# Clausius statement of the 2nd Law

I'm slightly messed up with the Clausius statement of the 2nd Law.

I've seen at least two versions, which seem to be conceptually different.

a) It is impossible to transfer heat from a colder body to a hotter body without any other effect.

b) It is impossible to transfer heat from a cold (thermal) reservoir to a hot (thermal) reservoir without any other effect.

I would like to use the Clausius statement to rule out heat transfer from a colder body to a hotter body, where both have finite thermal capacity. No work put in. If we suppose this transfer could happen, then the hotter body would become hotter and the colder body would became colder. So, there actually is some other effect (temperature changes). How does a) rule out this procces? Obviously b) does the job.

• I must admit that I also have problems with this pair of statements. The notion of "body" is not defined and this makes the discussion impossible. The proof of the equivalence of Clausius statement (a) and Kelvin one (which uses the notion of resevoir) is logically wrong in many textbooks just because there is a confusion between resevoir and body in the crucial point. – Valter Moretti Oct 29 '14 at 13:10
• For instance Fermi's textbook (which nevertheless I like very much), in my opinion, is wrong on this issue. The equivalence proof there is made of two statements: (a) not K implies not C, and it more or less is correct, (b) not C implies not K and the proof of this fact is wrong. This is because in stating not C, Fermi assumes that the bodies are resevoirs, contrarily to the original statement of Clausius postulate appearing few pages before where the generic term "body" appears. – Valter Moretti Oct 29 '14 at 13:14
• I think you're right. The problem is in the definition of what the system is. Let us try: Isolated system: Body A (hotter) + Body B (colder). Diathermal contact between them. Now suppose heat flows from B to A. A gets hotter (higher internal energy), B gets colder (lower internal energy), but by 1st law, the system's internal energy remains constant. So, no other effect at all (regarding the system). Therefore, this process must be ruled out by Kelvin. Is that correct? – Diracology Oct 29 '14 at 15:56

It is impossible to transfer heat from a colder body to a hotter body without any other effect...

in the surrounding bodies.

Every reservoir is a body too.

The statement

without any other effect.

means without external work being acted upon the system. Actually nobody uses the word "effect" and most textbooks use the correct terminology of external work.

Reservoirs are bodies too, therefore 1) directly applies to 2).

• I see what you mean. And I totally agree that if we use the term "without external work" the Clausius Statement rule out "spontaneous" heat transfer from a colder to a hotter body. However, Fermi's book says: A transformation whose only final result is to transfer heat from a body at a given temperature to a body at a higher temperature... – Diracology Jul 9 '15 at 0:54

Imagine two systems. A and B. A is in a higher energy state (say is warmer) than B.

Clausius says you cannot transfer energy from B to A without a corresponding change in another system which is neither A nor B (lets say 'C').

But you can transfer energy from A to B without affectng or invoking C.

The 'other affect' bit implies another system/state/body.

Simples.

A.