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.