Molecular Statistics
※
Degrees of freedom:
Degrees of freedom: the total number of independent variables
whose values have to be specified for a complete description
of the system.
(A) One atom: In the context of molecular motion these are the
spatial coordinates of all the particles.
Since we need
three coordinates to describe the position of an atom, we
say the atom has three degrees of freedom.
(B) Diatomic molecule: If the atoms are not bound to one
another, there will be no relation among the coordinates of
the two atoms. On the other hand, when the two atoms
are bound, the displacement of each other is coupled to the
other. The result is to give three translational, one
vibrational, and two rotational degrees of freedom for the
1
molecule.
The vibrational and rotational degrees of
freedom are also referred to as internal degrees of freedom.
(C) Polyatomic molecule with N atoms:
(i) Linear molecule: The 3N degrees of freedom of the
atoms become three translational degrees of freedom
and (3N-3) internal degrees of freedom.
By analogy
with diatomic molecules, we expect two rotational
degrees of freedom for any linear polyatomic molecule
(e.g. CO2 and C2H2).
The remaining (3N-5) internal
coordinates must correspond to vibrations.
(ii) Bent molecule: A bent molecule loses a vibrational
degree of freedom while gaining a rotational degree of
freedom.
Therefore, a nonlinear polyatomic molecule
has three rotational degrees of freedom, hence (3N-6)
vibrational degrees of freedom.
2
e.g. vibrational motions for CO2 and H2O:
Degrees of
freedom
Atom
Linear
Bent
polyatomic
polyatomic
Translational
3
3
3
Vibrational
0
3N-5
3N-6
Rotational
0
2
3
3
※Molecular energies:
(A) Translational energies:
n 2h 2
En 
8m
for one axis.
Where n = 1, 2, 3…., is the translational quantum number,
m is the mass of the particle, and l is the distance between
the walls.
Since a particle has three translational degrees of freedom,
the total translational energy is given by
2
h 2  nx2 n y nz2 
   
En 
8m  2x 2y 2z 
Where lx, ly, and lz denote the dimensions of the box.
(B) Vibrational energies:
For a diatomic molecule, the vibrational energy is given

by: E  h   1 2
V

Where h called Planck’s constant, is 6.626  10 J  sec
34
Andνis the frequency of oscillation.
The symbolυ
denotes the vibrational quantum number which can have
only integer values, 0, 1, 2, ….
(C) Rotational energies:
Rotational energies of diatomic molecules are represented
quite accurately by the quantum formula: E 
J
4
h
2
8 I
2
J  J  1
where J, which is restricted to values of 0, 1, 2, .. is called
the rotational quantum number.
The moment of inert I is
related to the bond length r of the diatomic molecule by
I = mrr2.
The mr is called the reduced mass, which is
obtained from the mass of the two atoms:
m 
r
mm
m m
1
2
1
2
※ The origin of quantum mechanics:
The origin of atomic structure:
Three great discoveries:
 J. J. Thomson’s discovery:
 
To measure e m value using magnet to deflect e- beam in
1897.
 R. A. Millikan’s discovery:
To determine electron charge using oil drop experiment in
1909.
 E. Rutherford’s discovery:
To define atomic structure usingαparticle to strike a thin
foil of gold in 1911.
Three important discoveries lead to following summary:
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 Each atom consists of a very small nucleus composed of
protons and neutrons, which is encircled by moving
electron.
 Both electrons and protons are electrically charged, the
charge magnitude being 1 . 6  10 C , which is negative in
19
sign for electrons and positive for protons; neutrons are
electrically neutral.
 The masses for proton and neutrons have approximately
the same value, 1 . 67  10 kg .
2 7
The mass of electron is
9 . 11  10 31 kg .
 Each chemical element is characterized by the number of
protons in the nucleus, or the atomic number (Z).
For an
electrically neutral, the atomic number is equal to the
number of electrons.
 The atomic mass (M) of a specific atom may be expressed
as the sum of the masses of protons and neutrons within
the nucleus, i.e. M=Z+N.
The quantization of energy:
Two important phenomena:
 Black-body radiation:
6
 A black-body radiator consists of a block of material
having an internal cavity connected to the outside
surface of the block by a small hole.
 The radiation emerging from the hole when the block
is raised to a uniform temperature is called black-body
radiation.
 The spectra radiation depends on wavelength.
 Planck proposed the concept of energy quantized in
1900.
 Photon with energy in h .
7
 Photoelectric effect:
 The electrons come out when a light is shining onto a
metal.
 The frequency of light controls whether or not
electrons are emitted, but not light intensity.
 The number of electrons emitted is proportional to the
intensity.
 Photon with energy in h
and the kinetic energy of the
electron emitted is determined by h  1 mv  w .
2
2
 Confirm Planck’s concept of energy quantized.
※Boltzmann distribution:
 The populations of energy levels:
Since any molecule can only posses certain energies at a
given temperature, each of molecules can distribute over all
the available energy levels.
Although we cannot keep track
of the energy state of a single molecule, we can speak of the
average numbers of molecules in each state, and these
average numbers are constant in time so long as the
temperature remains the same. The average number of
molecules in a state is called the population of the state.
8
Only the lowest energy level is ocupied at T = 0 K.
Raising the temperature excites some molecules into higher
energy levels, and more and more levels become accessible
as the temperature is raised further.
Nevertheless, whatever
the temperature, there is always a higher population in a state
of low energy than one of high energy.
 Boltzmann distribution:
The population in state n with energy En is determined by
the temperature T according to the Boltzmann distribution:
N Ne
n

En
k BT
0
This formula was derived by Ludwig Boltzmann towards the
end of the 19th century.
According to this formula, the
population ratio of the numbers of particles in various states
with energies Ei and Ej is determined by
Ni
Nj
e

 Ei  E j 
k BT
The fundamental constant kB is called Boltzmann’s constant,
.k  1.38  10
B
 23
J
K
Boltzmann’s constant is replaced by
the gas constant, R=NAkB, if the energies in the Boltzmann
distribution are replaced by molar energies.
9
The possibility fi of molecules occupying the i-th state is
determined to be

f 
i
Ei
k BT
N
N
e


N  N e
i
i

i
Ei
k BT
In consideration of degeneracy, the equation is rewrriten to
be:

f 
i
Ei
k BT
N
N
ge


N N g e
i
i
i
i

Ei
k BT
i
The sum in the denominator is called the partition
function, which means as “sum over states”.
The Boltzmann distribution law is written as
g
 e
N
g
Ni
i
j
j

 Ei  E j 
k BT
If fi are the fractions of molecules in different quantum states,
the average energy per particle <E> is given by
E fE
i
i
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