# Carnot's statement and thermal reservoirs

In Feynman's treatment of thermodynamics, Feynman formulated Carnot's principle as follows,

Carnot assumed that heat cannot be taken in at a certain temperature and converted into work with no other change in the system or the surroundings.

and also,

Carnot assumed that it is impossible to extract the energy of heat at a single temperature.

Aren't these statement at odds with the concept of heat baths wherein an amount of heat $$Q$$ can be extracted without changing the reservoir's temperature? And so is it the heat baths in the Carnot engine that make the engine impractical?

• IRL, the resevouir's temperature does change. What we do say is that the temperature of a heat bath is effectively constant, so you're assuming the change is insignificant (either that the resevoir is infinite or that it has an infinite heat capacity). So Carnot created an ideal engine and then went on to say since this is actually impossible, irl, if anyone tells you they've built an engine with an efficiency larger Carnot's, you will know that it is a lie.
– Rye
Commented Jan 8, 2020 at 18:04
• Also, maybe try to read Blundell's Concepts in Thermal Physics on this topic. Feynman isn't very good with Thermodynamics. academia.edu/36089704/Concepts_in_Thermal_Physics-Blundell.pdf
– Rye
Commented Jan 8, 2020 at 19:52

First I'd like to point out that Feynman is not your best resource on the subject of thermodynamics. Other posts have shown that some of his statements, while not necessarily incorrect, can be misleading. It's not his forte.

Carnot assumed that heat cannot be taken in at a certain temperature and converted into work with no other change in the system or the surroundings.

I don't think Carnot put it that way. Heat can be taken in at a certain temperature and converted to work. It's called a reversible isothermal expansion process. The changes that occur is there will be an increase in the entropy of the system due to the heat addition, and an equal decrease in the entropy of the surroundings due the same amount of heat being extracted from the surroundings at the same temperature. I think what Feynman meant to say is you can't convert heat into work while exchanging heat with a single temperature reservoir when operating in a cycle. If that is the case, then the statement would be consistent with the next statement, as I have clarified it.

Carnot assumed that it is impossible to extract the energy of heat at a single temperature.

Missing here is the reference to a complete cycle. The Kelvin Plank statement of the second law says

No heat engine can operate in a cycle while transferring heat with a single heat reservoir

A key phrase is "operate in a cycle". As I said, it is possible to extract heat from a single temperature and do work in a process (e.g., reversible isothermal expansion process), but it is not possible to convert heat entirely into work when operating in a cycle.

Aren't these statement at odds with the concept of heat baths wherein an amount of heat 𝑄 can be extracted without changing the reservoir's temperature?

First, with the corrections/clarifications I discussed above, the statements are consistent with each other. What's more Carnot's actual theorem and the Kelvin Plank statement both refer to heat engines operating between two fixed temperatures. The implication is the temperatures of the source and sink are constant during the heat transfer processes.

And so is it the heat baths in the Carnot engine that make the engine impractical?

It is not the heat baths that make the Carnot engine impractical. In practice thermodynamic cycles can operate between fixed temperatures. All that's required is the heat capacities be large enough relative to the amount of heat transfer so that the temperatures remain relatively constant.

What makes the Carnot engine, or for that matter any reversible engine, impractical is that the processes need to be carried out reversibly, or quasi-statically. The requires the temperature and pressure differentials between the operating fluid and the surroundings to be infinitesimally small. That, in turn, means the processes will occur infinitely slowly. So while the Carnot engine may be the most efficient in producing work, as a practical matter the rate at which work is produced (power) would be very low.

Someone said if you put a Carnot engine in your car you would get fantastic mileage, but pedestrians would be passing you by!.

Hope this helps.

• For Feynman's defense, he didn't really say heat cannot be taken in and converted into heat, alone. He added, and he even put emphasis on it, that this cannot happen without a change in the system or the surroundings, which you describe as the change of entropy. Just out of curiosity, did you read his treatment of thermodynamics? Commented Jan 8, 2020 at 20:50
• Quite a while ago I read it in the Feynman lectures series back in the 60’s while an undergraduate in physics (yeah, I’m that old!). I really liked the series over all. In a recent post though he was quoted saying in a an isothermal expansion you “pull” on the piston to reduce the pressure. Commented Jan 8, 2020 at 21:19

It is impossible to remove heat from a heat bath at constant temperature if the bath is isolated. It violates the Second Law of Thermodynamics since to remove heat energy would require work, but work cannot be done unless there is a temperature gradient.

If instead there is a very small temperature gradient, work can be done whilst maintaining constant temperature by altering the volume of the system. This is the principle behind the reverse heat engine, the most efficient heat engine.

The Carnot engine is impractical because it is very difficult to adjust the volume such that the temperature remains constant in practise.