PHYSICS MAXWELL BOLTZMANN STATISTICS SCIENCE.ppt

This is a ppt for both engineering and BSc.,PG physics student.

1. Chapter 3
Classical Statistics of
Maxwell-Boltzmann

2. 1-Boltzmann Statistics
-Goal:
Find the occupation number of each energy level (i.e. find
(N
1,N
2,…,N
n)) when the thermodynamic probability is a maximum.
-Constraints:)1(
1



i
iNN )2(
1



i
iiNU 
Nand Uare fixed
-Consider the first energy level, i=1. The number of ways of selecting
N
1particles from a total of N to be placed in the first level is)!(!
!
111 NNN
N
N
N











3. 1-Boltzmann Statistics
We ask:
In how many ways can these N
1particles be arranged in the first
level such that in this level there are g
1quantum states?
For each particle there are g
1choices. That is, there are
possibilities in all.
Thus the number of ways to put N
1particles into a level containing
g
1quantum states is
1
1
N
g  !!
!
11
1
1
NNN
gN
N


4. 1-Boltzmann Statistics
For the second energy level, the situation is the same, except that
there are only (N-N
1) particles remaining to deal with: 
 !!
!
212
21
2
NNNN
gNN
N


Continuing the process, we obtain the Boltzmann distribution ω
Bas: 
 
 
 
 









3213
321
212
21
11
1
21
!
!

!!
!
!!
!
)...,(
3
21
NNNNN
gNNN
NNNN
gNN
NNN
gN
NNN
N
NN
nB

5. 1-Boltzmann Statistics


!!!
!),(
321
321
21
321
NNN
ggg
NNNN
NNN
nB
 )3(
!
!),(
1
21 










n
i i
N
i
nB
N
g
NNNN
i


6. 2-The Boltzmann Distribution
Now our task is to maximize ω
Bof Equation (3)
At maximum, . Hence we can
choose to maximize ln(ω
B )instead of ω
Bitself, this turns the
products into sums in Equation (3). 0)(ln0 
B
B
BB
d
dd



Since the logarithm is a monotonic function of its argument, the
maxima of ω
Band ln(ω
B)occur at the same point.
From Equation (3), we have: 


1
)!ln()ln()!ln()ln(
i
iiiB NgNN

7. 2-The Boltzmann Distribution 


1
)!ln()ln()!ln()ln(
i
iiiB NgNN iiii NNNN  )ln()!ln(
Applying Stirling’s law: 


1
)ln()ln()!ln()ln(
i
iiiiiB NNNgNN )ln()ln(1
1
)ln()ln(
)ln(
ii
i
iii
i
B
Ng
N
NNg
N




8. 2-The Boltzmann Distribution
Now, we introduce the constraints
Introducing Lagrange multipliers (see chapter 2 in Classical
Mechanics (2)):0
)ln(
21









iii
B
NNN




 0)ln()ln( 
iii Ng  i
ii
ii
ii
i
N
NU
N
NN

















2
1
2
1
1
1
0
10

9. 2-The Boltzmann Distributioni
i
i
i
i
i
g
N
N
g
 

















lnln ii
eegNe
g
N
ii
i
i  
 ??

e 




11 i
i
i
i
i
eegNN
 






1
1
i
ii
i
i
i
eg
N
eegeN

 )4(
1





i
i
i
i
i
i
eg
eNg
N


We will prove later thatTk
B
1


10. 2-The Boltzmann Distribution(5)on)distributi (Boltzmann




i
ii
i
i
i
i
eg
Ne
g
N
f


and hence
The sum in the denominator is called the partition function for a
single particle(N=1) or sum-over-states, and is represented by the
symbol Z
sp:)6(
1




n
i
isp
i
egZ

where f
iis the probability of occupation of a single state belonging
to the ith energy level.

11. 2-The Boltzmann Distribution
or)7(
1





s
sp
s
eZ

Ω= total number of microstates of the system,
s = the index of the state (microstate) that the system can occupy,
ε
s= the total energy of the system when it is in microstate s.
Example:
A system possesses two identical and distinguishable particles (N=2),
and three energy levels (ε
1=0, ε
2=ε, and ε
3=2ε) with g
1=2, and
g
2=g
3=1. Calculate Z
spby using Eqs.(6) and (7)

12. 2-The Boltzmann Distribution
MacrostateNb. microstates
(N
1
,N
2
,N
3
) ω
B
ε
s
(1,0,0) 2 0
(0,1,0) 1 ε
(0,0,1) 1 2ε


2
)2()0(
321
3
1
2
2
321








ee
eee
egegegegZ
i
isp
i 

2
)1,0,0(
)2(
)0,1,0(
)(
)0,0,1(
)0(
)0,0,1(
)0(
4
1
2






ee
eeeeeZ
s
sp
s


13. 2-The Boltzmann Distribution
If the energy levels are crowded together very closely, as they are
for a gaseous system:
where
g(ε)dε: number of states in the energy range from εto ε+dε,
N(ε)dε: number of particles in the range εto ε+dε.
We then obtain the continuous distribution function:

dNN
dgg
byreplaced
i
byreplaced
i
)(
)(
 
  








deg
Ne
g
N
f
)()(
)(
)(

14. 3-Dilute gases and the Maxwell-
Boltzmann Distribution
The word “dilute” means N
i<< g
i, for all i.
This condition holds for real gases except at very low temperatures.
The Maxwell-Boltzmann statistics can be written in this case as:










n
i i
N
i
MB
N
g
i
1!


15. 3-Dilute gases and the Maxwell-
Boltzmann Distribution
The Maxwell-Boltzmann distribution corresponding to ω
MB,max:










n
i i
N
i
MB
N
g
i
1!
 










n
i i
N
i
B
N
g
N
i
1!
!
-ω
MBand ω
Bdiffer only by a constant–the factor N!,
-Since maximizing ωinvolves taking derivatives and the derivative
of a constant is zero, so we get precisely the Boltzmann distribution: on)distributiBoltzmann -(Maxwell
spi
i
i
Z
Ne
g
N
f
i



16. 3-Dilute gases and the Maxwell-
Boltzmann Distribution
-Boltzmann statisticsassumes distinguishable (localizable) particles
and therefore has limited application, largely solids and some liquids.
-Maxwell-Boltzmann statisticsis a very useful approximation for
the special case of a dilute gas, which is a good model for a real
gas under most conditions.

17. 4-Thermodynamic Properties from the
Partition Function
In this section, we will state the relationships between the partition
function and the various thermodynamic parameters of the system. 













1i
i
sp
i
eg
Z

 





1i
ii
sp
i
eg
Z


 NT
i
i
i
NT
i
i
NT
sp
V
egeg
VV
Z
i
i
,1
,
1
,



































 






 NT
i
i
i
NT
sp
V
eg
V
Z
i
,1
,


















 






18. 4-Thermodynamic Properties from the
Partition Function





11
1
,
1
i
ii
spsp
ii
i
ii
i
i
eg
Z
u
Z
e
NgNandN
N
u



-Calculation of average energy per particle:






 

 sp
spi
i
sp
Z
Z
eg
Z
u
i
11
1  )ln(
sp
Zu


 NuU
-Calculation of internal energy of the system: )ln(
spZNU




19. 4-Thermodynamic Properties from the
Partition Function
-Calculation of Entropy for Maxwell-Boltzmann statistics: i
i
e
N
Z
N
g
Z
e
NgN
sp
i
i
sp
ii



 


1
max
!
i i
N
i
N
g
i
  


1
)!ln()ln()ln(
i
iii NgN  


1
)ln()ln()ln(
i
iiiii NNNgN 











11
ln)ln(
i
i
i i
i
i N
N
g
N 











1
ln)ln(
i
sp
i Ne
N
Z
N
i
with

20. 4-Thermodynamic Properties from the
Partition FunctionNN
N
Z
N
i
ii
sp
i 















1
)(ln)ln(  NNN
N
Z
U
i
ii
N
i
i
sp
















11
ln)ln(  NUNNZN
sp
 )ln()ln()ln( UNZN
sp
 )1)ln()(ln()ln( )ln(
BkS   UkNZNkS
BspB
 1)ln()ln(

21. 4-Thermodynamic Properties from the
Partition Function
-Calculation of βVU
S
T
pdVdUTdS 








1 UkZNkS
BspB
 )ln( V
BB
V
sp
B
V U
Ukk
U
Z
Nk
U
S






























)ln( T
k
U
Ukk
U
Z
Nk
U
S
B
V
BB
V
NU
B
V
1)ln(
/
































 Tk
B
1


22. 4-Thermodynamic Properties from the
Partition Function
-Calculation of the Helmholtz free energy: 

U
BspB UTkNZTNkUATSUA

 1)ln()ln(  1)ln()ln(  NZTNkA
spB
-Calculation of the pressure:dV
V
A
dT
T
A
pdVSdTdA
TV
















 TV
A
p 







 T
sp
B
V
Z
TNkp











)ln(

23. 5-Partition Function for a Gas



1i
Tk
isp
Bi
egZ

The definition of the partition function is
For a sample of gas in a container of macroscopic size, the energy
levels are very closely spaced.
Consequences:
-The energy levels can be regarded as a continuum.
-We can use the result for the density of states derived in Chapter 2:

 dm
h
V
dg
s
2123
3
24
)( 

24. 5-Partition Function for a Gas
γ
s= 1 since the gas is composed of molecules rather than spin 1/2
particles. Thus

 dm
h
V
dg
2123
3
24
)( 
Then





0
2123
0
3
24
)( 



dem
h
V
degZ
TkTk
sp
BB
The integral can be found in tables and isTk
Tk
de
B
BTk
B


2
0
21


 23
2
2







h
Tmk
VZ
B
sp

Partition function depends on both the
volume V and the temperature T.

25. 6-Properties of a Monatomic Ideal Gas23
2
2
),( 






h
Tmk
VTVZ
B
sp
 






2
2
ln
2
3
)ln(
2
3
)ln()ln(
h
mk
TVZ
B
sp

-Calculation of pressure:V
TNk
p
V
Z
TNkp
B
T
sp
B 











)ln( R
N
R
N
n
knRNknRTpV
A
BB
1
Since 
where N
Ais the Avogadro’s number and n the number of moles. A
B
N
R
k

26. 6-Properties of a Monatomic Ideal Gas
-Calculation of internal energy: 
 







T
T
Z
NUZNU
sp
sp
)ln(
)ln( 2
2
11
Tk
T
TkTTk
B
BB









 















TT
Z
h
km
TVZ
spB
sp
1
2
3)ln(2
ln
2
3
)ln(
2
3
)ln()ln(
2
 






T
TNkU
B
2
3
2 TNkU
B
2
3


27. 6-Properties of a Monatomic Ideal Gas
-Calculation of heat capacity at constant volume:BVB
NV
V
NkCTNkU
T
U
C
2
3
2
3
and
,









C
Vis constant and independent of temperature in an ideal gas. BAV knNC
2
3
 nRC
V
2
3

-Calculation of entropy 
T
U
NZNkS
spB  1)ln()ln( T
TNk
N
h
mk
TVNkS
B
B
B
2
3
1)ln(
2
ln
2
3
)ln(
2
3
)ln(
2














 



























3
23
)2(
ln
2
5
)ln(
2
3
ln
h
mk
NkT
N
V
NkS
B
BB
