Thermodynamics and Statistical
Mechanics
Kinetic Theory of Gases
Thermo & Stat Mech Spring 2006 Class 14
1
Mixing of Two Ideal Gases
Change of Gibbs Function
Gi  n1 g1i  n2 g 2i
G f  n1 g1 f  n2 g 2 f
G f  Gi  G  n1 ( g1 f  g1i )  n2 ( g 2 f  g 2i )
An expression is needed for the
specific Gibbs function.
Thermo & Stat Mech - Spring 2006
Class 14
2
Specific Gibbs Function
g  u  Pv  Ts  h  Ts
dh
cP 
for an ideal gas, so
dT
h  cPT  h0
g  cPT  Ts  h0
Thermo & Stat Mech - Spring 2006
Class 14
3
Specific Entropy
Tds  du  Pdv  cv dT  Pdv
 (cP  R )dT  Pdv  cP dT  RdT  Pdv
 cP dT  Pdv  vdP  Pdv
Tds  cP dT  vdP
dT v
dT
dP
ds  cP
 dP  cP
R
T T
T
P
Thermo & Stat Mech - Spring 2006
Class 14
4
Specific Gibbs Function
dT
dP
ds  cP
R
T
P
s  cP ln T  R ln P  s0
g  cPT  Ts  h0
g  cPT  T (cP ln T  R ln P  s0 )  h0
g  RT ln P  cPT  cPT ln T  Ts0  h0
Thermo & Stat Mech - Spring 2006
Class 14
5
Specific Gibbs Function
g  RT ln P  cPT  cPT ln T  Ts0  h0
 cP  cP ln T  s0 h0 
g  RT ln P  RT 


R
RT 

g  RT (ln P   )
Thermo & Stat Mech - Spring 2006
Class 14
6
Mixing of Two Ideal Gases
g  RT (ln P   )
G  n1 ( g1 f  g1i )  n2 ( g 2 f  g 2i )
G  RT [n1 (ln P1 f  ln P1i )
 n2 (ln P2 f  ln P2i )]

 P1 f
G  RT n1 ln 
 P1i

 P2 f

  n2 ln 
 P2i

Thermo & Stat Mech - Spring 2006
Class 14



7
For the Same Pressure

 P1 f
G  RT n1 ln 
 P1i

P1i  P2i  P




 P2 f
  n2 ln 

 P2i
n1
n1
P1 f  x1 P 
P P
n1  n2
n
n2
n2
P2 f  x2 P 
P P
n1  n2
n
so
so
Thermo & Stat Mech - Spring 2006
Class 14
P1 f
P1i
P2 f
P2i
 x1
 x2
8
For the Same Pressure
G  RT n1 ln x1  n2 ln x2 
 n1 

 n2 
G  nRT   ln x1    ln x2 
n
 n 

G  nRT x1 ln x1  x2 ln x2 
  (G ) 
S  
  nRx1 ln x1  x2 ln x2 
 T  P
Thermo & Stat Mech - Spring 2006
Class 14
9
For the Same Volume

 P1 f 
 P2 f
  n2 ln 
G  RT n1 ln 
 P1i 
 P2i

T is constant, so P  V 1




 V1i
G  RT n1 ln 


 V1 f





 V2i
  n2 ln 

V

 2f

1
 1 
G  RT n1 ln    n2 ln  
2
 2 

Thermo & Stat Mech - Spring 2006
Class 14
10
For the Same Volume

1
 1 
G  RT n1 ln    n2 ln  
2
 2 

1
G  (n1  n2 ) RT ln  
2
G  (n1  n2 ) RT ln 2
S  (n1  n2 ) R ln 2
Thermo & Stat Mech - Spring 2006
Class 14
11
Basic Assumptions
1. A macroscopic volume contains a large
number of molecules.
2. The separation of molecules is large
compared to molecular dimensions.
3. No forces exist between molecules except
those associated with collisions
4. The collisions are elastic.
Thermo & Stat Mech - Spring 2006
Class 14
12
Basic Assumptions
When no external forces are applied:
5. The molecules are uniformly distributed
within a container.
6. The directions of the velocities of the
molecules are uniformly distributed.
The fraction of molecules with speeds in the
range v to v + dv is: f (v) dv
Thermo & Stat Mech - Spring 2006
Class 14
13
Molecular Speeds
f (v) is the probability density.


0
f (v)dv  1

v   vf (v)dv
0

Mean or average speed
v   v f (v)dv Mean square speed
2
2
0
vrms  v 2
Root mean square speed
Thermo & Stat Mech - Spring 2006
Class 14
14
Gas Pressure
Thermo & Stat Mech - Spring 2006
Class 14
15
Gas Pressure




F dF
dp
P 
F
p  mv
A dA
dt
p  mv cos   (mv cos  )
p  2mv cos 
dp (v,  ,  )
dN (v,  ,  )
 2mv cos 
dt
dt
dp
dN (v, ,  )
  2mv cos 
dvdd
dt
dt
Thermo & Stat Mech - Spring 2006
Class 14
16
Gas Pressure
Thermo & Stat Mech - Spring 2006
Class 14
17
Molecular Flux
N
n
V
sin dd
dN (v,  ,  )  (dA cos  )(vdt)n f (v)dv
4
dN (v,  ,  )  n dA 
[vf (v)dv] sin  cos dd
 
dt
 4 
Thermo & Stat Mech - Spring 2006
Class 14
18
Molecular Flux
Thermo & Stat Mech - Spring 2006
Class 14
19
Molecular Flux
 n
dN (v,  ,  )
 d  
dAdt
 4

n

 2
vf (v)dv 

0
4 0

[vf (v)dv] sin  cos dd

2
sin  cos d  d
0
1
1

(v ) (2 )  n v
4
4
2
n
Thermo & Stat Mech - Spring 2006
Class 14
20
Gas Pressure
dp
dN (v,  ,  )
  2mv cos 
dvdd
dt
dt
dN (v,  ,  )  n dA 
[vf (v)dv] sin  cos dd
 
dt
 4 
 2
2
dp 2m n dA  2
2

v f (v)dv  cos  sin d  d

0
0
0
dt
4
2m n 2  1 
dp
1
P
v  (2 )  n m v 2
dAdt
4
3
 3
 
Thermo & Stat Mech - Spring 2006
Class 14
21
Ideal Gas Law
1
P  n m v2
3
1
1
21
2
2
2
PV  n V m v  Nmv   Nmv 
3
3
32

N
PV  nRT
n
NA
N
PV 
RT  NkT
NA
R
k
NA
Thermo & Stat Mech - Spring 2006
Class 14
22
Molecular Kinetic Energy
21
2
PV   Nmv 
32

PV  NkT
21
2
 Nmv   NkT
32

1 2 3
mv  kT
2
2
Thermo & Stat Mech - Spring 2006
Class 14
23
Equipartition of Energy
1 2 3
mv  kT
2
2
1
3
2
2
2
m v x  v y  v z  kT
2
2
1 2 1
mvi  kT
2
2
1
kT  Energy per degree of freedom
2


Thermo & Stat Mech - Spring 2006
Class 14
24
Internal Energy
1
Degrees of Freedom
U  Nf kT
f 
2
Molecule
Monatomic gas f  3
3
3
U  NkT  nRT
2
2
Diatomic gas f  5
dU 3
CV 
 nR
dT 2
5
5
U  NkT  nRT
2
2
dU 5
CV 
 nR
dT 2
Thermo & Stat Mech - Spring 2006
Class 14
25
Heat Capacities
1
1
U  Nf kT  nf RT
2
2
dU f
CV 
 nR
dT 2
f
f

C P  CV  nR  nR  nR    1nR
2
2 
f


1

nR
C
2
2 
  P 
 1
f
CV
f
nR
2
Thermo & Stat Mech - Spring 2006
Class 14
26
Maxwell Velocity Distribution
Consider a gas at equilibrium. The number of
molecules in any range of velocity does not
change. Collisions cause individual molecules
to change velocity, but the distribution does not
change. For every collision that changes the
distribution, there must be one that changes it
back.
 



Let : F (v )dv  number between v and v  dv
Thermo & Stat Mech - Spring 2006
Class 14
27
Molecular Collisions
Thermo & Stat Mech - Spring 2006
Class 14
28
Molecular Collisions
   
v1  v2  v1 'v2 '


Probabilit y of such a collision  F (v1 ) F (v2 )


Probabilit y of reverse collision  F (v1 ' ) F (v2 ' )




F (v1 ) F (v2 )  F (v1 ' ) F (v2 ' )
Collisions are elastic so :
2 2  2  2
v1  v2  v '1 v '2


 v v
Then : F (v )  Ae
Thermo & Stat Mech - Spring 2006
Class 14
29
Maxwell Distribution
N (v)dv  Number of speeds from v to v  dv
N (v)dv  Ae
 v 2
4 v dv
2


0
0
N   N (v)dv  4 A v e
3
2

2  v 2
dv
NkT  m  v N (v)dv
1
2
2
0

3 NkT
4  v
 4 A v e dv
0
m
2
Thermo & Stat Mech - Spring 2006
Class 14
30
Evaluation of Constants

I2   v e
2  v 2
0

I4   v e
0
4  v 2

dv 
32
4
I 2 3 
dv  
 52
 8
3 
3kT I 4 8 5 2
3
 

m
I2
2

32
4
Thermo & Stat Mech - Spring 2006
Class 14
31
Maxwell Distribution
m

2kT
 
A  N 
 
32
32
 m 

 N 
 2 kT 
mv 2

2
2 kT
 m 
 v e
N (v)dv  4 N 
 2 kT 
dv
32
mv 2

2
2 kT
 m 
N (v)dv
 v e
 4 
f (v)dv 
N
 2 kT 
Thermo & Stat Mech - Spring 2006
Class 14
32
dv
32
Maxwell Distribution
Maxwell Distribution
0.0025
Probability
0.002
0.0015
0.001
0.0005
0
0
200
400
600
800
1000
1200
1400
v (m /s)
Thermo & Stat Mech - Spring 2006
Class 14
33
Some Molecular Speeds
v 

0
8kT
vf (v)dv 
m

3kT
v   v f (v)dv 
0
m
3kT
2
vrms  v 
m
2
vmax
2
2kT

m
Most Probable
Thermo & Stat Mech - Spring 2006
Class 14
34