Thermal Physics Lecture Note 10
Transcript of Thermal Physics Lecture Note 10
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10. Introduction to statistical mechanics
For a system consisting of large number of particles moving freely (gas system), there are two
ways to describe its behaviour.
(1) Consider individual particle motion. This is not possible for large number of particles.
(2) Consider the energy distribution of the particles.
If there areN number of particles in the system, where
n1particles each with energyE1
n2particles each with energyE2
and so on.
so that, N=
nii
and E=
niEii
(total energy)
Here E= niEii
is the kinetic energy only if there is no interaction between the particles;
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E= niEii
+ potential energy
if there is interaction between particles
potential energy = V12+ V13+....... + V23+ V24....
If this system of multi particles is left closed and isolated for a sufficiently long time,
an equilibrium state will be achieved, and the values for n1,n2 , ........ will not be changed again
The distribution of the values of ni will be expressed by a certain distribution function following
certain suitable distribution law. Three type of distribution laws are commonly used:
Maxwell-Boltzmann
Bose-Einstein
Fermi-Dirac
Studies on these distribution functions and how they vary with time is called
Statistical Mechanics
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In statistical mechanics, the behaviour of the particles in a system is described by usingdistribution function
f(r, v, t)which represents the number of particles at positionrat timethaving velocity between
vxdan vx+dvx, vydan vy+dvy , vzdan vz+dvz
r : (x , y , z ) is referred to as spatial space
v: ( vx, vy, vz) is referred to as velocity space
(r , v ) is referred to as phase space
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The number of particles at positionr at timet: n(r,t)=
f(r,v,t)dv
For any property of the system Q(r,v, t), its average value is :
=
1
Nf(r,v,t)Q(r,v,t)dv
, N : total number of particles
Example : For a gas at equilibrium, the overall distribution function is
f(v) function of v
Average velocity of particles: v=1
Nvf(v)dv
rms velocity v2=
1v2f(v)dv
Average kinetic energy E=1
2mv2
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For this course, we introduce three types of distribution functions,
Maxwell-Boltzmann distribution
This is often referred to as the classical distribution function
(where the kinetic energy of the electrons are assumed to be continuous)
It is suitable for system with high temperature (room temperature and above)
All particles are identical and distinguishable
The particles are considered to be distributed at energy stateE1,E2......
Every energy state has intrinsic probability of g
(statistical weight)
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Fermi-Dirac distribution
For quantum system each energy level is splitted into sub-levels
the energy state is definied by three quantum numbers : principle, orbit dan spin
The number of splitted levels is expressed by degeneracy gifor energy levelEi
Each energy state is filled up by one particle (Exclusive Pauli Principle)
Bose-Einstein Distribution
Similar to F-D distribution, but the Exclusive principle is not obeyed
The general form of the three types of distribution function for energy level ican be written as
i
ii
Egn
exp
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For M-B distribution:ZN
=
)exp(kT
1=
= 0
where
iinN and
i
ii
kT
EgZ exp
Z = electronic partition function
For F-D distribution:kT
F
kT
1= = 1
For B-E distribution:kT
kT
1= = -1
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M-B B-E F-D
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Consider ideal gas using M-B distribution,
0
)()exp( dEEgkT
EZ
here dEEh
mVdEEg 2
1
3
3 21
)2(4)(
= from quantum physics
Therefore,
0
2
1
3
3
exp)2(4 2
1
dEkT
EE
h
mVZ
3
0
2
1
2
1exp kTdE
kT
EE
3
23
)2(
h
mkTVZ
=
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Number of particles with energy EandE + dE :
dEkT
EE
h
mV
Z
NdEEg
kT
E
Z
Ndn
exp)2(4
)(exp 212
1
3
3
ReplaceZ:
kTEE
kTN
dEdn exp2 2
1
2/3
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This can also be expressed in terms of v:
TakedE
dnmv
dv
dE
dE
dn
dv
dn=
vfkT
mvv
kT
mN
dv
dn=
=
2exp
24
22
2/3
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Most probable value of v[peak of the functionf(v)] is given by
m
kTvm
2= which is when kTmvm=
2
2
1
By using f(v), we can obtain
Average velocity
=
02
23
3exp
4)(
1dv
v
vv
vdvvvf
Nv
mm
which ismkTvv m
82=
and the root mean square (rms) velocity
21
2
1
02
24
3
2 exp4)(1
=
=
dvv
vvv
dvvfvN
v
mm
rms
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Which is,mkTvv mrms 3
23
=
dv
dn
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We can also express Maxwell-Boltzmann distribution as
the number of particles having velocity between vand v+dvper unit volume in the velocity space,
or particle density in the velocity space
=
kT
mv
kT
m
Ndvv
dnv 2exp24
22/3
2
v is referred to asMaxwell-Boltzmann velocity distribution function
v has a maximum at v= 0