Clarification on 2nd law of thermodynamics

Question

I was reading Feynman lectures on the 2nd law of thermodynamics

Now, what about the second law of thermodynamics? We know that if we do work against friction, say, the work lost to us is equal to the heat produced. If we do work in a room at temperature T, and we do the work slowly enough, the room temperature does not change much, and we have converted work into heat at a given temperature.

I agree with this. Work if done using quasi-equilibrium states, would render the temperature to be constant.

Then Feynman considers the reverse possibility

What about the reverse possibility? Is it possible to convert the heat back into work at a given temperature?

Well I say yes, but Feynman says

The second law of thermodynamics asserts that it is not.

Now I don’t understand this. Isothermal expansion is a thing. If we have a system, initially we compress it isothermally and then isothermally expand it, it consists a cycle but it is theoretically allowed and would violate the second law.

I don’t understand this yet again:-

the whole world were at the same temperature, one could not convert any of its heat energy into work: while the process of making work go into heat can take place at a given temperature, one cannot reverse it to get the work back again.

Now considering isothermal process, I don’t understand this statement.

Answer 1

It's hard to follow what you have presented from Feynman (there is other relevant text that you did not include), but consider the following.

While it is theoretically possible for a heat engine to take heat from a single heat reservoir and convert it entirely into work in a process (E.g., a reversible isothermal expansion process), it is not possible to produce net work using a single heat reservoir in a cycle. That would violate the Kelvin-Planck statement of the second law:

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

Some heat must always be rejected to another, lower temperature, heat reservoir. The net work done is then the difference between the heat taken from the high temperature reservoir and heat rejected to the low temperature reservoir.

Hope this helps.

Answer 2

Answers from the comments, as suggested here. Answers should be posted as answers, and comments should be used for their intended purpose.

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From @naturallyinconsistent

When you isothermally compress something, you are doing a combination of two possibilities: 1) use work to push heat out from a system into a heat reservoir, 2) pull heat out of a system into a heat reservoir at infinitesimally lower temperature than the system. For simplicity, Feynman discussed a detail in (1) that implies excess heat generation that is also pushed out. When you isothermally expand, the two possibilities are 3) use work to pull heat from heat reservoir to system, 4) use an infinitesimally hotter reservoir. Consider this properly, then you cannot violate 2nd law

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From @hyportnex

Thermal energy transfer contributes to work only if entropy is transported over a thermal gradient, in isothermal entropy transport work is done by the system against its own stored internal energy.

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From @Flatterman

Are you assuming that you are converting the thermal energy back into a system that has no temperature itself, like an ideal spring? Well, there is your problem. Ideal springs do not exist and that's why they are technically not violating Feynman's statement. Neither do infinite collections of heat reservoirs, which also don't exist. All real world systems, however, can only come close to ideal energy conversion but they can never reach it.

A Hamiltonian potential (e.g. representing a spring) does not have a temperature. It ignores all internal degrees of freedom of a real spring and is therefor not a thermodynamic system. Can we violate thermodynamics by using non-thermodynamic systems? Absolutely.

Thermodynamics is a theory of (almost) homogeneous irreversible systems. We can construct almost reversible corner cases in the theory. We can not construct them on the bench. On the bench absolutely every system is irreversible.