Transcript Presentation453.18

Lecture 18 – Quantization of energy
Ch 9
pages 442-444
Summary of lecture 17
 We have found that the spectral distribution of radiation emitted
by a heated black body, modeled as a large number of atomic
oscillators is predicted to be
I ( , T ) 
8
4
 E 
8
4
kT
based on classical mechanical considerations. This result would
predict that short wavelength radiation should be emitted with
high intensity, contradicting experimental observations that
short wavelength radiation is emitted with low intensities (bodies
do not glow at low temperature).
Quantization of Energy
In 1901, Max Planck published a quantum theory of
radiation to explain the known spectral distribution of black
body radiators. Unlike Raleigh, he only allowed the
oscillators to adopt certain energy values, not all. Planck’s
quantum hypothesis can be constructed as follows.
He assumed that the black body is composed of a large
number of oscillators whose energies obey the harmonic
oscillator equation:
p x2 x 2
E

2m
2
Quantization of Energy
p x2 x 2
E

2m
2
The frequency of the harmonic oscillation is given in terms of
the constants m (mass) and  (spring force constant) as:
2 

m
Rearrange the equation for the harmonic oscillator as follows
p x2 x 2
p x2
p x2 x 2
x 2
E



 1 2  2  1
2m
2
2mE 2 E
a
b
2E
a  2mE and b 

Quantization of Energy
E
p
x 2

2m
2
2
x
p x2 x 2
p x2
p x2 x 2
x 2
E



 1 2  2  1
2m
2
2mE 2 E
a
b
2E
a  2mE and b 

The expression:
p x2 x 2
 2 1
2
a
b
Represents an ellipse with semiaxes a and b. Therefore, the
trajectory of a harmonic oscillator
can be represented in momentumcoordinate space as an ellipse:
Quantization of Energy
Classically, when an oscillating atom emits radiation, its
trajectory is modified as the momentum and the amplitude of
displacement change. The energy emitted by an oscillator has
no restricted values. But Planck assumed that in the black
body, oscillator trajectories are restricted in such a way that
only certain trajectories are possible.
Quantization of Energy
Stating that only certain trajectories are allowed means that
only ellipses with certain values of a and b may exist. The
area of an ellipse is ab and we can express this quantum
restriction on the motion and energy of an oscillator:
nh  ab   2mE 
1/ 2
2E /  1 / 2 
2E
k /m

E

Where  is the oscillator frequency, h is a constant (Planck’s
constant), and n is an integer. It follows that, under Plank’s
quantum hypothesis, the energy of an oscillator is restricted
by the quantization rule:
E=nh where n=0,1,2,3…
Quantization of Energy
E=nh where n=0,1,2,3…
We shall see next week that the correct expression for the
quantized energy levels of a harmonic oscillator is actually
E  ( n  1 / 2)hv
Quantization of Energy
E  ( n  1 / 2)hv
The second crucial hypothesis introduced by Planck was that,
if an oscillator emits energy, it must pass from E=(n+1)h to
say E=nh
The quantum hypothesis restricts energy changes to DE=h.
This means that energy is emitted into the cavity of the black
body in discrete amounts or quanta. These energy particles
are called photons and this hypothesis is crucial to explain
spectroscopy, as we shall see later.
Quantization of Energy
E  ( n  1 / 2)hv
To formalize Plank’s equations, we can write his hypothesis to
explain the black body phenomenon as follows. The intensity
of radiation is still governed by the equation:
I ( , T ) 
8

4
E
And we can still calculate the energy using the relationship
E  kT 2
 ln q
T
Quantization of Energy
E  ( n  1 / 2)hv
I ( , T ) 
8

4
E
E  kT 2
But q now has the ‘quantized’ form:


n0
n0
q   e  E ( n )/ kT   e  nh / kT
Therefore:
 
  nhe  nh / kT
 ln q
1 q  n  0
 E  kT 2
 kT 2

T
q T    nh / kT
 e
 n 0






 ln q
T
Quantization of Energy
I ( , T ) 
8
4
 
  nhe  nh / kT
 ln q
1 q  n  0
 E  kT 2
 kT 2

T
q T    nh / kT
 e
 n 0
E
By expanding in terms of
x  e  h / kT
1  2 x  3x 2  4 x 3
 E  hx
1  x  x2  x3
If we introduce the general expression:
1  x n  1  nx  n(n  1) x 2  .... (n  1  r )! x r
2!
We find:
(n  1)! r!
(1  x ) 2 hx
h
 E  hx


(1  x ) 1 1  x e h / kT  1






Quantization of Energy
I ( , T ) 
8
4
(1  x ) 2 hx
h
 E  hx


(1  x ) 1 1  x e h / kT  1
E
Planck’s Quantum Theory of Black Body Radiation is
summarized by the following expression for the light emitted
as a function of temperature and frequency:
I ( , T ) 
8

4
 E 
8
 e
4
h
h / kT
1

8hc

5
e
hc / kT
 1
Where c is the speed of light so that   c
By fitting the equation for I(,T) to experimental data, Planck
determined that the constant h=6.626x10-34 J-sec. The constant h
is now called Planck’s constant. At high temperatures (kT>>h,
Planck’s Radiation Law converges to the classical Jean’s Law.
Quantization of Energy
I ( , T ) 
8

4
 E 
8
 e
4
h
h / kT
1

8hc

5
e
hc / kT
 1
By fitting the equation for I(,T) to experimental data, Planck
determined that the constant h=6.626x10-34 J-sec. The constant h
is now called Planck’s constant. At high temperatures (kT>>h,
Planck’s Radiation Law converges to the classical Jean’s Law.
Heat Capacities Revisited
Heat capacities of diatomic gas molecules and crystalline solids
are predicted to be CV=7R/2 and CV=3R, respectively, at room
temperature
These predictions are based on the assumption that vibrational
motions contribute a factor of RT (per dimension) to the
energy in accordance with the equipartition principle
However, CV is closer to 5R/2 per mole of gas per diatomic
molecules and CV is almost zero at room temperature for many
solids.
Molecular Partition Function of a Diatomic Molecule
From the discussion of the last class, the classical energy of a
diatomic molecule is:
E  Etrans  Erotate  Evibrate
This expression can be used to calculate the molecular
partition function. First remember once again that, in Lecture
2, we mentioned the following fundamental property of the
partition function
To a high degree of approximation, the energy of a molecule in
a particular state is the simple sums of various types of energy
(translational, rotational, vibrational, electronic, etc.).
Molecular Partition Function of a Diatomic Molecule
E  Etrans  Erotate  Evibrate
If
then
q
 e
 Etr / kT
 e
 Erot / kT
 e
 Evib / kT
...  q
tr
qrot qvib ...
Using this fact we can rearrange the form for the molecular
partition function:
 p x2  p 2y  p z2 
 p2  p2
 dp dp exp 
q  V  dp x  dp y  dp z exp 



 2lkT
2 kT




 p R2 /   R 2
 d Re xp


2kT


We have partitioned q according to:
q  V  qtrans  qrotate  qvibrate



Molecular Partition Function of a Diatomic Molecule
 p x2  p 2y  p z2 
 p2  p2
 dp dp exp 
q  V  dp x  dp y  dp z exp 



 2lkT
2 kT




 p R2 /   R 2
 d Re xp


2kT


We can calculate the energy from the relationship
 E 
 ln Vxqtrans xqrot xqvib 
E
 ln q
 kT 2
 kT 2
N
T
T
Notice that the translational, rotational, and vibrational
partition functions all involve integrals of the form:

 ax
e
 dx 
2
0
Therefore:
1 
2 a
q trans  kT 3 / 2 ; q rot  kT ; q vib  kT



Molecular Partition Function of a Diatomic Molecule
 p x2  p 2y  p z2 
 p2  p2
 dp dp exp 
q  V  dp x  dp y  dp z exp 



 2lkT
2

kT



2

 2
 d Re xp p R /   R


2kT


q trans  kT 3 / 2 ; q rot  kT ; q vib  kT
From which it immediately follows that:
7/2
E
7
2  ln q
2  ln Vxqtrans xqrot xqvib 
2  ln T
 E   kT
 kT
 kT
 kT
N
T
T
T
2
The energy per mole E and the heat capacity CV are then:
E
7
RT
2
7R
 E 
CV  
 
2
 T V



Molecular Partition Function of a Diatomic Molecule
E
7
RT
2
7R
 E 
CV  
 
2
 T V
or approximately 29 J/K-mole for a diatomic gas. This
expression reflects the equipartition principle, each degree of
freedom contributes 1/2R to the heat capacity or 1/2RT to the
total energy of a system (per mole).
Molecular Partition Function of a Diatomic Molecule
However, almost no diatomic gas obeys this expression. For
example, for H2, CV is approximately 20J/K-mole or
approximately 5/2R
An even more serious situation arises when we attempt to
calculate the heat capacity of solids. If we regard the solid as a
three-dimensional array of atoms, the motions executed by
these atoms are vibrations
Therefore, the motions of the atoms may be regarded as
harmonic oscillations in three dimensions.
Molecular Partition Function of a Diatomic Molecule
From the equipartition principle, we would expect the
vibrational energy to be E=3RT and the contribution to the
heat capacity from vibrational motions should be CV=3R
In fact, at room temperature the vibrational heat capacity for
crystalline solids is almost 0 and only approaches 3R at high
temperatures
These observations indicate serious failures of classical
mechanics to accurately account for the behavior of
polyatomic gases and solids
These failures contributed to the birth of quantum mechanics.
Molecular Partition Function of a Diatomic Molecule
Plank’s hypothesis can also be used to reexamine the heat
capacities and their deviation from classical behavior as well
Let us focus on diatomic gases by defining the average energy
as:
E  E trans  E rotate  E vibrate
If we assume translational and rotational motions obey the
equipartition principle, but that the vibrational motions obey
quantum mechanical behavior, then we can write:
E  E
trans
 E
rotate
 E
vibrate

5kT
 E
2
vibrate
Molecular Partition Function of a Diatomic Molecule
E  E
trans
 E
rotate
 E
vibrate

5kT
 E
2
vibrate
Using Planck’s quantization hypothesis for harmonic
oscillations and applying it to bond vibrations (homework), we
can calculate the partition function to be:
q   e  En / kT   e  nhv / kT (1  e  hv / kT ) 1
n
n
To be correct, as we shall see next week, the energy levels for a
quantum mechanical 1-dimensional oscillator with
characteristic frequency v are given by:
1
E n  ( n  )hv
2
e  hv / 2 kT
q
1  e hv / kT 
Molecular Partition Function of a Diatomic Molecule
E  E
trans
 E
rotate
 E
vibrate

5kT
 E
2
vibrate
If we limit ourselves to Planck’s description at this stage, then
the partition function provided in the homework allows you to
calculate properties such as the vibrational energy and specific
heat:
E 

 E 
CV    
 T V T
5kT
 E
2
vibrate

5kT
h
 h / kT
2
e
1
2
Nh  5Nk
 5NkT
 h  h / kT h / kT





Nk
e
e

1




2
e h / kT  1 
 2
 kT 
2
Molecular Partition Function of a Diatomic Molecule

 E 
CV    
 T V T
2
Nh  5Nk
 5NkT
 h  h / kT h / kT





Nk
e
e

1




2
e h / kT  1 
 2
 kT 
2
In the low temperature limit (h>>kT):
2
5Nk
5Nk
5Nk
 h  h / kT h / kT
 h 

CV 
 Nk 
e
 1 
 Nke h / kT 
 e
 
2
2
2
 kT 
 kT 
2
2
In the high temperature limit (h<<kT, using
e hv / lT  1  hv / kT
2
5Nk
5Nk
h 
h
 h  h / kT h / kT
 h  


CV 
 Nk 
e
 1 
 Nk 
 1
 e
 1 
 1 
2
2
kT 
kT
 kT 
 kT  

2
2
2

5Nk
7 Nk
 Nk 
2
2
Molecular Partition Function of a Diatomic Molecule

 E 
CV    
 T V T
2
Nh  5Nk
 5NkT
 h  h / kT h / kT





Nk
e
e

1




2
e h / kT  1 
 2
 kT 
2
The high temperature value of CV for diatomic molecules
agrees with the equipartition principle, which is the result
obtained using classical statistical mechanics. In this limit:
kT>>h (notice that h is the separation between the
vibrational energy levels). Classical statistical mechanics
correctly predicts the vibrational heat capacity if the
separation between the vibrational energy levels (i.e. energy
quantization) is negligible compared to kT. But at low
temperature, where quantization of energy levels is important,
classical statistical mechanics fails and quantum effects
become significant.
Molecular Partition Function of a Diatomic Molecule

 E 
CV    
 T V T
2
Nh  5Nk
 5NkT
 h  h / kT h / kT





Nk
e
e

1




2
e h / kT  1 
 2
 kT 
2
Later, when we develop a theory of quantum wave mechanics,
we will show why quantization for vibrational motions is much
more important at low temperatures than for translational and
rotational motions
Before we do that we will consider another problem that
classical physics fails to explain: the stability of the hydrogen
atom.