How does the 2nd Law of Thermodynamics demand the existence of at least two thermal reservoirs in a cycle?

Question

In my studies, I've come across Kelvin's statement of the 2nd law:

It is impossible to undergo a cyclic process whose only effect is removing heat from a reservoir and generating an equivalent amount of work.

Later on, when introducing heat engines, the following is mentioned:

A heat engine produces work from heat, operating in cycles. By Kelvin's statement of the 2nd Law, that cannot be achieved with a single thermal reservoir: at least two reservoirs at different temperatures are needed.

I can see why an engine working with a single reservoir would go against Kelvin's statement. However, I can't quite see why does the 2nd law imply the necessity of a second reservoir at a temperature lower than the first. Why must that be so?

Answer 1

You need the second lower temperature thermal reservoir in order to complete the cycle and do net work.

For example, take the case of the Carnot cycle. Heat is absorbed from the high temperature reservoir during the reversible isothermal expansion and work is obtained from the gas. But you need a path to get back to the initial state. You could reverse the process to isothermally compress the gas and get back using the same single reservoir, but the work done on the gas during the compression would equal the work done by the gas during the expansion and no net work would be done! Therefore you need to isentropically expand the gas to get to a lower temperature and then isothermally compress the gas rejecting heat to the second, lower temperature reservoir. The cycle then wraps up with an isentropic compression to the original state and net work is done.

Hope this helps.