Semiclassical Statistical Mechanics
4.17
4.2 Reduced Distribution Functions (D.F.)
can be reduced (5 types)
1. One particle or singlet D.F.
normalized to N
= single-particle energy (potential energy is constant-no field)
Should be normalized to N,
let Z=
General Quantum Statistical Thermodynamics: Perfect Crystals
3.1 Perfect Crystals
Crystal / One large molecule
Metals (Cu, Ni, .), diamond, .
Natoms Y 3N-6 vibrations (normal modes)
Since 3N>6, use 3N
Use Harmonic Oscillator approximation
i = normal mode
2.50
Ideal Gases: Chemical Equilibrium
2.5 Chemical Equilibrium
For example,
= stoichiometric coefficients
Consider Q for a mixture of H2, O2 and H2O, must include ALL
possible cases for number of molecules, Na, Nb, Nc, . Ni =
number of molecules of type
Ideal Gases: Translational motion
2.31
2.2. Translational Motion of Molecules
qtrans
trans (solving particle in 3-D Box)
with proper boundary conditions
a x a x a box
Y
n = translational quantum numbers = 1, 2, 3,.
Quantum number space Y
Ideal Gases: Tran
Ideal Gases: Internal Molecular Motions
2.3 Internal Molecular Energies
Start with a reasonable first approximation that can separate.
and
Rotation (Rigid rotor approximation) (linear molecules):
I = moment of inertia about the C.M.
j = rotational quantum
Ideal gases
2.1
Chapter 2
2.1 FERMI ! DIRAC, BOSE !EINSTEIN &
BOLTZMANN STATISTICS
1.
IDEAL GAS
No mechanical interaction between the particles.
Each particle described as if alone in the box (quantum
mechanically)
Y
r ! coordinate of one particle
6 for o
Metals: The electron gas
3.10
3.2 METALS: The electron gas
Simple Model
+
+
+
+
+
+
+
+
+
Free valence electrons
IDEAL ELECTRON GAS (FERMIONS)
1.
CAN BOLTZMANN STATISTICS BE USED?
Like translational case (factor of 2 due to spin)
/ temperature at which r
Semiclassical Statistical Mechanics
4.25
4.3 Real Gases
For ideal gases had,
For a real gas,
2-body +
interactions
Consider 2-body interactions as,
3-body +
interactions
.
Semiclassical Statistical Mechanics
substituting in P,
translation only,
and
If A2
Radiation: The photon gas
3.19
3.3 RADIATION OR PHOTON GAS
1.
Electromagnetic radiation
2.
Taken as non interacting Photons
3.
Photons = BOSONS
4.
Photons can be
(a) created
(b) adsorbed (annihilated)
Y
CAN HAVE ANY number of photons at given time.
Q must
The Grand Canonical Ensemble
3.32
3.4 GRAND CANONICAL ENSEMBLE (, V, T)
Particle bath + system =>
= system + particles in bath
! particles with states
System
! particles with states
Particle Bath
particles with states
System+Bath
The Grand Canonical Ensem
Semiclassical Statistical Mechanics
4.1
4.1 Semiclassical Statistical Mechanics
For a N particle system,
distribution function
For a canonical ensemble,
Normalization,
N! => indistinguishable
1 => distinguishable
Therefore,
and,
Semiclassical Statistical
Thermodynamic Properties
2.48
2.4 Computing Thermodynamic Properties:
Entropy:
where,
n = moles of molecules,
N0 = Avogadros number,
k = Boltzmanns constant
T = temperature,
R = gas constant
m = mass of molecule
h = Plancks constant
P = pressure
IA , IB ,