Chapter 3 Classical Statistics of Maxwell-Boltzmann
Chapter 3
Classical Statistics of Maxwell-Boltzmann
1- Boltzmann Statistics
- Goal:
Find the occupation number of each energy level (i.e. find(N1,N2,…,Nn)) when the thermodynamic probability is a maximum.
- Constraints: )1(1
i
iNN )2(1
i
iiNU
N and U are fixed
- Consider the first energy level, i=1. The number of ways of selectingN1 particles from a total of N to be placed in the first level is
)!(!
!
111 NNN
NN
N
1- Boltzmann Statistics
We ask:
In how many ways can these N1 particles be arranged in the firstlevel such that in this level there are g1 quantum states?
For each particle there are g1 choices. That is, there are
possibilities in all.
Thus the number of ways to put N1 particles into a level containingg1 quantum states is
1
1Ng
!!
!
11
11
NNN
gN N
1- Boltzmann Statistics
For the second energy level, the situation is the same, except thatthere are only (N-N1) particles remaining to deal with:
!!
!
212
212
NNNN
gNN N
Continuing the process, we obtain the Boltzmann distribution ωB as:
3213
321
212
21
11
121
!
!
!!
!
!!
!)...,(
3
21
NNNNN
gNNN
NNNN
gNN
NNN
gNNNN
N
NN
nB
1- Boltzmann Statistics
!!!
!),(321
32121
321
NNN
gggNNNN
NNN
nB
)3(!
!),(1
21
n
i i
Ni
nB N
gNNNN
i
2- The Boltzmann Distribution
Now our task is to maximize ωB of Equation (3)
At maximum, . Hence we can
choose to maximize ln(ωB ) instead of ωB itself, this turns theproducts into sums in Equation (3).
0)(ln0 B
BBB
ddd
Since the logarithm is a monotonic function of its argument, themaxima of ωB and ln(ωB) occur at the same point.
From Equation (3), we have:
1
)!ln()ln()!ln()ln(i
iiiB NgNN
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(11
)ln()ln()ln(
iii
iiii
B NgN
NNgN
2- The Boltzmann Distribution
Now, we introduce the constraints
Introducing Lagrange multipliers (see chapter 2 in ClassicalMechanics (2)):
0)ln( 21
iii
B
NNN
0)ln()ln( iii Ng
iii
ii
iii
NNU
NNN
2
12
1
11
0
10
2- The Boltzmann Distribution
ii
ii
i
i
g
N
N
g
lnln
ii eegNeg
Nii
i
i ??e
11 i
ii
iieegNN
11
iii
i i
i
eg
NeegeN
)4(
1
ii
ii i
i
eg
eNgN
We will prove later that TkB
1
2- The Boltzmann Distribution
(5)on)distributi (Boltzmann
iii
ii i
i
eg
Ne
g
Nf
and hence
The sum in the denominator is called the partition function for asingle particle (N=1) or sum-over-states, and is represented by thesymbol Zsp:
)6(1
n
iisp
iegZ
where fi is the probability of occupation of a single state belongingto the ith energy level.
2- The Boltzmann Distribution
or )7(1
s
spseZ
Ω = total number of microstates of the system,s = the index of the state (microstate) that the system can occupy,εs = 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 (ε1=0, ε2=ε, and ε3=2ε) with g1=2, andg2=g3=1. Calculate Zsp by using Eqs.(6) and (7)
2- The Boltzmann Distribution
Macrostate Nb. microstates
(N1,N2,N3) ωB εs
(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
egegegegZi
ispi
2
)1,0,0(
)2(
)0,1,0(
)(
)0,0,1(
)0(
)0,0,1(
)0(4
1
2
ee
eeeeeZs
sps
2- The Boltzmann Distribution
If the energy levels are crowded together very closely, as they arefor a gaseous system:
whereg(ε)dε: number of states in the energy range from ε to ε+dε,N(ε)dε: number of particles in the range ε to ε+dε.
We then obtain the continuous distribution function:
dNN
dggbyreplaced
i
byreplacedi
)(
)(
deg
Ne
g
Nf
)()(
)()(
3- Dilute gases and the Maxwell-Boltzmann Distribution
The word “dilute” means Ni << gi, 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
Ni
MB N
g i
1 !
3- Dilute gases and the Maxwell-Boltzmann Distribution
The Maxwell-Boltzmann distribution corresponding to ωMB,max:
n
i i
Ni
MB N
g i
1 !
n
i i
Ni
B N
gN
i
1 !!
- ωMB and ωB differ only by a constant – the factor N!,
- Since maximizing ω involves taking derivatives and the derivativeof a constant is zero, so we get precisely the Boltzmann distribution:
on)distributiBoltzmann -(Maxwellspi
ii Z
Ne
g
Nf
i
3- Dilute gases and the Maxwell-Boltzmann Distribution
- Boltzmann statistics assumes distinguishable (localizable) particlesand therefore has limited application, largely solids and some liquids.
- Maxwell-Boltzmann statistics is a very useful approximation forthe special case of a dilute gas, which is a good model for a realgas under most conditions.
4- Thermodynamic Properties from the Partition Function
In this section, we will state the relationships between the partitionfunction and the various thermodynamic parameters of the system.
1ii
sp iegZ
1iii
sp iegZ
NT
i
ii
NTii
NT
sp
Vegeg
VV
Zii
,1,1,
NT
i
ii
NT
sp
Veg
V
Zi
,1,
4- Thermodynamic Properties from the Partition Function
11
1,
1
iii
spspii
iii
i
i
egZ
uZ
eNgNandN
Nu
- Calculation of average energy per particle:
sp
spii
sp
Z
Zeg
Zu i
11
1
)ln( spZu
NuU
- Calculation of internal energy of the system:
)ln( spZNU
4- Thermodynamic Properties from the Partition Function
- Calculation of Entropy for Maxwell-Boltzmann statistics:
i
i
eN
Z
N
g
Z
eNgN sp
i
i
spii
1
max !i i
Ni
N
g i
1
)!ln()ln()ln(i
iii NgN
1
)ln()ln()ln(i
iiiii NNNgN
11
ln)ln(i
ii i
ii N
N
gN
1
ln)ln(i
spi Ne
N
ZN i
with
4- Thermodynamic Properties from the Partition Function
NNN
ZN
iii
spi
1
)(ln)ln(
NNNN
Z
U
iii
N
ii
sp
11
ln)ln(
NUNNZN sp )ln()ln()ln(
UNZN sp )1)ln()(ln()ln( )ln( BkS
UkNZNkS BspB 1)ln()ln(
4- Thermodynamic Properties from the Partition Function
- Calculation of β
VU
S
TpdVdUTdS
1
UkZNkS BspB )ln(
VBB
V
spB
V UUkk
U
ZNk
U
S
)ln(
Tk
UUkk
U
ZNk
U
SB
VBB
V
NU
BV
1)ln(
/
TkB
1
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:
dVV
AdT
T
ApdVSdTdA
TV
TV
Ap
T
spB V
ZTNkp
)ln(
5- Partition Function for a Gas
1i
Tkisp
BiegZ
The definition of the partition function is
For a sample of gas in a container of macroscopic size, the energylevels 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:
dmh
Vdg s
21233
24)(
5- Partition Function for a Gasγs = 1 since the gas is composed of molecules rather than spin 1/2particles. Thus
dmh
Vdg 2123
3
24)(
Then
0
2123
03
24)( dem
h
VdegZ TkTk
spBB
The integral can be found in tables and is
TkTk
de BBTk B
20
21
23
2
2
h
TmkVZ B
sp
Partition function depends on both thevolume V and the temperature T.
6- Properties of a Monatomic Ideal Gas23
2
2),(
h
TmkVTVZ B
sp
2
2ln
2
3)ln(
2
3)ln()ln(
h
mkTVZ B
sp
- Calculation of pressure:
V
TNkp
V
ZTNkp B
T
spB
)ln(
RN
RN
nknRNknRTpV
ABB
1 Since
where NA is the Avogadro’s number and n the number of moles.
AB N
Rk
6- Properties of a Monatomic Ideal Gas
- Calculation of internal energy:
T
T
ZNUZNU sp
sp
)ln()ln(
22
11Tk
T
TkTTk BBB
TT
Z
h
kmTVZ spB
sp
1
2
3)ln(2ln
2
3)ln(
2
3)ln()ln(
2
TTNkU B 2
32
TNkU B2
3
6- Properties of a Monatomic Ideal Gas- Calculation of heat capacity at constant volume:
BVBNV
V NkCTNkUT
UC
2
3
2
3and
,
CV is constant and independent of temperature in an ideal gas.
BAV knNC2
3 nRCV 2
3
- Calculation of entropy T
UNZNkS spB 1)ln()ln(
T
TNkN
h
mkTVNkS
BB
B2
3
1)ln(2
ln2
3)ln(
2
3)ln( 2
3
23)2(ln
2
5)ln(
2
3ln
h
mkNkT
N
VNkS B
BB