Chapter 3 
The Second Law of Thermodynamics
(Feb. 2008)
We’ll use a different sequence than the book.
First, we will get familiar with a new state
function, the entropy S.
After a brief interlude about the efficiency of heat engines, we
will discuss how
S > 0 is a signature of spontaneous processes in isolated systems.
3.1) Entropy S, a State Function Related to q
rev
(2
nd
half of 202)
dU =
q
rev
+
w
rev
with
w
rev
=  P dV,
so for an ideal gas
q
rev
= f(T)/2 nR dT + P dV = f(T)/2 nR dT + n RT/V dV
As shown in Homework 4.2), for an ideal gas,
dS =
q
rev
/T = 1/2 nR f(T)/T dT + n R 1/V dV is an exact differential.
{Generalize this result:
Consider an isolated system (dU = 0,
w = 0,
q = 0) consisting of two rigid subsystems
w
A
=
w
B
= 0
A is an ideal gas,
B is any system in thermal contact with A.
Heat produced in A is absorbed in B and vice versa:
q
rev,A
+
q
rev,B
=
q = 0
Since T
A
= T
B
,
q
rev,B
/T = 
q
rev,A
/T , the same exact differential for both systems.}
The Second Law generalizes this result, postulating that S is a state function for any
system.
dS =
q
rev
/T
S is a state function like U, so we can calculate it for any state of the system.
3.2)
S(T, V) for an Ideal Gas
dS is an exact differential
S is a state function
S(T, V) can be found
Let’s start with some simple integrals:
For isothermal volume change:
U = 0, so
q
rev
= 
w
rev
S =
1
∫
2
dS =
1
∫
2
q
rev
/T = 
1
∫
2
w
rev
/T =
=
nR
V1
∫
V2
1/V dV =
nR (lnV
2
– lnV
1
) = nR ln(V
2
/V
1
)
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View Full DocumentFor constantvolume temperature change: w = 0, so
q
rev
= dU = C
V
(T) dT
S =
1
∫
2
dS =
1
∫
2
q
rev
/T =
1
∫
2
dU/T =
=
T1
∫
T2
f/2 n R/T dT =
f/2 nR (lnT
2
– lnT
1
) = f/2 nR ln(T
2
/T
1
)
(for Tindependent f; otherwise, this is more complicated).
Logarithmic dependence is typical for S
= k
B
ln
= R/N
A
ln
Combine these results to
S(T,V) = nR ln(V/V
0
) + f/2 nR ln(T/T
0
) + S(T
0
,V
0
)
for an ideal gas with constant f
(this works so simply only because the T and V dependencies are unrelated)
Calculate
dS = (
∂
S/
∂
T)
V
dT + (
∂
S/
∂
V)
T
dV
Use the chain rule of derivatives:
(
∂
S/
∂
T)
V
= f/2 nR 1/(T/T
0
) 1/T
0
= f/2 nR 1/T
(
∂
S/
∂
V)
T
= nR 1/(V/V
0
) 1/V
0
= nR 1/V
Put these together: dS = f/2 nR 1/T
dT + nR 1/V dV
Just as derived from the First Law.
Problem for T = 0 (dS becomes infinite).
That this may not happen will be discussed in
conjunction with the Third Law of thermodynamics.
3.3)
Entropy and Disorder
In statistical mechanics, entropy is recognized as a measure of
disorder/randomness/number of configurations of the system with the same energy.
Formally,
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 Spring '08
 SCHMIDTROHR
 Thermodynamics, state function, reversible process, spontaneous processes, nR ln

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