Would a paddle wheel on a ratchet slowly spin in a pressurised room?

Question

Imagine you have a wheel that is on a ratchet, which only allows it to turn clockwise. Attached to the wheel is a single paddle(to keep things simple). Let's also say this is in a room with no gravity, but air pressure(e.g. on the ISS). Let's assume the ratchet is extremely fine, having no "give".

I was thinking about it and wondering whether or not this wheel would slowly turn. I am not talking about perpetual motion, but it seems to me that it will slowly extract energy from the heat in the air.

On average, the air molecules apply equal pressure to both sides of the paddle, so it seems that no torque would be applied. However, think about this on the scale of individual particles. Some particles are hitting the wheel clockwise, and some are hitting the wheel counter-clockwise, but they aren't doing so in exact pairs at the same time, since it is random.

So suppose the wheel is at rest. There are 2 cases as to what happens next:

Case 1: A particle hits the paddle from the left, imparting a counter-clockwise torque. This is stopped by the ratchet, so the wheel remains at rest. Return to the start of the proof and consider the 2 cases again.

Case 2: A particle hits the paddle from the right, imparting a clockwise torque. The wheel spins clockwise for the small, but non-zero, amount of time that passes before a particle hits the paddle from the left again.

By repeating these two cases, the wheel will only move clockwise, if slowly. It's getting the energy from the heat in the room, since it steals energy from the particles hitting the paddle from the right.

On the other hand, this seems surely impossible to me. If we attached a generator to the wheel, we would be reversing entropy. So where is the fault in my argument?

Answer 1

As mentioned by another poster, the Feynman lectures contains an analysis of this exact scenario.

Feynman considers the mechanism of the ratchet itself. Containing a jagged wheel and a pawl which latches the wheel and prevents it turning in one direction. Here is the relevant excerpt.

When the vanes get kicked, sometimes the pawl lifts up and goes over the end. But sometimes, when it tries to turn the other way, the pawl has already lifted due to the fluctuations of the motions on the wheel side, and the wheel goes back the other way! The net result is nothing. It is not hard to demonstrate that when the temperature on both sides is equal, there will be no net average motion of the wheel. Of course the wheel will do a lot of jiggling this way and that way, but it will not do what we would like, which is to turn just one way.

Let us look at the reason. It is necessary to do work against the spring in order to lift the pawl to the top of a tooth. Let us call this energy ϵ , and let θ be the angle between the teeth. The chance that the system can accumulate enough energy, ϵ, to get the pawl over the top of the tooth, is e−ϵ/kT. But the probability that the pawl will accidentally be up is also e−ϵ/kT. So the number of times that the pawl is up and the wheel can turn backwards freely is equal to the number of times that we have enough energy to turn it forward when the pawl is down. We thus get a “balance,” and the wheel will not go around.