Entropy
This suggests that the following generalization to be true for any reversible
processes,
dQr
 T  0
cycle
where,
dQr is the infinitesimal heat absorbed/released by the system at an
infinitesimal reversible step at temp T.

cycle
denotes the integration evaluated over one complete cycle.
Entropy
We have seen this property previously,

dU  0
cycle
Changes in the internal energy U over a
closed cycle is zero!
This is a consequence of the fact that U is a state variable and dU for any
processes depends on the initial and final states only.
Thus the result
such that,
dQr
 0 indicates that there is another state variable S


T
cycle
dS 
dQ
T
and
 dS  0
cycle
This new state variable S is called the entropy of the system.
Entropy: Disorder
Recall that the 2nd Law of Thermodynamics is a statement on nature’s
preferential direction for systems to move toward the state of disorder.
Let see how Entropy is a quantitative measure of disorder.
From our previous derivation, the quantity d Q / T was from the isothermal
branch of the infinitesimal Carnot cycle. Let look at an isothermal expansion
of an ideal gas microscopically:
Intuitively, as the gas expands into a bigger
volume, the degree of randomness for the
system increases since molecules now have
more choices (spaces) for them to move around.
One can associate the increase in randomness to
the ratio:
V
dV
or
V
V
(T stays the same  avg. KE stays the same)
Entropy: Disorder
Since this is an isothermal process, we have the following relation from the 1st Law:
nRT
dQ  dW  PdV 
dV
V
 dU  0 
1 dQ dV

nR T
V
So, the newly introduced macroscopic variable S (entropy),
dS 
dQ
T
S   J / K
is directly proportion to the degree of disorder of the system.
dS is an infinitesimal entropy change for a reversible process at temperature T.
For any finite reversible process, the total entropy change S is,
f
S  
i
dQ
T
2nd Law (Quantitative Form)
W
system
environment
Q
Stot  S sys  Senv  0
(Stot  0 reversible; Stot  0 irreversible)
“The total entropy (disorder) of an isolated system in any
processes can never decrease.”
“Nature always tends toward the macrostate with the highest S
(disorder) [most probable (later)] in any processes.”
Entropy Changes for Different
Processes
1.
General Reversible Processes:
f
f
i
i
S   dS  
dQr
T
Note: S is a state variable, S is the same for all processes
(including irreversible ones) with the same initial and final
states!
P
NOTE: in most applications,
it is the change in entropy S
which one typically needs to
calculate and not S itself.
i
f
V
Entropy Changes for Different
Processes
2.
Reversible Cycles:
Scycle
3.
dQr
  ds  
0
T
cycle
cycle
Any Reversible Processes for an Idea Gas:
(Ti , Vi )  (T f , V f )
(Note: Thru the Ideal Gas Law, P
is fixed for a given pair of T & V.)
1st Law gives,
dU  dQr  dW
dQr  dU  dW  nCV dT  PdV
dQr  nCV dT 
nRT
dV
V
Entropy Changes for Different
Processes
Dividing T on both sides and integrating,
f
f
dQ
dV 
 nC dT
S   r    V
 nR

T
T
V


i
i
We have,
 Tf
S  nCV ln 
 Ti

 Vf 
  nR ln  

 Vi 
Entropy Changes for Different
Processes
4.
Calorimetric Changes:
dQ  mcdT
f
f
dQ
mcdT
S  

T
T
i
i
 Tf 
If c is constant within temperature range, S  mc ln  
 Ti 
If c T  is a function of T, S  m  c T  dT
f
i
T
Entropy Changes for Different
Processes
5.
During Phase Changes (or other isothermal
Processes):
dQ 1
S  
  dQ
T
T
Q mL
S  
T
T
(T stays constant during a
phase change.)
Entropy Changes for Different
Processes
6.
Irreversible Processes:
Although for a given irreversible process, we cannot write
dS = dQr/T, S between a well definite initial state a and final state
b can still be calculated using a surrogate reversible process
connecting a and b. (S is a state variable!)
Example 20.8: (free expansion of an ideal gas)
Initial State a: (V,T)
Since Q=W=0, U=0.
For an ideal gas, this
means that T=0 also.
Although Q=0, but S
is not zero!
Final State b: (2V,T)
S in a Free Expansion
Important point: Since S is a state variable, S is the same for
any processes connecting the same initial a and final b states.
In this case, since T does not change, we can use an surrogate isothermal
process to take the ideal gas from state a (V,T) to state b (2V,T) to calculate S.
Applying our general formula,
 Tf
S  nCV ln 
 Ti

 Vf 
  nR ln  

 Vi 
we have,
Surrogate
Isothermal Expansion
T 
 2V
S  nCV ln    nR ln 
T 
V

  nR ln 2  5.76 J / K

(n=1)
2nd Law (Quantitative Form)
W
system
environment
Q
Stot  S sys  Senv  0
(Stot  0 reversible; Stot  0 irreversible)
“The total entropy (disorder) of an isolated system in any
processes can never decrease.”
“Nature always tends toward the macrostate with the highest S
(disorder) [most probable] in any processes.”
2nd Law (S > 0 & Clausius Statement)
Clausius Statement: Heat can’t spontaneously transfer from TC to TH.
We will prove this by contradiction using Stot > 0.
Assume the contrary,
cold
S H 
SC 
Q
TH
Q
(heat absorbed into TH)
Q
(heat released by TC)
TC
(with the explicit signs,
|Q| is taken to be +.)
hot
Q Q
Stot 

TH
TC
 1
1 
Stot  Q 
 
 TH TC 
TH
 TL 
0
Not Possible!
(violated Stot > 0)
2nd Law (S > 0 & Kelvin-Planck
Statement)
Kelvin-Planck Statement: No heat engine can convert heat from TH
completely into W.
We will prove this by contradiction again using the Stot > 0.
Assume the contrary,
Sengine  0
(engine operates in a cycle)
 QH
S H 
TH
|QH|
SC  0 (no heat exchange)
So,
Stot  Sengine  S H  SC 
Not Possible!
(again violated Stot > 0)
 QH
TH
0
2nd Law (S > 0 & Carnot Theorem)
For any reversible heat engines,
Sengine  0
Senv 
 QC
TC

 QH
2nd Law
TH
Stot  Sengine  Senv 
QC
QH

TC
TH
QC
TC
or

QH
TH
0
QC TC

QH TH
2nd Law (S > 0 & Carnot Theorem)
Consider the efficiency of a heat engine in general,
QC
e  1
QH
Applying the inequality from the previous slide,
QC
T
 C
QH TH
QC
T
e  1
 1 C
QH
TH
Recall that the efficiency of a Carnot Cycle is given by ecarnot  1 
TC
.
TH
This gives our desired result, e  ecarnot . Carnot engine is the most efficient!
Extra:engines