Maxwell- Boltzman Distribution of Speeds

Maxwell- Boltzman Distribution of Speeds

1. -6 -4 -2 0 2 4 6
0.00
0.25
0.50
0.75
1.00
T

3
/
2
e
x
p
(

v
2
/
T
)
v
T = 1
T = 2
T = 4
T = 8
T = 16
Maxwell-Boltzmann Maxwell-Boltzmann velocityvelocity distribution distribution
-6 -4 -2 0 2 4 6
0.00
0.25
0.50
0.75
1.00
e
x
p
(

v
2
/
T
)
v
T = 1
T = 2
T = 4
T = 8
T = 16
  
  
3/2 2
1/2 2
exp /2
exp /2
ˆˆ ˆ( , , )
i i
x y z x y y
f v T mv kT
f v T mv kT
v v v v vi v j v k


 
 
   
 


2. Maxwell-Boltzmann Maxwell-Boltzmann speedspeed distribution distribution
  
3/2
2 2
4 exp /2 ; | |
2
m
f v v mv kT v v
kT


 
  
 
 


3. Maxwell-Boltzmann Maxwell-Boltzmann speedspeed distribution distribution
  
3/2
2 2
4 exp /2 ; ( ) ( )
2
m
f v v mv kT N v N f v
kT


 
   
 
 
1/2 1/2 1/2
2 8 3
;
m rms
kT kT kT
v v v
m m m 
     
    
     
     
v
m

4. Gas pressure and the ideal gas law Gas pressure and the ideal gas law
Kinetic theory provides a natural interpretation of the
absolute temperature of a dilute gas. Namely, the
temperature is proportional to the mean kinetic energy (

)
of the gas molecules.
•The mean kinetic energy is independent of pressure,
volume, and the molecular species, i.e. it is the same for
all molecules.
2 21 2 1 2
3 3 2 3
PV Nmv N mv N NkT 
 
    
 
 

6. The probability density function The probability density function
•The random motions of the molecules can be
characterized by a probability distribution function.
•Since the velocity directions are uniformly distributed, we
can reduce the problem to a speed distribution function
which is isotropic.

0
1f v dv


•Let f(v)dv be the fractional number of molecules in the
speed range from v to v + dv.
•A probability distribution function has to satisfy the
condition

0
v vf v dv

 
2 2
0
v v f v dv


2
rms
v v
•We can then use the distribution function to compute the
average behavior of the molecules: