to expand slowly to B, in contact with that reservoir, pushing out the piston while its temperature remains just less than that of the reservoir. [I'm afraid I can't understand the distinction between your (a) and (b).]

2. Agreed. A pedant would add to your last sentence, "between the same two volumes".

3. The piston is pushed in by an outside agency. Strictly the force exerted on the piston by that agency must be only slightly greater than the opposite force on it due to the gas pressure.

4. Same applies about force from outside on piston.

Your last paragraph does, if I may say so, show some confusion. The Carnot cycle is a thought-experiment illustrating the first and second laws of thermodynamics. These are all about heat into or out of a system, work into or out of a system (in this case via the piston) and the internal energy of the system. The piston is vital to the thought-experiment!