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Consider the two statements below:

(C1) It is impossible for a cyclic engine/heat-pump to transfer heat from a colder body to a hotter body without any third work/heat channel.

(C2) It is impossible for a cyclic engine/heat-pump to transfer heat from a colder heat reservoir to a hotter heat reservoir without any third work/heat channel.

Clearly (C1) implies (C2) almost automatically, because a heat reservoir is counted as a body. However, does (C2) imply (C1)? If so, what is the proof that (C2) implies (C1)? Is there some extra fact that is needed?

I find this an important question, because the Kelvin-Planck statement (as I know it) involves heat reservoirs. You can easily derive (C2) using the Kelvin-Planck statement, but the Clausius statement is usually stated as (C1).

I've seen only one link that acknowledges the difference between the two, and it is this question: Clausius statement of the 2nd Law. However, that question doesn't ask what I'm wondering about.

Edit: For clarification, let's say we negate (C1) (to get a negation of (C2)). Then there exists at least one situation where you have heat flowing from a cold body to a hot body.

My question is, how does the existence of this one situation imply that there exists a situation where heat flows from a cold reservoir to a hot reservoir?

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  • $\begingroup$ If you admit that a reservoir is a body, then you can prove "C2 implies C1" by proving its contrapositive "(not C1) implies (not C2)". $\endgroup$
    – Deep
    Jul 4, 2018 at 4:42
  • $\begingroup$ @Deep I disagree. When you negate (C1), it becomes an "existence" statement rather than a "for all" statement. I added an edit to my question discussing this. $\endgroup$
    – Maximal Ideal
    Jul 4, 2018 at 5:05
  • $\begingroup$ Suppose that your hot body and cold body have infinite heat capacity. Then, any amount of heat added to or removed from them does not change their temperatures. So, in this limit, they become heat reservoirs. $\endgroup$
    – Chet Miller
    Jul 4, 2018 at 12:15
  • $\begingroup$ @SpiralRain Perhaps you are right. Reservoir-bodies are a subset of all bodies. So a statement specifically about reservoir-bodies cannot be carried over to all bodies. So (C1) is indeed the most general statement. $\endgroup$
    – Deep
    Jul 5, 2018 at 6:09

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A cleaner version of Clausius's statement is "It is impossible to construct a device which operates on a cycle and produces no other effect than the transfer of heat from a cooler body to a hotter body". The basis of all the statements is that the total entropy change (system + surroundings) is > 0 for all real (irreversible) processes and may approach 0 for an ideal reversible process. The only difference between C1 and C2, as I see it, is that C2 describes an isothermal (constant temperature) heat transfer process, which simplifies the calculation of the entropy changes for the high and low temperature reservoirs. For a heat transfer Q, in the absence of work, the change in entropy for the high temperature reservoir is simply -Q/TH and for the low temperature is +Q/TL. You can see that for all TH>TL the sum of the two (total entropy change) is >0. It can only be 0 if TH=TL, but if the temperatures are the same then heat transfer can not naturally occur. Work is required. Hope that helps.

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