# Clausius statement — Reservoir vs Body

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?

• 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)".
– Deep
Jul 4, 2018 at 4:42
• @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. Jul 4, 2018 at 5:05
• 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. Jul 4, 2018 at 12:15
• @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.
– Deep
Jul 5, 2018 at 6:09