4Classical thermodynamics

II Statistical Physics

4.2 The second law

We want to talk about the second law, but we don’t know about entropy yet.

The “original” versions of the second law were much more primitive and direct.

There are two versions of the second law, which we will show are equivalent.

Law

(Kelvin’s second law)

.

There exists no process whose sole effect is to extract

heat from a heat reservoir and convert it into work.

Law

(Clausius’s second law)

.

There is no process whose sole effect is to transfer

heat from a colder body to a hotter body.

Both of these statements are motivated by experiments, but from the point

of view of our course, we are just treating them as axioms.

Note that the words “sole effect” in the statement of the second laws are

important. For example, a fridge transfers heat from a colder body to a hotter

body. But of course, a fridge needs work input — we need to get electricity from

somewhere.

Since they are given the same name, the two statements are actually equiv-

alent! This relies on some rather mild assumptions, namely that fridges and

heat engines exist. As mentioned, fridges take in work, and transfer heat from a

colder body to a hotter body. Heat engines transfer heat from a hotter body to a

cooler body, and does work in the process. We will later construct some explicit

examples of such machines, but for now we will just take it that they exist.

Proposition. Clausius’s second law implies Kelvin’s second law.

Proof.

Suppose there were some process that violated Kelvin’s second law. Let’s

use it to run a fridge:

hot reservoir

cold reservoir

not

Kelvin

fridge

Q

H

W =Q

H

Q

C

Q

C

+Q

H

and this violates Clausius’s law.

Similarly, we can prove the other direction.

Proposition. Kelvin’s second law implies Clausius’s second law.

Proof. If we have a not Clausius machine, then we can do

hot reservoir

cold reservoir

heat

engine

not

Clausius

Q

H

Q

H

− W

W

Q

H

− W

Q

H

− W

Then this has a net effect of taking heat

W

from the hot reservoir and done

work W . This violates Kelvin’s law.