Thermodynamics and Statistical
Mechanics
Open Systems
and
Chemical Potential
Thermo & Stat Mech Spring 2006 Class 13
1
Diffusive Interaction
If particles are added to a system, the energy of
the system can change, because of the chemical
potential of the added particles in their new
environment. A term is needed to account for
this effect.
dU = TdS – PdV + mdn
Thermo & Stat Mech - Spring 2006
Class 13
2
Chemical Potential
dU = TdS – PdV + mdn
In this equation, m is the chemical energy
per kilomole, and dn is the change in the
number of kilomoles.
Thermo & Stat Mech - Spring 2006
Class 13
3
Chemical Potential
Suppose 2.0 ×10-5 kilomoles of acid is
added to a 1.0 liter of water at room
temperature. The temperature of the water
rises 0.15ºC. From this data the chemical
potential of the acid in water can be
calculated.
Thermo & Stat Mech - Spring 2006
Class 13
4
Chemical Potential
Q  mc P  T
 (1 . 0 kg)(1.0 kcal  kg
 (0.15 kcal)(4184
m 
Q
n

-1
 K )(0.15 K)
-1
J/kcal)  627.6 J
627.6 J
2.0  10
-5
kmoles
m   3 . 138  10 J/kmole
7
Thermo & Stat Mech - Spring 2006
Class 13
5
Chemical Potential
 3 . 138  10 J/kmole
7
m 
m 
6 . 02  10
26
molecules/
 5 . 213  10
 20
1 . 6  10
 19
kmole
J/molecule
J/eV
m   0 . 326 eV/molecul
Thermo & Stat Mech - Spring 2006
Class 13
e
6
Chemical Potential
dU  TdS  PdV  m dn
 U 
 U 
 U 
dU  
 dS  
 dV  
 dn
  S V ,n
  V  S ,n
  n  S ,V
 U 
T 

  S V ,n
 U 
P  

  V  S ,n
Thermo & Stat Mech - Spring 2006
Class 13
 U 
m 

  n  S ,V
7
More Than One Component
m
dU  TdS  PdV 
m
j
dn
j
j 1
 U 
 U 
dU  
 dS  
 dV 
  S V ,n
  V  S ,n
 U 
T 

  S V ,n
 U 
P  

  V  S ,n
 U

  n
j 1 
j
m
 U
mj 
 n
j

Thermo & Stat Mech - Spring 2006
Class 13


dn

 S ,V , n
j
i



 S ,V , n
i
8
Gibbs Function
 U 
 U 

V 
U  S


  V  S ,n
  S V ,n
i
i
 U 
 n j   n 
j 1
j 

S ,V , n
m
i
m
U  ST  PV 
m
j
nj
j 1
G  U  TS  PV
m
G 
m
j
nj
j 1
Thermo & Stat Mech - Spring 2006
Class 13
9
Equilibrium Conditions
Consider two systems, A1 and A2, that can
interact thermally, mechanically, and
diffusively. For either system,
U  T  S  P  V  m  n
or
S 
1
T
U 
P
T
V 
m
n
T
Thermo & Stat Mech - Spring 2006
Class 13
10
Equilibrium Conditions
The change in entropy for the combined
system is given by, S0 = S1 + S2,
where S1 and S2 are given by the
expression on the previous slide. Then,
S0 
1
T1

U 1 
1
T2
P1
T1
 V1 
U 2 
P2
T2
m1
T1
 n1
V2 
Thermo & Stat Mech - Spring 2006
Class 13
m2
T2
n2
11
Equilibrium Conditions
Since the two systems are interacting only
with each other, we have,
U2 = – U1
V2 = – V1
n2 = – n1
Thermo & Stat Mech - Spring 2006
Class 13
12
Equilibrium Conditions
Then,
S0 
1
T1

U 1 
1
T2
P1
T1
U 1 
 V1 
P2
T2
m1
T1
 V1 
 n1
m2
T2
 n1
 1
 P1
 m1
1 
P2 
m2 
U 1  
  V1  
  n1
 S 0  








T
T
T
T
T
T
2 
2 
2 
 1
 1
 1
Thermo & Stat Mech - Spring 2006
Class 13
13
Equilibrium Conditions
When the two systems come to equilibrium, S0
will be a maximum. That means that S0 will
be zero for any small variations of U1, V1, or
n1. That is possible only if the coefficients of
U1, V1, and n1 are all zero.
Thermo & Stat Mech - Spring 2006
Class 13
14
Equilibrium Conditions
 1
 P1
 m1
1 
P2 
m2





 T  T   U 1   T  T   V1   T  T
2 
2 
2
 1
 1
 1
1

1
T1
T2
P1
P2

T1
T2
m1
m2
T1

T2

  n1  0


 0 , so T1  T 2
 0 , so P1  P2
 0 , so m 1  m 2
Thermo & Stat Mech - Spring 2006
Class 13
15
Approach to Equilibrium
To examine the approach to equilibrium, we
shall replace U1 by Q1. To do so, use
U1 = Q1 – P1V1 + m1n1. Then,
 1
 P1 P2 
 m1 m 2 
1 
U 1   
  V1  
  n1
 S 0   


T

T

T
T
T
T
2 
2 
2 
 1
 1
 1
 1
1 
( P1  P2 )
( m1  m 2 )
 Q1 
  
 V1 
 n1

T2
T2
 T1 T 2 
Thermo & Stat Mech - Spring 2006
Class 13
16
Approach to Equilibrium
S0 > 0, so each term must be positive.
 1
1 
( P1  P2 )
( m1  m 2 )
 Q1 
 S 0   
 V1 
 n1

T2
T2
 T1 T 2 
If T1 > T2 , Q1 < 0
If P1 > P2 , V1 > 0
If m1 > m2 , n1 < 0
Thermo & Stat Mech - Spring 2006
Class 13
17
Approach to Equilibrium
When two systems interact,
• Heat flows from the hotter to the cooler.
• The system at higher pressure expands at
the expense of the other.
• Particles flow from the system of higher
chemical potential to the other system.
Thermo & Stat Mech - Spring 2006
Class 13
18