The fundamental physics Atomic models theory
DrWalidFemtosecond
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Sep 24, 2024
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About This Presentation
Overview of Atomic Models in Physics
The theory of atomic models has evolved significantly over time, reflecting advancements in scientific understanding and experimentation. This evolution is marked by several key models, each contributing to our current comprehension of atomic structure.
Slide Content
Bohr’s Atom
electrons in orbits
nucleus
Atomic ModelsAtomic Models
Prof. Walid TawfikProf. Walid Tawfik
electrons in orbits
nucleus
Atomic ModelsAtomic Models
Prof. Walid TawfikProf. Walid Tawfik
22
INTRODUCTIONINTRODUCTION
The purpose of this chapter is to build a simplest The purpose of this chapter is to build a simplest
atomic model that will help us to understand the atomic model that will help us to understand the
structure of atomsstructure of atoms
This is attained by referring to some basic This is attained by referring to some basic
experimental facts that have been gathered since experimental facts that have been gathered since
1900’s (e.g. Rutherford scattering experiment, 1900’s (e.g. Rutherford scattering experiment,
atomic spectral lines etc.)atomic spectral lines etc.)
In order to build a model that well describes the In order to build a model that well describes the
atoms which are consistent with the experimental atoms which are consistent with the experimental
facts, we need to take into account the wave facts, we need to take into account the wave
nature of electronnature of electron
This is one of the purpose we explore the wave This is one of the purpose we explore the wave
nature of particles in previous chaptersnature of particles in previous chapters
INTRODUCTIONINTRODUCTION
The purpose of this chapter is to build a simplest The purpose of this chapter is to build a simplest
atomic model that will help us to understand the atomic model that will help us to understand the
structure of atomsstructure of atoms
This is attained by referring to some basic This is attained by referring to some basic
experimental facts that have been gathered since experimental facts that have been gathered since
1900’s (e.g. Rutherford scattering experiment, 1900’s (e.g. Rutherford scattering experiment,
atomic spectral lines etc.)atomic spectral lines etc.)
In order to build a model that well describes the In order to build a model that well describes the
atoms which are consistent with the experimental atoms which are consistent with the experimental
facts, we need to take into account the wave facts, we need to take into account the wave
nature of electronnature of electron
This is one of the purpose we explore the wave This is one of the purpose we explore the wave
nature of particles in previous chaptersnature of particles in previous chapters
33
Basic properties of atomsBasic properties of atoms
1)1)Atoms are of microscopic size, ~ 10Atoms are of microscopic size, ~ 10
-10-10
m. Visible m. Visible
light is not enough to resolve (see) the detail structure of light is not enough to resolve (see) the detail structure of
an atom as its wavelength is only of the order of 100 nm. an atom as its wavelength is only of the order of 100 nm.
2)2)Atoms are stableAtoms are stable
3)3)Atoms contain negatively charges, electrons, but Atoms contain negatively charges, electrons, but
are electrically neutral. An atom with are electrically neutral. An atom with ZZ electrons must also electrons must also
contain a net positive charge of +contain a net positive charge of +ZeZe..
4)4)Atoms emit and absorb EM radiation (in other Atoms emit and absorb EM radiation (in other
words, atoms interact with light quite readily)words, atoms interact with light quite readily)
Because atoms interacts with EM radiation quite strongly, Because atoms interacts with EM radiation quite strongly,
it is usually used to probe the structure of an atom. The it is usually used to probe the structure of an atom. The
typical of such EM probe can be found in the atomic typical of such EM probe can be found in the atomic
spectrum as we will see nowspectrum as we will see now
Basic properties of atomsBasic properties of atoms
1)1)Atoms are of microscopic size, ~ 10Atoms are of microscopic size, ~ 10
-10-10
m. Visible m. Visible
light is not enough to resolve (see) the detail structure of light is not enough to resolve (see) the detail structure of
an atom as its wavelength is only of the order of 100 nm. an atom as its wavelength is only of the order of 100 nm.
2)2)Atoms are stableAtoms are stable
3)3)Atoms contain negatively charges, electrons, but Atoms contain negatively charges, electrons, but
are electrically neutral. An atom with are electrically neutral. An atom with ZZ electrons must also electrons must also
contain a net positive charge of +contain a net positive charge of +ZeZe..
4)4)Atoms emit and absorb EM radiation (in other Atoms emit and absorb EM radiation (in other
words, atoms interact with light quite readily)words, atoms interact with light quite readily)
Because atoms interacts with EM radiation quite strongly, Because atoms interacts with EM radiation quite strongly,
it is usually used to probe the structure of an atom. The it is usually used to probe the structure of an atom. The
typical of such EM probe can be found in the atomic typical of such EM probe can be found in the atomic
spectrum as we will see nowspectrum as we will see now
HELIUM ATOM
+
N
N
+
-
-
proton
electron
neutron
Shell
What do these particles consist of?
+
N
N
+
-
-
proton
electron
neutron
Shell
What do these particles consist of?
ATOMIC STRUCTUREATOMIC STRUCTURE
the number of protons in an atom
the number of protons and
neutrons in an atom
HeHe
22
44
Atomic mass
Atomic number
number of electrons = number of protons
the number of protons in an atom
the number of protons and
neutrons in an atom
HeHe
22
44
Atomic mass
Atomic number
number of electrons = number of protons
ATOMIC STRUCTUREATOMIC STRUCTURE
Electrons are arranged in Energy Levels or
Shells around the nucleus of an atom.
• first shell a maximum of 2 electrons
• second shell a maximum of 8 electrons
• third shell a maximum of 8 electrons
Electrons are arranged in Energy Levels or
Shells around the nucleus of an atom.
• first shell a maximum of 2 electrons
• second shell a maximum of 8 electrons
• third shell a maximum of 8 electrons
ATOMIC STRUCTUREATOMIC STRUCTURE
Particle
proton
neutron
electron
Charge
+ ve charge
-ve charge
No charge
1.6726219 ×
10
-27
kilograms
1.6726219 ×
10
-27
kilograms
9.10938356 ×
10
-31
kilograms
Mass
Particle
proton
neutron
electron
Charge
+ ve charge
-ve charge
No charge
1.6726219 ×
10
-27
kilograms
1.6726219 ×
10
-27
kilograms
9.10938356 ×
10
-31
kilograms
Mass
ATOMIC STRUCTUREATOMIC STRUCTURE
There are two ways to represent the atomic
structure of an element or compound;
1.Electronic Configuration
2.Dot & Cross Diagrams
There are two ways to represent the atomic
structure of an element or compound;
1.Electronic Configuration
2.Dot & Cross Diagrams
ELECTRONIC CONFIGURATIONELECTRONIC CONFIGURATION
With electronic configuration elements are represented
numerically by the number of electrons in their shells
and number of shells. For example;
N
Nitrogen
7
14
2 in 1
st
shell
5 in 2
nd
shell
configuration = 2 , 5
2 + 5 = 7
With electronic configuration elements are represented
numerically by the number of electrons in their shells
and number of shells. For example;
N
Nitrogen
7
14
2 in 1
st
shell
5 in 2
nd
shell
configuration = 2 , 5
2 + 5 = 7
ELECTRONIC CONFIGURATIONELECTRONIC CONFIGURATION
Write the electronic configuration for the following
elements;
Ca O
Cl Si
Na
20
40
11
23
8
17
16
35
14
28
B
11
5
a) b) c)
d) e) f)
2,8,8,2 2,8,1
2,8,7 2,8,4 2,3
2,6
Write the electronic configuration for the following
elements;
Ca O
Cl Si
Na
20
40
11
23
8
17
16
35
14
28
B
11
5
a) b) c)
d) e) f)
2,8,8,2 2,8,1
2,8,7 2,8,4 2,3
2,6
DOT & CROSS DIAGRAMSDOT & CROSS DIAGRAMS
With Dot & Cross diagrams elements and compounds
are represented by Dots or Crosses to show electrons,
and circles to show the shells. For example;
Nitrogen
N XX X
X
XX
X
N
7
14
With Dot & Cross diagrams elements and compounds
are represented by Dots or Crosses to show electrons,
and circles to show the shells. For example;
Nitrogen
N XX X
X
XX
X
N
7
14
DOT & CROSS DIAGRAMSDOT & CROSS DIAGRAMS
Draw the Dot & Cross diagrams for the following
elements;
O Cl
8 17
16
35
a) b)
O
X
X
X
X
X
X
X
X
Cl
X
X
X
X X
X
X
X
X
X
X
X
X
X
X
X
X
X
Draw the Dot & Cross diagrams for the following
elements;
O Cl
8 17
16
35
a) b)
O
X
X
X
X
X
X
X
X
Cl
X
X
X
X X
X
X
X
X
X
X
X
X
X
X
X
X
X
There are 3 types of subatomic particles:
Electrons (e
–
)Protons (p
+
) andNeutrons
(n
0
).
Neutrons have no charge and a mass similar
to protons
Elements are often symbolized with their
mass number and atomic number
E.g. Oxygen:O
16
8
These values are given on the periodic table.
For now, round the mass # to a whole number.
These numbers tell you a lot about atoms.
# of protons = # of electrons = atomic number
# of neutrons = mass number – atomic number
Electrons (e
–
)Protons (p
+
) andNeutrons
(n
0
).
Neutrons have no charge and a mass similar
to protons
Elements are often symbolized with their
mass number and atomic number
E.g. Oxygen:O
16
8
These values are given on the periodic table.
For now, round the mass # to a whole number.
These numbers tell you a lot about atoms.
# of protons = # of electrons = atomic number
# of neutrons = mass number – atomic number
Atomic Information
1.The Atomic Number of an atom
= number of protons in the nucleus.
2.The Atomic Mass of an atom
= number of Protons + Neutrons in the nucleus
3.The number of Protons
= Number of Electrons.
4.Electrons orbit the nucleus in shells.
5.Each shell can only carry a set number of
electrons.
1.The Atomic Number of an atom
= number of protons in the nucleus.
2.The Atomic Mass of an atom
= number of Protons + Neutrons in the nucleus
3.The number of Protons
= Number of Electrons.
4.Electrons orbit the nucleus in shells.
5.Each shell can only carry a set number of
electrons.
Isotopes
•Atoms with the same number of protons but
different numbers of neutrons
•Another way to say – atoms of the same element
with different numbers of neutrons
•An elements mass number is the number of
protons plus the number of neutrons
•REVIEW: to determine the number of neutrons
subtract the atomic number from the mass
number
•Mass # – atomic # = # of neutrons
•Atoms with the same number of protons but
different numbers of neutrons
•Another way to say – atoms of the same element
with different numbers of neutrons
•An elements mass number is the number of
protons plus the number of neutrons
•REVIEW: to determine the number of neutrons
subtract the atomic number from the mass
number
•Mass # – atomic # = # of neutrons
Isotopes of Lithium
Isotopic symbols
Isotopic symbols
NOTE: Unlike on the periodic
table where the atomic number
is at the top of the box and the
average atomic mass is at the
bottom in isotopic symbols the
mass number is at the top and
the atomic number is at the
bottom.
If X is Hydrogen and its mass number is 2 then the isotopic symbol
would be
2
H
1
NOTE: Unlike on the periodic
table where the atomic number
is at the top of the box and the
average atomic mass is at the
bottom in isotopic symbols the
mass number is at the top and
the atomic number is at the
bottom.
If X is Hydrogen and its mass number is 2 then the isotopic symbol
would be
2
H
1
Isotopic symbols
If X is Fluorine and its mass
number is 20 then the correct
isotopic symbol is
Fluorine – 20.
If X is Fluorine and its mass
number is 20 then the correct
isotopic symbol is
Fluorine – 20.
If the atom has a charge the charge is written to
the upper right side of the symbol. If the charge
is either negative or positive 1 the 1 is not
written but understood to be 1.
the upper right side of the symbol. If the charge
is either negative or positive 1 the 1 is not
written but understood to be 1.
Determine the number of neutron in each isotope
1.238 – 92 = 146 neutrons
2. 84 – 36 =48 neutrons
3. 35 – 17 =18 neutrons
4. 14 – 6 = 8 neutrons
1.238 – 92 = 146 neutrons
2. 84 – 36 =48 neutrons
3. 35 – 17 =18 neutrons
4. 14 – 6 = 8 neutrons
ISOTOPES
e
–
MassAtomic
n
0
p
+
Review
Ca 20 40 20 20 20
Ar 18 40 18 22 18
Br 35 80 35 45 35
–
MassAtomic
n
0
p
+
Review
Ca 20 40 20 20 20
Ar 18 40 18 22 18
Br 35 80 35 45 35
SUMMARYSUMMARY
1. The Atomic Number of an atom = number of
protons in the nucleus.
2. The Atomic Mass of an atom = number of
Protons + Neutrons in the nucleus.
3. The number of Protons = Number of Electrons.
4. Electrons orbit the nucleus in shells.
5. Each shell can only carry a set number of electrons.
1. The Atomic Number of an atom = number of
protons in the nucleus.
2. The Atomic Mass of an atom = number of
Protons + Neutrons in the nucleus.
3. The number of Protons = Number of Electrons.
4. Electrons orbit the nucleus in shells.
5. Each shell can only carry a set number of electrons.
2525
Emission spectral linesEmission spectral lines
Experimental fact: Experimental fact: A single atom or molecule in a very A single atom or molecule in a very
diluted sample of gas emits radiation characteristic of the diluted sample of gas emits radiation characteristic of the
particular atom/molecule speciesparticular atom/molecule species
The emission is due to the de-excitation of the atoms The emission is due to the de-excitation of the atoms
from their excited statesfrom their excited states
e.g. if heating or passing electric current through the gas e.g. if heating or passing electric current through the gas
sample, the atoms get excited into higher energy statessample, the atoms get excited into higher energy states
When a excited electron in the atom falls back to the When a excited electron in the atom falls back to the
lower energy states (de-excites), EM wave is emittedlower energy states (de-excites), EM wave is emitted
The spectral lines are analysed with The spectral lines are analysed with spectrometerspectrometer, ,
which give important physical information of the which give important physical information of the
atom/molecules by analysing the wavelengths atom/molecules by analysing the wavelengths
composition and pattern of these lines. composition and pattern of these lines.
Emission spectral linesEmission spectral lines
Experimental fact: Experimental fact: A single atom or molecule in a very A single atom or molecule in a very
diluted sample of gas emits radiation characteristic of the diluted sample of gas emits radiation characteristic of the
particular atom/molecule speciesparticular atom/molecule species
The emission is due to the de-excitation of the atoms The emission is due to the de-excitation of the atoms
from their excited statesfrom their excited states
e.g. if heating or passing electric current through the gas e.g. if heating or passing electric current through the gas
sample, the atoms get excited into higher energy statessample, the atoms get excited into higher energy states
When a excited electron in the atom falls back to the When a excited electron in the atom falls back to the
lower energy states (de-excites), EM wave is emittedlower energy states (de-excites), EM wave is emitted
The spectral lines are analysed with The spectral lines are analysed with spectrometerspectrometer, ,
which give important physical information of the which give important physical information of the
atom/molecules by analysing the wavelengths atom/molecules by analysing the wavelengths
composition and pattern of these lines. composition and pattern of these lines.
2626
Line spectrum of an atomLine spectrum of an atom
The light given off by individual atoms, as in a The light given off by individual atoms, as in a
low-pressure gas, consist of a series of discrete low-pressure gas, consist of a series of discrete
wavelengths corresponding to different colour.wavelengths corresponding to different colour.
Line spectrum of an atomLine spectrum of an atom
The light given off by individual atoms, as in a The light given off by individual atoms, as in a
low-pressure gas, consist of a series of discrete low-pressure gas, consist of a series of discrete
wavelengths corresponding to different colour.wavelengths corresponding to different colour.
2727
Comparing continuous and line Comparing continuous and line
spectrumspectrum
(a) continuous (a) continuous
spectrum produced spectrum produced
by a glowing light-by a glowing light-
bulbbulb
(b) Emission line (b) Emission line
spectrum by lamp spectrum by lamp
containing heated gascontaining heated gas
Comparing continuous and line Comparing continuous and line
spectrumspectrum
(a) continuous (a) continuous
spectrum produced spectrum produced
by a glowing light-by a glowing light-
bulbbulb
(b) Emission line (b) Emission line
spectrum by lamp spectrum by lamp
containing heated gascontaining heated gas
2828
Absorption line spectrumAbsorption line spectrum
We also have absorption spectral line, in We also have absorption spectral line, in
which white light is passed through a gas. which white light is passed through a gas.
The absorption line spectrum consists of a The absorption line spectrum consists of a
bright background crossed by dark lines bright background crossed by dark lines
that correspond to the absorbed that correspond to the absorbed
wavelengths by the gas atom/molecules.wavelengths by the gas atom/molecules.
Absorption line spectrumAbsorption line spectrum
We also have absorption spectral line, in We also have absorption spectral line, in
which white light is passed through a gas. which white light is passed through a gas.
The absorption line spectrum consists of a The absorption line spectrum consists of a
bright background crossed by dark lines bright background crossed by dark lines
that correspond to the absorbed that correspond to the absorbed
wavelengths by the gas atom/molecules.wavelengths by the gas atom/molecules.
2929
Experimental arrangement for the Experimental arrangement for the
observation of the absorptions lines observation of the absorptions lines
of a sodium vapourof a sodium vapour
Experimental arrangement for the Experimental arrangement for the
observation of the absorptions lines observation of the absorptions lines
of a sodium vapourof a sodium vapour
3030
Comparing emission and Comparing emission and
absorption spectrumabsorption spectrum
The emitted and absorption radiation displays The emitted and absorption radiation displays
characteristic discrete sets of spectrum which characteristic discrete sets of spectrum which
contains certain discrete wavelengths onlycontains certain discrete wavelengths only
(a) shows ‘finger print’ emission spectral lines of H, (a) shows ‘finger print’ emission spectral lines of H,
Hg and Ne. (b) shows absorption lines for HHg and Ne. (b) shows absorption lines for H
Comparing emission and Comparing emission and
absorption spectrumabsorption spectrum
The emitted and absorption radiation displays The emitted and absorption radiation displays
characteristic discrete sets of spectrum which characteristic discrete sets of spectrum which
contains certain discrete wavelengths onlycontains certain discrete wavelengths only
(a) shows ‘finger print’ emission spectral lines of H, (a) shows ‘finger print’ emission spectral lines of H,
Hg and Ne. (b) shows absorption lines for HHg and Ne. (b) shows absorption lines for H
3131
A successful atomic model must be A successful atomic model must be
able to explain the observed able to explain the observed
discrete atomic spectrumdiscrete atomic spectrum
We are going to study two attempts We are going to study two attempts
to built model that describes the to built model that describes the
atoms: the Thompson Plum-atoms: the Thompson Plum-
pudding model (which fails) and the pudding model (which fails) and the
Rutherford-Bohr model (which Rutherford-Bohr model (which
succeeds)succeeds)
A successful atomic model must be A successful atomic model must be
able to explain the observed able to explain the observed
discrete atomic spectrumdiscrete atomic spectrum
We are going to study two attempts We are going to study two attempts
to built model that describes the to built model that describes the
atoms: the Thompson Plum-atoms: the Thompson Plum-
pudding model (which fails) and the pudding model (which fails) and the
Rutherford-Bohr model (which Rutherford-Bohr model (which
succeeds)succeeds)
3232
The Thompson model – Plum-The Thompson model – Plum-
pudding modelpudding model
Sir J. J. Thompson (1856-
1940) is the Cavandish
professor in Cambridge who
discovered electron in cathode
rays. He was awarded Nobel
prize in 1906 for his research
on the conduction of electricity
by bases at low pressure.
He is the first person to He is the first person to
establish the particle nature of establish the particle nature of
electron. Ironically his son, electron. Ironically his son,
another renown physicist another renown physicist
proves experimentally electron proves experimentally electron
behaves like wave…behaves like wave…
The Thompson model – Plum-The Thompson model – Plum-
pudding modelpudding model
Sir J. J. Thompson (1856-
1940) is the Cavandish
professor in Cambridge who
discovered electron in cathode
rays. He was awarded Nobel
prize in 1906 for his research
on the conduction of electricity
by bases at low pressure.
He is the first person to He is the first person to
establish the particle nature of establish the particle nature of
electron. Ironically his son, electron. Ironically his son,
another renown physicist another renown physicist
proves experimentally electron proves experimentally electron
behaves like wave…behaves like wave…
3333
Plum-pudding modelPlum-pudding model
An atom consists of An atom consists of ZZ electrons is embedded in electrons is embedded in
a cloud of positive charges that exactly a cloud of positive charges that exactly
neutralise that of the electrons’neutralise that of the electrons’
The positive cloud is heavy and comprising most The positive cloud is heavy and comprising most
of the atom’s massof the atom’s mass
Inside a stable atom, the electrons sit at their Inside a stable atom, the electrons sit at their
respective equilibrium position where the respective equilibrium position where the
attraction of the positive cloud on the electrons attraction of the positive cloud on the electrons
balances the electron’s mutual repulsionbalances the electron’s mutual repulsion
Plum-pudding modelPlum-pudding model
An atom consists of An atom consists of ZZ electrons is embedded in electrons is embedded in
a cloud of positive charges that exactly a cloud of positive charges that exactly
neutralise that of the electrons’neutralise that of the electrons’
The positive cloud is heavy and comprising most The positive cloud is heavy and comprising most
of the atom’s massof the atom’s mass
Inside a stable atom, the electrons sit at their Inside a stable atom, the electrons sit at their
respective equilibrium position where the respective equilibrium position where the
attraction of the positive cloud on the electrons attraction of the positive cloud on the electrons
balances the electron’s mutual repulsionbalances the electron’s mutual repulsion
3434
One can treat the One can treat the
electron in the electron in the
pudding like a point pudding like a point
mass stressed by mass stressed by
two springstwo springs
SHM
One can treat the One can treat the
electron in the electron in the
pudding like a point pudding like a point
mass stressed by mass stressed by
two springstwo springs
SHM
3535
The “electron plum” stuck on the The “electron plum” stuck on the
pudding vibrates and executes pudding vibrates and executes
SHMSHM
The electron at the EQ position shall vibrate like a The electron at the EQ position shall vibrate like a
simple harmonic oscillator with a frequency simple harmonic oscillator with a frequency
Where , Where , R R radius of the atom, radius of the atom, mm mass of mass of
the electronthe electron
From classical EM theory, we know that an From classical EM theory, we know that an
oscillating charge will emit radiation with frequency oscillating charge will emit radiation with frequency
identical to the oscillation frequency identical to the oscillation frequency as given as given
aboveabove
m
k
2
1
3
2
4R
Ze
k
o
The “electron plum” stuck on the The “electron plum” stuck on the
pudding vibrates and executes pudding vibrates and executes
SHMSHM
The electron at the EQ position shall vibrate like a The electron at the EQ position shall vibrate like a
simple harmonic oscillator with a frequency simple harmonic oscillator with a frequency
Where , Where , R R radius of the atom, radius of the atom, mm mass of mass of
the electronthe electron
From classical EM theory, we know that an From classical EM theory, we know that an
oscillating charge will emit radiation with frequency oscillating charge will emit radiation with frequency
identical to the oscillation frequency identical to the oscillation frequency as given as given
aboveabove
m
k
2
1
3
2
4R
Ze
k
o
3636
The plum-pudding model predicts The plum-pudding model predicts
unique oscillation frequencyunique oscillation frequency
Radiation with frequency identical to the Radiation with frequency identical to the
oscillation frequency. oscillation frequency.
Hence light emitted from the atom in the plum-Hence light emitted from the atom in the plum-
pudding model is predicted to have exactly pudding model is predicted to have exactly one one
unique unique frequency as given in the previous frequency as given in the previous
slide. slide.
This prediction has been falsified because This prediction has been falsified because
observationally, light spectra from all atoms observationally, light spectra from all atoms
(such as the simplest atom, hydrogen,) have (such as the simplest atom, hydrogen,) have
sets of discrete spectral lines correspond to sets of discrete spectral lines correspond to
many different frequencies (already discussed many different frequencies (already discussed
earlier).earlier).
The plum-pudding model predicts The plum-pudding model predicts
unique oscillation frequencyunique oscillation frequency
Radiation with frequency identical to the Radiation with frequency identical to the
oscillation frequency. oscillation frequency.
Hence light emitted from the atom in the plum-Hence light emitted from the atom in the plum-
pudding model is predicted to have exactly pudding model is predicted to have exactly one one
unique unique frequency as given in the previous frequency as given in the previous
slide. slide.
This prediction has been falsified because This prediction has been falsified because
observationally, light spectra from all atoms observationally, light spectra from all atoms
(such as the simplest atom, hydrogen,) have (such as the simplest atom, hydrogen,) have
sets of discrete spectral lines correspond to sets of discrete spectral lines correspond to
many different frequencies (already discussed many different frequencies (already discussed
earlier).earlier).
3737
Experimental verdict on the plum Experimental verdict on the plum
pudding modelpudding model
Theoretically one expect the deviation angle of a scattered Theoretically one expect the deviation angle of a scattered
particle by the plum-pudding atom to be small:particle by the plum-pudding atom to be small:
This is a prediction of the model that can be checked This is a prediction of the model that can be checked
experimentallyexperimentally
Rutherford was the first one to carry out such experimentRutherford was the first one to carry out such experiment
1~
aveN
Experimental verdict on the plum Experimental verdict on the plum
pudding modelpudding model
Theoretically one expect the deviation angle of a scattered Theoretically one expect the deviation angle of a scattered
particle by the plum-pudding atom to be small:particle by the plum-pudding atom to be small:
This is a prediction of the model that can be checked This is a prediction of the model that can be checked
experimentallyexperimentally
Rutherford was the first one to carry out such experimentRutherford was the first one to carry out such experiment
1~
aveN
3838
Ernest RutherfordErnest Rutherford
British physicist Ernest Rutherford, winner of the
1908 Nobel Prize in chemistry, pioneered the field
of nuclear physics with his research and
development of the nuclear theory of atomic
structure
Born in New Zealand, teachers to many physicists
who later become Nobel prize laureates
Rutherford stated that an atom consists largely of
empty space, with an electrically positive nucleus
in the center and electrically negative electrons
orbiting the nucleus. By bombarding nitrogen gas
with alpha particles (nuclear particles emitted
through radioactivity), Rutherford engineered the
transformation of an atom of nitrogen into both
an atom of oxygen and an atom of hydrogen.
This experiment was an early stimulus to the
development of nuclear energy, a form of energy
in which nuclear transformation and
disintegration release extraordinary power.
Ernest RutherfordErnest Rutherford
British physicist Ernest Rutherford, winner of the
1908 Nobel Prize in chemistry, pioneered the field
of nuclear physics with his research and
development of the nuclear theory of atomic
structure
Born in New Zealand, teachers to many physicists
who later become Nobel prize laureates
Rutherford stated that an atom consists largely of
empty space, with an electrically positive nucleus
in the center and electrically negative electrons
orbiting the nucleus. By bombarding nitrogen gas
with alpha particles (nuclear particles emitted
through radioactivity), Rutherford engineered the
transformation of an atom of nitrogen into both
an atom of oxygen and an atom of hydrogen.
This experiment was an early stimulus to the
development of nuclear energy, a form of energy
in which nuclear transformation and
disintegration release extraordinary power.
3939
Rutherford experimental setupRutherford experimental setup
Alpha particles from Alpha particles from
source is used to be source is used to be
scattered by atoms scattered by atoms
from the thin foil from the thin foil
made of goldmade of gold
The scattered alpha The scattered alpha
particles are detected particles are detected
by the background by the background
screenscreen
Rutherford experimental setupRutherford experimental setup
Alpha particles from Alpha particles from
source is used to be source is used to be
scattered by atoms scattered by atoms
from the thin foil from the thin foil
made of goldmade of gold
The scattered alpha The scattered alpha
particles are detected particles are detected
by the background by the background
screenscreen
4040
4141
“…“…fire a 15 inch artillery shell at a fire a 15 inch artillery shell at a
tissue paper and it came back and tissue paper and it came back and
hit you”hit you”
In the scattering experiment Rutherford In the scattering experiment Rutherford
saw some electrons being bounced back saw some electrons being bounced back
at 180 degree. at 180 degree.
He said this is like firing “a 15-inch shell at He said this is like firing “a 15-inch shell at
a piece of a tissue paper and it came back a piece of a tissue paper and it came back
and hit you”and hit you”
Hence Thompson plum-pudding model
fails in the light of these experimental
result
“…“…fire a 15 inch artillery shell at a fire a 15 inch artillery shell at a
tissue paper and it came back and tissue paper and it came back and
hit you”hit you”
In the scattering experiment Rutherford In the scattering experiment Rutherford
saw some electrons being bounced back saw some electrons being bounced back
at 180 degree. at 180 degree.
He said this is like firing “a 15-inch shell at He said this is like firing “a 15-inch shell at
a piece of a tissue paper and it came back a piece of a tissue paper and it came back
and hit you”and hit you”
Hence Thompson plum-pudding model
fails in the light of these experimental
result
4242
So, is the plum pudding model So, is the plum pudding model
utterly useless?utterly useless?
So the plum pudding model does not work as its So the plum pudding model does not work as its
predictions fail to fit the experimental data as well as predictions fail to fit the experimental data as well as
other observationsother observations
Nevertheless it’s a perfectly sensible scientific theory Nevertheless it’s a perfectly sensible scientific theory
because:because:
It is a mathematical model built on sound and rigorous It is a mathematical model built on sound and rigorous
physical arguments physical arguments
It predicts some physical phenomenon with definitenessIt predicts some physical phenomenon with definiteness
It can be verified or falsified with experimentsIt can be verified or falsified with experiments
It also serves as a prototype to the next model which is It also serves as a prototype to the next model which is
built on the experience gained from the failure of this built on the experience gained from the failure of this
model model
So, is the plum pudding model So, is the plum pudding model
utterly useless?utterly useless?
So the plum pudding model does not work as its So the plum pudding model does not work as its
predictions fail to fit the experimental data as well as predictions fail to fit the experimental data as well as
other observationsother observations
Nevertheless it’s a perfectly sensible scientific theory Nevertheless it’s a perfectly sensible scientific theory
because:because:
It is a mathematical model built on sound and rigorous It is a mathematical model built on sound and rigorous
physical arguments physical arguments
It predicts some physical phenomenon with definitenessIt predicts some physical phenomenon with definiteness
It can be verified or falsified with experimentsIt can be verified or falsified with experiments
It also serves as a prototype to the next model which is It also serves as a prototype to the next model which is
built on the experience gained from the failure of this built on the experience gained from the failure of this
model model
4343
How to interpret the Rutherford How to interpret the Rutherford
scattering experiment?scattering experiment?
The large deflection of The large deflection of
alpha particle as seen in alpha particle as seen in
the scattering experiment the scattering experiment
with a thin gold foil must with a thin gold foil must
be produced by a close be produced by a close
encounter between the encounter between the
alpha particle and a very alpha particle and a very
small but massive kernel small but massive kernel
inside the atom inside the atom
In contrast, a diffused In contrast, a diffused
distribution of the positive distribution of the positive
charge as assumed in charge as assumed in
plum-pudding model plum-pudding model
cannot do the jobcannot do the job
How to interpret the Rutherford How to interpret the Rutherford
scattering experiment?scattering experiment?
The large deflection of The large deflection of
alpha particle as seen in alpha particle as seen in
the scattering experiment the scattering experiment
with a thin gold foil must with a thin gold foil must
be produced by a close be produced by a close
encounter between the encounter between the
alpha particle and a very alpha particle and a very
small but massive kernel small but massive kernel
inside the atom inside the atom
In contrast, a diffused In contrast, a diffused
distribution of the positive distribution of the positive
charge as assumed in charge as assumed in
plum-pudding model plum-pudding model
cannot do the jobcannot do the job
4444
Comparing model with nucleus Comparing model with nucleus
concentrated at a point-like nucleus concentrated at a point-like nucleus
and model with nucleus that has and model with nucleus that has
large sizelarge size
Comparing model with nucleus Comparing model with nucleus
concentrated at a point-like nucleus concentrated at a point-like nucleus
and model with nucleus that has and model with nucleus that has
large sizelarge size
4545
RecapRecap
the atomic model building storythe atomic model building story
Plum-pudding model by
Thompson
It fails to explain the
emission and absorption
line spectrum from atoms
because it predicts only a
single emission frequency
Most importantly it fails to
explain the back-scattering
of alpha particle seen in
Rutherford’s scattering
experiment because the
model predicts only
1~
aveN
m
k
2
1
RecapRecap
the atomic model building storythe atomic model building story
Plum-pudding model by
Thompson
It fails to explain the
emission and absorption
line spectrum from atoms
because it predicts only a
single emission frequency
Most importantly it fails to
explain the back-scattering
of alpha particle seen in
Rutherford’s scattering
experiment because the
model predicts only
1~
aveN
m
k
2
1
4646
Rutherford put forward an Rutherford put forward an
model to explain the result model to explain the result
of the scattering of the scattering
experiment: the Rutherford experiment: the Rutherford
modelmodel
An atom consists of a very An atom consists of a very
small nucleus of charge +small nucleus of charge +ZeZe
containing almost all of the containing almost all of the
mass of the atom; this mass of the atom; this
nucleus is surrounded by a nucleus is surrounded by a
swarm of swarm of ZZ electrons electrons
The atom is largely The atom is largely
comprised of empty spacecomprised of empty space
RR
atom atom ~ ~ 1010
-10-10
mm
RR
nucleus nucleus ~ ~ 1010
-13 -13
- 10- 10
-15 -15
mm
The Rutherford model The Rutherford model
(planetary model)(planetary model)
Rutherford put forward an Rutherford put forward an
model to explain the result model to explain the result
of the scattering of the scattering
experiment: the Rutherford experiment: the Rutherford
modelmodel
An atom consists of a very An atom consists of a very
small nucleus of charge +small nucleus of charge +ZeZe
containing almost all of the containing almost all of the
mass of the atom; this mass of the atom; this
nucleus is surrounded by a nucleus is surrounded by a
swarm of swarm of ZZ electrons electrons
The atom is largely The atom is largely
comprised of empty spacecomprised of empty space
RR
atom atom ~ ~ 1010
-10-10
mm
RR
nucleus nucleus ~ ~ 1010
-13 -13
- 10- 10
-15 -15
mm
The Rutherford model The Rutherford model
(planetary model)(planetary model)
4747
Infrared catastrophe: insufficiency Infrared catastrophe: insufficiency
of the Rutherford modelof the Rutherford model
According to classical According to classical
EM, the Rutherford model EM, the Rutherford model
for atom (a classical for atom (a classical
model) has a fatal flaw: it model) has a fatal flaw: it
predicts the collapse of predicts the collapse of
the atom within 10the atom within 10
-10-10
s s
A accelerated electron A accelerated electron
will radiate EM radiation, will radiate EM radiation,
hence causing the hence causing the
orbiting electron to loss orbiting electron to loss
energy and consequently energy and consequently
spiral inward and impact spiral inward and impact
on the nucleuson the nucleus
Infrared catastrophe: insufficiency Infrared catastrophe: insufficiency
of the Rutherford modelof the Rutherford model
According to classical According to classical
EM, the Rutherford model EM, the Rutherford model
for atom (a classical for atom (a classical
model) has a fatal flaw: it model) has a fatal flaw: it
predicts the collapse of predicts the collapse of
the atom within 10the atom within 10
-10-10
s s
A accelerated electron A accelerated electron
will radiate EM radiation, will radiate EM radiation,
hence causing the hence causing the
orbiting electron to loss orbiting electron to loss
energy and consequently energy and consequently
spiral inward and impact spiral inward and impact
on the nucleuson the nucleus
4848
4949
Rutherford model also can’t explain Rutherford model also can’t explain
the discrete spectrumthe discrete spectrum
The Rutherford model also cannot The Rutherford model also cannot
explain the pattern of discrete explain the pattern of discrete
spectral lines as the radiation spectral lines as the radiation
predicted by Rutherford model is a predicted by Rutherford model is a
continuous burst.continuous burst.
Rutherford model also can’t explain Rutherford model also can’t explain
the discrete spectrumthe discrete spectrum
The Rutherford model also cannot The Rutherford model also cannot
explain the pattern of discrete explain the pattern of discrete
spectral lines as the radiation spectral lines as the radiation
predicted by Rutherford model is a predicted by Rutherford model is a
continuous burst.continuous burst.
5050
So how to fix up the problem?So how to fix up the problem?
NEILS BOHR COMES TO NEILS BOHR COMES TO
THE RESCUETHE RESCUE
Niels BohrNiels Bohr (1885 to (1885 to
1962) is best known for 1962) is best known for
the investigations of the investigations of
atomic structure and also atomic structure and also
for work on radiation, for work on radiation,
which won him the 1922 which won him the 1922
Nobel Prize for physicsNobel Prize for physics
He was sometimes He was sometimes
dubbed “the God Father” dubbed “the God Father”
in the physicist in the physicist
communitycommunity
http://www-gap.dcs.st-http://www-gap.dcs.st-
and.ac.uk/~history/and.ac.uk/~history/
Mathematicians/Bohr_Niels.htmlMathematicians/Bohr_Niels.html
So how to fix up the problem?So how to fix up the problem?
NEILS BOHR COMES TO NEILS BOHR COMES TO
THE RESCUETHE RESCUE
Niels BohrNiels Bohr (1885 to (1885 to
1962) is best known for 1962) is best known for
the investigations of the investigations of
atomic structure and also atomic structure and also
for work on radiation, for work on radiation,
which won him the 1922 which won him the 1922
Nobel Prize for physicsNobel Prize for physics
He was sometimes He was sometimes
dubbed “the God Father” dubbed “the God Father”
in the physicist in the physicist
communitycommunity
http://www-gap.dcs.st-http://www-gap.dcs.st-
and.ac.uk/~history/and.ac.uk/~history/
Mathematicians/Bohr_Niels.htmlMathematicians/Bohr_Niels.html
5151
To fix up the infrared catastrophe …To fix up the infrared catastrophe …
Neils Bohr put forward a model which is Neils Bohr put forward a model which is
a a hybrid of the Rutherford model with the hybrid of the Rutherford model with the
wave nature of electron taken into wave nature of electron taken into
accountaccount
To fix up the infrared catastrophe …To fix up the infrared catastrophe …
Neils Bohr put forward a model which is Neils Bohr put forward a model which is
a a hybrid of the Rutherford model with the hybrid of the Rutherford model with the
wave nature of electron taken into wave nature of electron taken into
accountaccount
5252
Bohr’s model of hydrogen-like atomBohr’s model of hydrogen-like atom
We shall consider a simple atom We shall consider a simple atom
consists of a nucleus with charge consists of a nucleus with charge
ZeZe and mass of and mass of MM
nucleusnucleus >> m >> m
ee
The nucleus is surrounded by The nucleus is surrounded by
only only a single electrona single electron
We will assume the centre of the We will assume the centre of the
circular motion of the electron circular motion of the electron
coincides with the centre of the coincides with the centre of the
nucleusnucleus
We term such type of simple We term such type of simple
system: hydrogen-like atomssystem: hydrogen-like atoms
For example, hydrogen atom For example, hydrogen atom
corresponds to corresponds to Z = Z = 1; a singly 1; a singly
ionised Helium atom Heionised Helium atom He
+ +
corresponds to corresponds to Z = Z = 2 etc2 etc
Diagram representing
the model of a
hydrogen-like atom
+Ze
M >>m
Bohr’s model of hydrogen-like atomBohr’s model of hydrogen-like atom
We shall consider a simple atom We shall consider a simple atom
consists of a nucleus with charge consists of a nucleus with charge
ZeZe and mass of and mass of MM
nucleusnucleus >> m >> m
ee
The nucleus is surrounded by The nucleus is surrounded by
only only a single electrona single electron
We will assume the centre of the We will assume the centre of the
circular motion of the electron circular motion of the electron
coincides with the centre of the coincides with the centre of the
nucleusnucleus
We term such type of simple We term such type of simple
system: hydrogen-like atomssystem: hydrogen-like atoms
For example, hydrogen atom For example, hydrogen atom
corresponds to corresponds to Z = Z = 1; a singly 1; a singly
ionised Helium atom Heionised Helium atom He
+ +
corresponds to corresponds to Z = Z = 2 etc2 etc
Diagram representing
the model of a
hydrogen-like atom
+Ze
M >>m
5353
Bohr’s postulate, 1913Bohr’s postulate, 1913
r
vm
r
eZe
e
2
2
0
4
1
Coulomb’s attraction = centripetal force
Assumption: the mass of the
nucleus is infinitely heavy
compared to the electron’s
Postulate No.1: Postulate No.1: Mechanical Mechanical
stability (classical mechanics)stability (classical mechanics)
An electron in an atom An electron in an atom
moves in a circular orbit moves in a circular orbit
about the nucleus under about the nucleus under
Coulomb attraction obeying Coulomb attraction obeying
the law of classical the law of classical
mechanicsmechanics
Bohr’s postulate, 1913Bohr’s postulate, 1913
r
vm
r
eZe
e
2
2
0
4
1
Coulomb’s attraction = centripetal force
Assumption: the mass of the
nucleus is infinitely heavy
compared to the electron’s
Postulate No.1: Postulate No.1: Mechanical Mechanical
stability (classical mechanics)stability (classical mechanics)
An electron in an atom An electron in an atom
moves in a circular orbit moves in a circular orbit
about the nucleus under about the nucleus under
Coulomb attraction obeying Coulomb attraction obeying
the law of classical the law of classical
mechanicsmechanics
5454
Postulate 2: condition for orbit Postulate 2: condition for orbit
stabilitystability
Instead of the infinite orbit which could Instead of the infinite orbit which could
be possible in classical mechanics (c.f be possible in classical mechanics (c.f
the orbits of satellites), it is only possible the orbits of satellites), it is only possible
for an electron to move in an orbit that for an electron to move in an orbit that
contains an integral number of de Broglie contains an integral number of de Broglie
wavelengths, wavelengths,
nn
nn = = 22rr
nn, , nn = 1,2,3... = 1,2,3...
Postulate 2: condition for orbit Postulate 2: condition for orbit
stabilitystability
Instead of the infinite orbit which could Instead of the infinite orbit which could
be possible in classical mechanics (c.f be possible in classical mechanics (c.f
the orbits of satellites), it is only possible the orbits of satellites), it is only possible
for an electron to move in an orbit that for an electron to move in an orbit that
contains an integral number of de Broglie contains an integral number of de Broglie
wavelengths, wavelengths,
nn
nn = = 22rr
nn, , nn = 1,2,3... = 1,2,3...
5555
Bohr’s 2Bohr’s 2
ndnd
postulate means that postulate means that nn
de Broglie wavelengths must fit into de Broglie wavelengths must fit into
the circumference of an orbitthe circumference of an orbit
Bohr’s 2Bohr’s 2
ndnd
postulate means that postulate means that nn
de Broglie wavelengths must fit into de Broglie wavelengths must fit into
the circumference of an orbitthe circumference of an orbit
5656
Electron that don’t form standing Electron that don’t form standing
wave wave
Since the electron must form Since the electron must form
standing waves in the orbits, standing waves in the orbits,
the the orbits of the electron the the orbits of the electron
for each for each nn is quantised is quantised
Orbits with the perimeter that Orbits with the perimeter that
do not conform to the do not conform to the
quantisation condition cannot quantisation condition cannot
persistpersist
All this simply means: all All this simply means: all
orbits of the electron in the orbits of the electron in the
atom must be quantised, and atom must be quantised, and
orbit that is not quantised is orbit that is not quantised is
not allowed (hence can’t not allowed (hence can’t
exist)exist)
Electron that don’t form standing Electron that don’t form standing
wave wave
Since the electron must form Since the electron must form
standing waves in the orbits, standing waves in the orbits,
the the orbits of the electron the the orbits of the electron
for each for each nn is quantised is quantised
Orbits with the perimeter that Orbits with the perimeter that
do not conform to the do not conform to the
quantisation condition cannot quantisation condition cannot
persistpersist
All this simply means: all All this simply means: all
orbits of the electron in the orbits of the electron in the
atom must be quantised, and atom must be quantised, and
orbit that is not quantised is orbit that is not quantised is
not allowed (hence can’t not allowed (hence can’t
exist)exist)
5757
Quantisation of angular momentumQuantisation of angular momentum
As a result of the orbit As a result of the orbit
quantisation, the angular quantisation, the angular
momentum of the orbiting momentum of the orbiting
electron is also electron is also
quantised: quantised:
L = L = ((mm
eevv)) r = pr r = pr
(definition)(definition)
nn= = 22 r r (orbit (orbit
quantisation)quantisation)
Combining both:Combining both:
p= h/p= h/ = nh/ = nh/ 22 r r
L = mL = m
eevr = p r = nh/ vr = p r = nh/ 22
p = mv
Angular momentum of the electron,
L = p x r. It is a vector quantity with
its direction pointing to the
direction perpendicular to the
plane defined by p and r
Quantisation of angular momentumQuantisation of angular momentum
As a result of the orbit As a result of the orbit
quantisation, the angular quantisation, the angular
momentum of the orbiting momentum of the orbiting
electron is also electron is also
quantised: quantised:
L = L = ((mm
eevv)) r = pr r = pr
(definition)(definition)
nn= = 22 r r (orbit (orbit
quantisation)quantisation)
Combining both:Combining both:
p= h/p= h/ = nh/ = nh/ 22 r r
L = mL = m
eevr = p r = nh/ vr = p r = nh/ 22
p = mv
Angular momentum of the electron,
L = p x r. It is a vector quantity with
its direction pointing to the
direction perpendicular to the
plane defined by p and r
5858
Third postulateThird postulate
Despite the fact that it is constantly Despite the fact that it is constantly
accelerating, an electron moving in such an accelerating, an electron moving in such an
allowed orbit does not radiate EM energy allowed orbit does not radiate EM energy
(hence total energy remains constant) (hence total energy remains constant)
As far as the stability of atoms is concerned, As far as the stability of atoms is concerned,
classical physics is invalid hereclassical physics is invalid here
My Comment: At the quantum scale (inside the My Comment: At the quantum scale (inside the
atoms) some of the classical EM predictions atoms) some of the classical EM predictions
fail (e.g. an accelerating charge radiates EM fail (e.g. an accelerating charge radiates EM
wave)wave)
Third postulateThird postulate
Despite the fact that it is constantly Despite the fact that it is constantly
accelerating, an electron moving in such an accelerating, an electron moving in such an
allowed orbit does not radiate EM energy allowed orbit does not radiate EM energy
(hence total energy remains constant) (hence total energy remains constant)
As far as the stability of atoms is concerned, As far as the stability of atoms is concerned,
classical physics is invalid hereclassical physics is invalid here
My Comment: At the quantum scale (inside the My Comment: At the quantum scale (inside the
atoms) some of the classical EM predictions atoms) some of the classical EM predictions
fail (e.g. an accelerating charge radiates EM fail (e.g. an accelerating charge radiates EM
wave)wave)
5959
Quantisation of velocity and radiusQuantisation of velocity and radius
Combining the quantisation of angular Combining the quantisation of angular
momentum and the equation of momentum and the equation of
mechanical stability we arrive at the result mechanical stability we arrive at the result
that:that:
the allowed radius and velocity at a given the allowed radius and velocity at a given
orbit are also quantised:orbit are also quantised:
2
22
0
4
Zem
n
r
e
n
n
Ze
v
n
2
0
4
1
Quantisation of velocity and radiusQuantisation of velocity and radius
Combining the quantisation of angular Combining the quantisation of angular
momentum and the equation of momentum and the equation of
mechanical stability we arrive at the result mechanical stability we arrive at the result
that:that:
the allowed radius and velocity at a given the allowed radius and velocity at a given
orbit are also quantised:orbit are also quantised:
2
22
0
4
Zem
n
r
e
n
n
Ze
v
n
2
0
4
1
6060
Some mathematical steps leading Some mathematical steps leading
to quantisation of orbits, to quantisation of orbits,
2
22
04
Zem
n
r
e
n
2 2
2
2
0 0
1 1
(Eq. 2)
4 4
e
e
Ze emv Ze
v
r r m r
(Eq.2)(Eq.2) (Eq.1) (Eq.1)
22
,,
((mm
eevrvr))
22
= = ((nh/ nh/ 22))
22
LHS: LHS: mm
ee
22
rr
22
vv
2 2
= = mm
ee
22
rr
22
( (ZeZe
2 2
/ 4/ 4
mm
e e rr) )
= = mm
e e r Zer Ze
2 2
/ 4/ 4
= = RHSRHS = = ((nh/ nh/ 22))
22
r = nr = n
22
((h/ h/ 22))
22
44
/ / ZeZe
22
mm
e e ≡ ≡ rr
nn , ,
nn = 1,2,3… = 1,2,3…
(Eq.1)
2
e
nh
mvr
Some mathematical steps leading Some mathematical steps leading
to quantisation of orbits, to quantisation of orbits,
2
22
04
Zem
n
r
e
n
2 2
2
2
0 0
1 1
(Eq. 2)
4 4
e
e
Ze emv Ze
v
r r m r
(Eq.2)(Eq.2) (Eq.1) (Eq.1)
22
,,
((mm
eevrvr))
22
= = ((nh/ nh/ 22))
22
LHS: LHS: mm
ee
22
rr
22
vv
2 2
= = mm
ee
22
rr
22
( (ZeZe
2 2
/ 4/ 4
mm
e e rr) )
= = mm
e e r Zer Ze
2 2
/ 4/ 4
= = RHSRHS = = ((nh/ nh/ 22))
22
r = nr = n
22
((h/ h/ 22))
22
44
/ / ZeZe
22
mm
e e ≡ ≡ rr
nn , ,
nn = 1,2,3… = 1,2,3…
(Eq.1)
2
e
nh
mvr
6161
Prove it yourself the quantisation of Prove it yourself the quantisation of
the electron velocity the electron velocity
using Eq.(1) and Eq.(2)using Eq.(1) and Eq.(2)
n
Ze
v
n
2
0
4
1
Prove it yourself the quantisation of Prove it yourself the quantisation of
the electron velocity the electron velocity
using Eq.(1) and Eq.(2)using Eq.(1) and Eq.(2)
n
Ze
v
n
2
0
4
1
6262
Important commentsImportant comments
The smallest orbit charaterised by The smallest orbit charaterised by
ZZ = 1, = 1, nn=1 is the ground state orbit of the hydrogen=1 is the ground state orbit of the hydrogen
It’s called the Bohr’s radius = the typical size of an atomIt’s called the Bohr’s radius = the typical size of an atom
In general, the radius of an hydrogen-like ion/atom with In general, the radius of an hydrogen-like ion/atom with
charge charge Ze Ze in the nucleus is expressed in terms of the in the nucleus is expressed in terms of the
Bohr’s radius as Bohr’s radius as
Note also that the ground state velocity of the electron in Note also that the ground state velocity of the electron in
the hydrogen atom is m/s << the hydrogen atom is m/s << cc
non-relativisticnon-relativistic
0
2
2
0
0 5.0
4
A
em
r
e
Z
r
nr
n
02
6
0
102.2v
Important commentsImportant comments
The smallest orbit charaterised by The smallest orbit charaterised by
ZZ = 1, = 1, nn=1 is the ground state orbit of the hydrogen=1 is the ground state orbit of the hydrogen
It’s called the Bohr’s radius = the typical size of an atomIt’s called the Bohr’s radius = the typical size of an atom
In general, the radius of an hydrogen-like ion/atom with In general, the radius of an hydrogen-like ion/atom with
charge charge Ze Ze in the nucleus is expressed in terms of the in the nucleus is expressed in terms of the
Bohr’s radius as Bohr’s radius as
Note also that the ground state velocity of the electron in Note also that the ground state velocity of the electron in
the hydrogen atom is m/s << the hydrogen atom is m/s << cc
non-relativisticnon-relativistic
0
2
2
0
0 5.0
4
A
em
r
e
Z
r
nr
n
02
6
0
102.2v
6363
PYQ 7 Test II 2003/04PYQ 7 Test II 2003/04
In Bohr’s model for hydrogen-like atoms, an electron In Bohr’s model for hydrogen-like atoms, an electron
(mass (mass mm) revolves in a circle around a nucleus with ) revolves in a circle around a nucleus with
positive charges positive charges ZeZe. How is the electron’s velocity . How is the electron’s velocity
related to the radius related to the radius r r of its orbit?of its orbit?
A. A. B.B. C. C.
D.D. E.E. Non of the above Non of the above
Solution: I expect your to be able to derive it from scratch without Solution: I expect your to be able to derive it from scratch without
memorisationmemorisation
ANS: D, Schaum’s series 3000 solved problems, Q39.13, pg 722 ANS: D, Schaum’s series 3000 solved problems, Q39.13, pg 722
modifiedmodified
mr
Ze
v
2
04
1
2
2
0
4
1
mr
Ze
v
2
0
4
1
mr
Ze
v
mr
Ze
v
2
0
2
4
1
PYQ 7 Test II 2003/04PYQ 7 Test II 2003/04
In Bohr’s model for hydrogen-like atoms, an electron In Bohr’s model for hydrogen-like atoms, an electron
(mass (mass mm) revolves in a circle around a nucleus with ) revolves in a circle around a nucleus with
positive charges positive charges ZeZe. How is the electron’s velocity . How is the electron’s velocity
related to the radius related to the radius r r of its orbit?of its orbit?
A. A. B.B. C. C.
D.D. E.E. Non of the above Non of the above
Solution: I expect your to be able to derive it from scratch without Solution: I expect your to be able to derive it from scratch without
memorisationmemorisation
ANS: D, Schaum’s series 3000 solved problems, Q39.13, pg 722 ANS: D, Schaum’s series 3000 solved problems, Q39.13, pg 722
modifiedmodified
mr
Ze
v
2
04
1
2
2
0
4
1
mr
Ze
v
2
0
4
1
mr
Ze
v
mr
Ze
v
2
0
2
4
1
6464
The quantised orbits of hydrogen-The quantised orbits of hydrogen-
like atom (not to scale)like atom (not to scale)
+Ze
r
0
r
2r
3
r
4
Z
r
nr
n
02
The quantised orbits of hydrogen-The quantised orbits of hydrogen-
like atom (not to scale)like atom (not to scale)
+Ze
r
0
r
2r
3
r
4
Z
r
nr
n
02
6565
Strongly recommending the Strongly recommending the
Physics 2000 interactive physics Physics 2000 interactive physics
webpage by the University of webpage by the University of
ColoradoColorado
For example the page
http://www.colorado.edu/physics/2000/quantu
mzone/bohr.html
provides a very interesting explanation and
simulation on atom and Bohr model in
particular.
Please visit this page if you go online
Strongly recommending the Strongly recommending the
Physics 2000 interactive physics Physics 2000 interactive physics
webpage by the University of webpage by the University of
ColoradoColorado
For example the page
http://www.colorado.edu/physics/2000/quantu
mzone/bohr.html
provides a very interesting explanation and
simulation on atom and Bohr model in
particular.
Please visit this page if you go online
6666
RecapRecap
The hydrogen-like atom’s radii are quantised The hydrogen-like atom’s radii are quantised
according to:according to:
The quantisation is a direct consequence of the The quantisation is a direct consequence of the
postulate that electron wave forms stationary states postulate that electron wave forms stationary states
(standing waves) at the allowed orbits(standing waves) at the allowed orbits
The smallest orbit or hydrogen, the Bohr’s radiusThe smallest orbit or hydrogen, the Bohr’s radius
0
2
2
0
0
5.0
4
A
em
r
e
Z
r
nr
n
02
+Ze
RecapRecap
The hydrogen-like atom’s radii are quantised The hydrogen-like atom’s radii are quantised
according to:according to:
The quantisation is a direct consequence of the The quantisation is a direct consequence of the
postulate that electron wave forms stationary states postulate that electron wave forms stationary states
(standing waves) at the allowed orbits(standing waves) at the allowed orbits
The smallest orbit or hydrogen, the Bohr’s radiusThe smallest orbit or hydrogen, the Bohr’s radius
0
2
2
0
0
5.0
4
A
em
r
e
Z
r
nr
n
02
+Ze
6767
Postulate 4Postulate 4
Similar to Einstein’s Similar to Einstein’s
postulate of the energy of a postulate of the energy of a
photonphoton
EM radiation is emitted if
an electron initially moving
in an orbit of total energy
E
i
, discontinuously changes
it motion so that it moves in
an orbit of total energy E
f
.
The frequency of the
emitted radiation,
= (E
f
- E
i
)/h
Postulate 4Postulate 4
Similar to Einstein’s Similar to Einstein’s
postulate of the energy of a postulate of the energy of a
photonphoton
EM radiation is emitted if
an electron initially moving
in an orbit of total energy
E
i
, discontinuously changes
it motion so that it moves in
an orbit of total energy E
f
.
The frequency of the
emitted radiation,
= (E
f
- E
i
)/h
6868
Energies in the hydrogen-like atomEnergies in the hydrogen-like atom
Potential energyPotential energy of the electron at a distance of the electron at a distance r r from from
the nucleus is, as we learned from standard the nucleus is, as we learned from standard
electrostatics, ZCT 102, form 6, matriculation etc. is electrostatics, ZCT 102, form 6, matriculation etc. is
simplysimply
r
r
Ze
dr
r
Ze
V
0
2
2
0
2
44
-ve means that the EM force is attractive-ve means that the EM force is attractive
F
e
-e
+Ze
Energies in the hydrogen-like atomEnergies in the hydrogen-like atom
Potential energyPotential energy of the electron at a distance of the electron at a distance r r from from
the nucleus is, as we learned from standard the nucleus is, as we learned from standard
electrostatics, ZCT 102, form 6, matriculation etc. is electrostatics, ZCT 102, form 6, matriculation etc. is
simplysimply
r
r
Ze
dr
r
Ze
V
0
2
2
0
2
44
-ve means that the EM force is attractive-ve means that the EM force is attractive
F
e
-e
+Ze
6969
Kinetic energy in the hydrogen-like Kinetic energy in the hydrogen-like
atomatom
According to definition, the KE of the According to definition, the KE of the
electron is electron is
r
Zevm
K
e
0
22
82
Adding up KE + V, we obtain the total mechanical energy of Adding up KE + V, we obtain the total mechanical energy of
the atom:the atom:
The last step follows from the equation
2
0
22
4r
Ze
r
vm
e
n
e
e
E
n
eZm
n
ZemZe
r
Ze
r
Ze
r
Ze
VKE
222
0
42
22
0
2
0
2
0
2
0
2
0
2
1
24
48
1
848
Kinetic energy in the hydrogen-like Kinetic energy in the hydrogen-like
atomatom
According to definition, the KE of the According to definition, the KE of the
electron is electron is
r
Zevm
K
e
0
22
82
Adding up KE + V, we obtain the total mechanical energy of Adding up KE + V, we obtain the total mechanical energy of
the atom:the atom:
The last step follows from the equation
2
0
22
4r
Ze
r
vm
e
n
e
e
E
n
eZm
n
ZemZe
r
Ze
r
Ze
r
Ze
VKE
222
0
42
22
0
2
0
2
0
2
0
2
0
2
1
24
48
1
848
7070
The ground state energyThe ground state energy
For the hydrogen atom (For the hydrogen atom (ZZ = 1), the ground = 1), the ground
state energy (which is characterised by state energy (which is characterised by n n =1)=1)
eV6.13
24
)1(
22
0
4
0
em
nEE
e
n
In general the energy level of a hydrogen
like atom with Ze nucleus charges can be
expressed in terms of
eV
6.13
2
2
2
0
2
n
Z
n
EZ
E
n
The ground state energyThe ground state energy
For the hydrogen atom (For the hydrogen atom (ZZ = 1), the ground = 1), the ground
state energy (which is characterised by state energy (which is characterised by n n =1)=1)
eV6.13
24
)1(
22
0
4
0
em
nEE
e
n
In general the energy level of a hydrogen
like atom with Ze nucleus charges can be
expressed in terms of
eV
6.13
2
2
2
0
2
n
Z
n
EZ
E
n
7171
Quantization of H energy levelsQuantization of H energy levels
The energy level of the The energy level of the
electrons in the atomic electrons in the atomic
orbit is quantisedorbit is quantised
The quantum number, The quantum number, nn, ,
that characterises the that characterises the
electronic states is called electronic states is called
principle quantum principle quantum
numbernumber
Note that the energy state Note that the energy state
is –ve (because it’s a is –ve (because it’s a
bounded system)bounded system)
eV
6.13
22
0
nn
E
E
n
Quantization of H energy levelsQuantization of H energy levels
The energy level of the The energy level of the
electrons in the atomic electrons in the atomic
orbit is quantisedorbit is quantised
The quantum number, The quantum number, nn, ,
that characterises the that characterises the
electronic states is called electronic states is called
principle quantum principle quantum
numbernumber
Note that the energy state Note that the energy state
is –ve (because it’s a is –ve (because it’s a
bounded system)bounded system)
eV
6.13
22
0
nn
E
E
n
7272
Energy of the electron at very large Energy of the electron at very large nn
An electron occupying an orbit An electron occupying an orbit
with very large with very large nn is “almost free” is “almost free”
because its energy approaches because its energy approaches
zero: zero:
EE = 0 means the electron is free = 0 means the electron is free
from the bondage of the nucleus’ from the bondage of the nucleus’
potential fieldpotential field
Electron at high Electron at high nn is not tightly is not tightly
bounded to the nucleus by the bounded to the nucleus by the
EM forceEM force
Energy levels at high Energy levels at high nn
approaches to that of a approaches to that of a
continuum, as the energy gap continuum, as the energy gap
between adjacent energy levels between adjacent energy levels
become infinitesimal in the large become infinitesimal in the large
nn limit limit
0nE
n
Energy of the electron at very large Energy of the electron at very large nn
An electron occupying an orbit An electron occupying an orbit
with very large with very large nn is “almost free” is “almost free”
because its energy approaches because its energy approaches
zero: zero:
EE = 0 means the electron is free = 0 means the electron is free
from the bondage of the nucleus’ from the bondage of the nucleus’
potential fieldpotential field
Electron at high Electron at high nn is not tightly is not tightly
bounded to the nucleus by the bounded to the nucleus by the
EM forceEM force
Energy levels at high Energy levels at high nn
approaches to that of a approaches to that of a
continuum, as the energy gap continuum, as the energy gap
between adjacent energy levels between adjacent energy levels
become infinitesimal in the large become infinitesimal in the large
nn limit limit
0nE
n
7373
Ionisation energy of the hydrogen Ionisation energy of the hydrogen
atomatom
The energy input required to remove the The energy input required to remove the
electron from its ground state to infinity (ie. electron from its ground state to infinity (ie.
to totally remove the electron from the to totally remove the electron from the
bound of the nucleus) is simply bound of the nucleus) is simply
this is the ionisation energy of hydrogen this is the ionisation energy of hydrogen
eV6.13
00
EEEE
ionisation
F
e
-e
+Ze
Ionisation energy to pull
the electron off from the
attraction of the +ve
nucleus
+Ze
-e
Free electron (= free from the attraction of
the +ve nuclear charge, E= 0)
E=E
0
=-13.6 eV
+
Ionisation energy of the hydrogen Ionisation energy of the hydrogen
atomatom
The energy input required to remove the The energy input required to remove the
electron from its ground state to infinity (ie. electron from its ground state to infinity (ie.
to totally remove the electron from the to totally remove the electron from the
bound of the nucleus) is simply bound of the nucleus) is simply
this is the ionisation energy of hydrogen this is the ionisation energy of hydrogen
eV6.13
00
EEEE
ionisation
F
e
-e
+Ze
Ionisation energy to pull
the electron off from the
attraction of the +ve
nucleus
+Ze
-e
Free electron (= free from the attraction of
the +ve nuclear charge, E= 0)
E=E
0
=-13.6 eV
+
7474
Two important quantities to Two important quantities to
rememberremember
As a practical rule, it is strongly advisable As a practical rule, it is strongly advisable
to remember the two very important values to remember the two very important values
(i) the Bohr radius, (i) the Bohr radius, rr
00 = 0.53A and = 0.53A and
(ii) the ground state energy of the (ii) the ground state energy of the
hydrogen atom, hydrogen atom, EE
00 = -13.6 eV = -13.6 eV
Two important quantities to Two important quantities to
rememberremember
As a practical rule, it is strongly advisable As a practical rule, it is strongly advisable
to remember the two very important values to remember the two very important values
(i) the Bohr radius, (i) the Bohr radius, rr
00 = 0.53A and = 0.53A and
(ii) the ground state energy of the (ii) the ground state energy of the
hydrogen atom, hydrogen atom, EE
00 = -13.6 eV = -13.6 eV
7575
Bohr’s 4th postulate explains the Bohr’s 4th postulate explains the
line spectrum of Hydrogen seriesline spectrum of Hydrogen series
Bohr’s 4th postulate explains the Bohr’s 4th postulate explains the
line spectrum of Hydrogen seriesline spectrum of Hydrogen series
7676
Bohr’s 4th postulate explains the Bohr’s 4th postulate explains the
line spectrumline spectrum
When atoms are excited to an When atoms are excited to an
energy state above its ground state, energy state above its ground state,
they shall radiate out energy (in they shall radiate out energy (in
forms of photon) within at the time forms of photon) within at the time
scale of ~10scale of ~10
-8-8
s upon their de- s upon their de-
excitations to lower energy states –excitations to lower energy states –
emission spectrum explainedemission spectrum explained
When a beam of light with a range of When a beam of light with a range of
wavelength from sees an atom, the few wavelength from sees an atom, the few
particular wavelengths that matches the particular wavelengths that matches the
allowed energy gaps of the atom will be allowed energy gaps of the atom will be
absorbed, leaving behind other unabsorbed absorbed, leaving behind other unabsorbed
wavelengthsto become the bright wavelengthsto become the bright
background in the absorption spectrum. background in the absorption spectrum.
Hence absorption spectrum explainedHence absorption spectrum explained
Bohr’s 4th postulate explains the Bohr’s 4th postulate explains the
line spectrumline spectrum
When atoms are excited to an When atoms are excited to an
energy state above its ground state, energy state above its ground state,
they shall radiate out energy (in they shall radiate out energy (in
forms of photon) within at the time forms of photon) within at the time
scale of ~10scale of ~10
-8-8
s upon their de- s upon their de-
excitations to lower energy states –excitations to lower energy states –
emission spectrum explainedemission spectrum explained
When a beam of light with a range of When a beam of light with a range of
wavelength from sees an atom, the few wavelength from sees an atom, the few
particular wavelengths that matches the particular wavelengths that matches the
allowed energy gaps of the atom will be allowed energy gaps of the atom will be
absorbed, leaving behind other unabsorbed absorbed, leaving behind other unabsorbed
wavelengthsto become the bright wavelengthsto become the bright
background in the absorption spectrum. background in the absorption spectrum.
Hence absorption spectrum explainedHence absorption spectrum explained
7777
Balmer series and the empirical Balmer series and the empirical
emission spectrum equationemission spectrum equation
Since 1860 – 1898 Balmer have found an Since 1860 – 1898 Balmer have found an
empiricalempirical formula that correctly predicted the formula that correctly predicted the
wavelength of four wavelength of four visible linesvisible lines of hydrogen: of hydrogen:
where n = 3,4,5,….R
H
is called the Rydberg constant,
experimentally measured to be R
H
= 1.0973732 x 10
7
m
-1
HH
HH
HH
HH
22
1
2
11
n
R
H
n = 6
n = 5n = 4
n = 3
HH
n
Balmer series and the empirical Balmer series and the empirical
emission spectrum equationemission spectrum equation
Since 1860 – 1898 Balmer have found an Since 1860 – 1898 Balmer have found an
empiricalempirical formula that correctly predicted the formula that correctly predicted the
wavelength of four wavelength of four visible linesvisible lines of hydrogen: of hydrogen:
where n = 3,4,5,….R
H
is called the Rydberg constant,
experimentally measured to be R
H
= 1.0973732 x 10
7
m
-1
HH
HH
HH
HH
22
1
2
11
n
R
H
n = 6
n = 5n = 4
n = 3
HH
n
7878
ExampleExample
For example, for the H
(486.1 nm) line, n = 4 in the
empirical formula
According to the empirical formula the wavelength of
the hydrogen beta line is
which is consistent with the observed value
22
1
2
11
n
R
H
nm486
16
)m101.0973732(3
16
3
4
1
2
11
1-7
22
HH RR
ExampleExample
For example, for the H
(486.1 nm) line, n = 4 in the
empirical formula
According to the empirical formula the wavelength of
the hydrogen beta line is
which is consistent with the observed value
22
1
2
11
n
R
H
nm486
16
)m101.0973732(3
16
3
4
1
2
11
1-7
22
HH RR
7979
Other spectra seriesOther spectra series
Apart from the Balmer series others Apart from the Balmer series others
spectral series are also discovered: spectral series are also discovered:
Lyman, Paschen and Brackett seriesLyman, Paschen and Brackett series
The wavelengths of these spectral lines The wavelengths of these spectral lines
are also given by the similar empirical are also given by the similar empirical
equation as equation as
,...7,6,5,
1
4
11
,...6,5,4,
1
3
11
,...4,3,2,
1
1
11
22
22
2
n
n
R
n
n
R
n
n
R
H
H
H
Lyman series, ultraviolet region
Paschen series, infrared region
Brackett series, infrared region
Other spectra seriesOther spectra series
Apart from the Balmer series others Apart from the Balmer series others
spectral series are also discovered: spectral series are also discovered:
Lyman, Paschen and Brackett seriesLyman, Paschen and Brackett series
The wavelengths of these spectral lines The wavelengths of these spectral lines
are also given by the similar empirical are also given by the similar empirical
equation as equation as
,...7,6,5,
1
4
11
,...6,5,4,
1
3
11
,...4,3,2,
1
1
11
22
22
2
n
n
R
n
n
R
n
n
R
H
H
H
Lyman series, ultraviolet region
Paschen series, infrared region
Brackett series, infrared region
8080
These are experimentally measured These are experimentally measured
spectral linespectral line
...4,3,2,
1
1
11
2
n
n
R
H
...6,5,4,
1
3
11
22
n
n
R
H
,6,5,4,3,
1
2
11
22
n
n
R
H
These are experimentally measured These are experimentally measured
spectral linespectral line
...4,3,2,
1
1
11
2
n
n
R
H
...6,5,4,
1
3
11
22
n
n
R
H
,6,5,4,3,
1
2
11
22
n
n
R
H
8181
2 2
1 1 1
For Lyman series, 1, 2,3,4,...
For Balmer series, 2, 3,4,5...
For Paschen series, 3, 4,5,6...
For Brackett series, 4, 5,6,7...
For Pfund series, 5,
H
f i
f i
f i
f i
f i
f i
R
n n
n n
n n
n n
n n
n n
6,7,8...
2 2
1 1 1
For Lyman series, 1, 2,3,4,...
For Balmer series, 2, 3,4,5...
For Paschen series, 3, 4,5,6...
For Brackett series, 4, 5,6,7...
For Pfund series, 5,
H
f i
f i
f i
f i
f i
f i
R
n n
n n
n n
n n
n n
n n
6,7,8...
8282
The empirical formula needs a The empirical formula needs a
theoretical explanationtheoretical explanation
22
111
if
H
nn
R
is an empirical formula with R
H measured
to be R
H
= 1.0973732 x 10
7
m
-1
.
Can the Bohr model provide a sound
theoretical explanation to the form of this
formula and the numerical value of R
H
in
terms of known physical constants?
The answer is: YES
The empirical formula needs a The empirical formula needs a
theoretical explanationtheoretical explanation
22
111
if
H
nn
R
is an empirical formula with R
H measured
to be R
H
= 1.0973732 x 10
7
m
-1
.
Can the Bohr model provide a sound
theoretical explanation to the form of this
formula and the numerical value of R
H
in
terms of known physical constants?
The answer is: YES
8383
Theoretical derivation of the Theoretical derivation of the
empirical formula from Bohr’s empirical formula from Bohr’s
modelmodel
According to the 4According to the 4
thth
postulate:postulate:
E = EE = E
i i – E– E
f f = h= h = h = hc/c/, and , and
EE
k k = E= E
0 0 / / nn
kk
22
= = - -13.6 eV / 13.6 eV / nn
kk
2 2
where where kk = = ii or or jj
Hence we can easily obtain Hence we can easily obtain
the theoretical expression for the theoretical expression for
the emission line spectrum of the emission line spectrum of
hydrogen-like atomhydrogen-like atom
0
2 2
4
2 2 2 2 23
0
1 1 1
1 1 1 1
4 4
i f
i f
e
f i f i
E E E
c ch ch n n
me
R
n n n nc
Theoretical derivation of the Theoretical derivation of the
empirical formula from Bohr’s empirical formula from Bohr’s
modelmodel
According to the 4According to the 4
thth
postulate:postulate:
E = EE = E
i i – E– E
f f = h= h = h = hc/c/, and , and
EE
k k = E= E
0 0 / / nn
kk
22
= = - -13.6 eV / 13.6 eV / nn
kk
2 2
where where kk = = ii or or jj
Hence we can easily obtain Hence we can easily obtain
the theoretical expression for the theoretical expression for
the emission line spectrum of the emission line spectrum of
hydrogen-like atomhydrogen-like atom
0
2 2
4
2 2 2 2 23
0
1 1 1
1 1 1 1
4 4
i f
i f
e
f i f i
E E E
c ch ch n n
me
R
n n n nc
8484
The theoretical Rydberg constantThe theoretical Rydberg constant
The theoretical Rydberg constant, The theoretical Rydberg constant, RR
∞∞, agrees with the , agrees with the
experimental one up a precision of less than 1%experimental one up a precision of less than 1%
1-7
2
0
3
4
m100984119.1
44
c
em
R
e
-17
m100973732.1
HR
This is a remarkable experimental verification of
the correctness of the Bohr model
The theoretical Rydberg constantThe theoretical Rydberg constant
The theoretical Rydberg constant, The theoretical Rydberg constant, RR
∞∞, agrees with the , agrees with the
experimental one up a precision of less than 1%experimental one up a precision of less than 1%
1-7
2
0
3
4
m100984119.1
44
c
em
R
e
-17
m100973732.1
HR
This is a remarkable experimental verification of
the correctness of the Bohr model
8585
Real life example of atomic Real life example of atomic
emissionemission
AURORA are caused AURORA are caused
by streams of fast by streams of fast
photons and electrons photons and electrons
from the sun that from the sun that
excite atoms in the excite atoms in the
upper atmosphere. upper atmosphere.
The green hues of an The green hues of an
auroral display come auroral display come
from oxygenfrom oxygen
Real life example of atomic Real life example of atomic
emissionemission
AURORA are caused AURORA are caused
by streams of fast by streams of fast
photons and electrons photons and electrons
from the sun that from the sun that
excite atoms in the excite atoms in the
upper atmosphere. upper atmosphere.
The green hues of an The green hues of an
auroral display come auroral display come
from oxygenfrom oxygen
8686
ExampleExample
Suppose that, as a result of a collision, the Suppose that, as a result of a collision, the
electron in a hydrogen atom is raised to the electron in a hydrogen atom is raised to the
second excited state (second excited state (nn = 3). = 3).
What is (i) the energy and (ii) wavelength of the What is (i) the energy and (ii) wavelength of the
photon emitted if the electron makes a direct photon emitted if the electron makes a direct
transition to the ground state?transition to the ground state?
What are the energies and the wavelengths of What are the energies and the wavelengths of
the two photons emitted if, instead, the electron the two photons emitted if, instead, the electron
makes a transition to the first excited state (makes a transition to the first excited state (nn=2) =2)
and from there a subsequent transition to the and from there a subsequent transition to the
ground state? ground state?
ExampleExample
Suppose that, as a result of a collision, the Suppose that, as a result of a collision, the
electron in a hydrogen atom is raised to the electron in a hydrogen atom is raised to the
second excited state (second excited state (nn = 3). = 3).
What is (i) the energy and (ii) wavelength of the What is (i) the energy and (ii) wavelength of the
photon emitted if the electron makes a direct photon emitted if the electron makes a direct
transition to the ground state?transition to the ground state?
What are the energies and the wavelengths of What are the energies and the wavelengths of
the two photons emitted if, instead, the electron the two photons emitted if, instead, the electron
makes a transition to the first excited state (makes a transition to the first excited state (nn=2) =2)
and from there a subsequent transition to the and from there a subsequent transition to the
ground state? ground state?
8787
n = 3
n = 2
n = 1, ground state
E = E
3 - E
1
E = E
3 - E
2
E = E
2 - E
1
The energy of the proton emitted in the
transition from the n = 3 to the n = 1 state is
12.1eVeV
1
1
3
1
6.13
2213
EEE
the wavelength of this photon is
nm102
1.12
1242
eV
nmeV
E
chc
Likewise the energies of the two photons
emitted in the transitions from n = 3 n = 2
and n = 2 n = 1 are, respectively,
eV89.1
2
1
3
1
6.13
2223
EEE with wavelength
nm657
89.1
1242
eV
nmeV
E
ch
eV2.10
1
1
2
1
6.13
2212
EEE with wavelength
nm121
2.10
1242
eV
nmeV
E
ch
Make use ofMake use of E E
k k = E= E
0 0 / / nn
kk
22
= = - -13.6 eV / 13.6 eV / nn
kk
22
n = 3
n = 2
n = 1, ground state
E = E
3 - E
1
E = E
3 - E
2
E = E
2 - E
1
The energy of the proton emitted in the
transition from the n = 3 to the n = 1 state is
12.1eVeV
1
1
3
1
6.13
2213
EEE
the wavelength of this photon is
nm102
1.12
1242
eV
nmeV
E
chc
Likewise the energies of the two photons
emitted in the transitions from n = 3 n = 2
and n = 2 n = 1 are, respectively,
eV89.1
2
1
3
1
6.13
2223
EEE with wavelength
nm657
89.1
1242
eV
nmeV
E
ch
eV2.10
1
1
2
1
6.13
2212
EEE with wavelength
nm121
2.10
1242
eV
nmeV
E
ch
Make use ofMake use of E E
k k = E= E
0 0 / / nn
kk
22
= = - -13.6 eV / 13.6 eV / nn
kk
22
8888
ExampleExample
The series limit of the Paschen (The series limit of the Paschen (nn
ff = 3) = 3)is is
820.1 nm (The series limit of a spectral 820.1 nm (The series limit of a spectral
series is the wavelength corresponds to series is the wavelength corresponds to
nn
ii∞∞).).
What are two longest wavelengths of the What are two longest wavelengths of the
Paschen series? Paschen series?
ExampleExample
The series limit of the Paschen (The series limit of the Paschen (nn
ff = 3) = 3)is is
820.1 nm (The series limit of a spectral 820.1 nm (The series limit of a spectral
series is the wavelength corresponds to series is the wavelength corresponds to
nn
ii∞∞).).
What are two longest wavelengths of the What are two longest wavelengths of the
Paschen series? Paschen series?
8989
SolutionSolution
Note that the Rydberg constant is not providedNote that the Rydberg constant is not provided
But by definition the series limit and the Rydberg constant But by definition the series limit and the Rydberg constant
is closely related is closely related
We got to make use of the series limit to solve that problemWe got to make use of the series limit to solve that problem
By referring to the definition of the series limit, By referring to the definition of the series limit,
222
1111
f
Hn
if
H
n
R
nn
R
i
Hence we can substitute Hence we can substitute RR
HH = = nn
ff
22
/ /
∞∞ into into
and express it in terms of the series limit as and express it in terms of the series limit as
n = 4,5,6…n = 4,5,6…
22
111
if
H
nn
R
2
2
1
11
i
f
n
n
SolutionSolution
Note that the Rydberg constant is not providedNote that the Rydberg constant is not provided
But by definition the series limit and the Rydberg constant But by definition the series limit and the Rydberg constant
is closely related is closely related
We got to make use of the series limit to solve that problemWe got to make use of the series limit to solve that problem
By referring to the definition of the series limit, By referring to the definition of the series limit,
222
1111
f
Hn
if
H
n
R
nn
R
i
Hence we can substitute Hence we can substitute RR
HH = = nn
ff
22
/ /
∞∞ into into
and express it in terms of the series limit as and express it in terms of the series limit as
n = 4,5,6…n = 4,5,6…
22
111
if
H
nn
R
2
2
1
11
i
f
n
n
9090
For Paschen series, For Paschen series, nn
ff = 3, = 3,
= 820.1 nm = 820.1 nm
2
2
3
1
nm1.820
11
i
n
The two longest wavelengths The two longest wavelengths
correspond to transitions of the two correspond to transitions of the two
smallest energy gaps from the energy smallest energy gaps from the energy
levels closest to levels closest to nn = 3 state (i.e the = 3 state (i.e the n n
= = 4, 4, n = n = 5 states) to the 5 states) to the n n = 3 state= 3 state
nm1875
94
4
nm1.820
9
nm1.820:4
2
2
2
2
i
i
i
n
n
n
nm1281
95
5
nm1.820
9
nm1.820:5
2
2
2
2
i
i
i
n
n
n
n
f=3
n
i
=4
n
i
=5
For Paschen series, For Paschen series, nn
ff = 3, = 3,
= 820.1 nm = 820.1 nm
2
2
3
1
nm1.820
11
i
n
The two longest wavelengths The two longest wavelengths
correspond to transitions of the two correspond to transitions of the two
smallest energy gaps from the energy smallest energy gaps from the energy
levels closest to levels closest to nn = 3 state (i.e the = 3 state (i.e the n n
= = 4, 4, n = n = 5 states) to the 5 states) to the n n = 3 state= 3 state
nm1875
94
4
nm1.820
9
nm1.820:4
2
2
2
2
i
i
i
n
n
n
nm1281
95
5
nm1.820
9
nm1.820:5
2
2
2
2
i
i
i
n
n
n
n
f=3
n
i
=4
n
i
=5
9191
ExampleExample
Given the ground state energy of hydrogen Given the ground state energy of hydrogen
atom -13.6 eV, what is the longest atom -13.6 eV, what is the longest
wavelength in the hydrogen’s Balmer series?wavelength in the hydrogen’s Balmer series?
Solution:Solution:
E = EE = E
ii – E – E
ff = = -13.6 eV-13.6 eV (1/(1/nn
ii
22
- 1/ - 1/nn
ff
22
) = ) = hchc//
Balmer series: Balmer series: nn
ff
= 2. Hence, in terms of 13.6 = 2. Hence, in terms of 13.6
eV the wavelengths in Balmer series is given eV the wavelengths in Balmer series is given
byby
...5,4,3,
1
4
1
1nm9
1
4
1
eV6.13
nmeV1240
1
4
1
eV6.13
222
i
iii
Balmer
n
nnn
hc
ExampleExample
Given the ground state energy of hydrogen Given the ground state energy of hydrogen
atom -13.6 eV, what is the longest atom -13.6 eV, what is the longest
wavelength in the hydrogen’s Balmer series?wavelength in the hydrogen’s Balmer series?
Solution:Solution:
E = EE = E
ii – E – E
ff = = -13.6 eV-13.6 eV (1/(1/nn
ii
22
- 1/ - 1/nn
ff
22
) = ) = hchc//
Balmer series: Balmer series: nn
ff
= 2. Hence, in terms of 13.6 = 2. Hence, in terms of 13.6
eV the wavelengths in Balmer series is given eV the wavelengths in Balmer series is given
byby
...5,4,3,
1
4
1
1nm9
1
4
1
eV6.13
nmeV1240
1
4
1
eV6.13
222
i
iii
Balmer
n
nnn
hc
9292
longest wavelength corresponds to the transition longest wavelength corresponds to the transition
from the from the nn
ii = 3 states to the = 3 states to the nn
ff = 2 states = 2 states
HenceHence
...5,4,3,
1
4
1
1nm9
2
i
i
Balmer
n
n
nm2.655
3
1
4
1
1nm9
2
max,
Balmer
This is the red This is the red HH
line in the hydrogen’s Balmer line in the hydrogen’s Balmer
seriesseries
Can you calculate the shortest wavelength (the Can you calculate the shortest wavelength (the
series limit) for the Balmer series? Ans = 364 nmseries limit) for the Balmer series? Ans = 364 nm
longest wavelength corresponds to the transition longest wavelength corresponds to the transition
from the from the nn
ii = 3 states to the = 3 states to the nn
ff = 2 states = 2 states
HenceHence
...5,4,3,
1
4
1
1nm9
2
i
i
Balmer
n
n
nm2.655
3
1
4
1
1nm9
2
max,
Balmer
This is the red This is the red HH
line in the hydrogen’s Balmer line in the hydrogen’s Balmer
seriesseries
Can you calculate the shortest wavelength (the Can you calculate the shortest wavelength (the
series limit) for the Balmer series? Ans = 364 nmseries limit) for the Balmer series? Ans = 364 nm
9393
PYQ 2.18 Final Exam 2003/04PYQ 2.18 Final Exam 2003/04
Which of the following statements are true?Which of the following statements are true?
I.I.. . the ground states are states with lowest energy the ground states are states with lowest energy
II.II. ionisation energy is the energy required to raise an ionisation energy is the energy required to raise an
electron from ground state to free stateelectron from ground state to free state
IIIIII..Balmer series is the lines in the spectrum of atomic Balmer series is the lines in the spectrum of atomic
hydrogen that corresponds to the transitions to the hydrogen that corresponds to the transitions to the nn = 1 = 1
state from higher energy statesstate from higher energy states
A.A. I,IV I,IV B.B. I,II, IV I,II, IVC. C. I, III,IVI, III,IV
D.D. I, II I, II E. E. II,IIIII,III
ANS: D, My own questionANS: D, My own question
(note: this is an obvious typo error with the statement (note: this is an obvious typo error with the statement
IV missing. In any case, only statement I, II are true.)IV missing. In any case, only statement I, II are true.)
PYQ 2.18 Final Exam 2003/04PYQ 2.18 Final Exam 2003/04
Which of the following statements are true?Which of the following statements are true?
I.I.. . the ground states are states with lowest energy the ground states are states with lowest energy
II.II. ionisation energy is the energy required to raise an ionisation energy is the energy required to raise an
electron from ground state to free stateelectron from ground state to free state
IIIIII..Balmer series is the lines in the spectrum of atomic Balmer series is the lines in the spectrum of atomic
hydrogen that corresponds to the transitions to the hydrogen that corresponds to the transitions to the nn = 1 = 1
state from higher energy statesstate from higher energy states
A.A. I,IV I,IV B.B. I,II, IV I,II, IVC. C. I, III,IVI, III,IV
D.D. I, II I, II E. E. II,IIIII,III
ANS: D, My own questionANS: D, My own question
(note: this is an obvious typo error with the statement (note: this is an obvious typo error with the statement
IV missing. In any case, only statement I, II are true.)IV missing. In any case, only statement I, II are true.)
9494
PYQ 1.15 KSCP 2003/04PYQ 1.15 KSCP 2003/04
Which of the following statement(s) is (are) true?Which of the following statement(s) is (are) true?
I.I. TheThe experimental proof for which electron posses a experimental proof for which electron posses a
wavelength was first verified by Davisson and wavelength was first verified by Davisson and
Germer Germer
II.II.The experimental proof of the existence of discrete The experimental proof of the existence of discrete
energy levels in atoms involving their excitation by energy levels in atoms involving their excitation by
collision with low-energy electron was confirmed in collision with low-energy electron was confirmed in
the Frank-Hertz experimentthe Frank-Hertz experiment
III.III. Compton scattering experiment establishes that light Compton scattering experiment establishes that light
behave like particles behave like particles
IV. IV. Photoelectric experiment establishes that electrons Photoelectric experiment establishes that electrons
behave like wave behave like wave
A.A. I,III,IIB.B. I,II,III,IVI,II,III,IVC. I, II, IIIC. I, II, III
D.D. III,IVIII,IVE. E. Non of the above Non of the above
Ans: CAns: C Serway and Moses, pg. 127 (for I), pg. 133 Serway and Moses, pg. 127 (for I), pg. 133
(for II), own options (for III,IV)(for II), own options (for III,IV)
PYQ 1.15 KSCP 2003/04PYQ 1.15 KSCP 2003/04
Which of the following statement(s) is (are) true?Which of the following statement(s) is (are) true?
I.I. TheThe experimental proof for which electron posses a experimental proof for which electron posses a
wavelength was first verified by Davisson and wavelength was first verified by Davisson and
Germer Germer
II.II.The experimental proof of the existence of discrete The experimental proof of the existence of discrete
energy levels in atoms involving their excitation by energy levels in atoms involving their excitation by
collision with low-energy electron was confirmed in collision with low-energy electron was confirmed in
the Frank-Hertz experimentthe Frank-Hertz experiment
III.III. Compton scattering experiment establishes that light Compton scattering experiment establishes that light
behave like particles behave like particles
IV. IV. Photoelectric experiment establishes that electrons Photoelectric experiment establishes that electrons
behave like wave behave like wave
A.A. I,III,IIB.B. I,II,III,IVI,II,III,IVC. I, II, IIIC. I, II, III
D.D. III,IVIII,IVE. E. Non of the above Non of the above
Ans: CAns: C Serway and Moses, pg. 127 (for I), pg. 133 Serway and Moses, pg. 127 (for I), pg. 133
(for II), own options (for III,IV)(for II), own options (for III,IV)
9595
PYQ 1.5 KSCP 2003/04PYQ 1.5 KSCP 2003/04
An electron collides with a hydrogen atom An electron collides with a hydrogen atom
in its ground state and excites it to a state in its ground state and excites it to a state
of of nn =3. How much energy was given to =3. How much energy was given to
the hydrogen atom in this collision?the hydrogen atom in this collision?
A. A. -12.1 eV-12.1 eVB.B. 12.1 eV 12.1 eVC. C. -13.6 eV-13.6 eV
D.D. 13.6 eV 13.6 eVE.E. Non of the above Non of the above
Solution:
ANS: B, Modern Technical Physics, Beiser, Example
25.6, pg. 786
0
3 0 0 2 2
13.6eV
( 13.6eV) 12.1eV
3 3
E
E E E E
PYQ 1.5 KSCP 2003/04PYQ 1.5 KSCP 2003/04
An electron collides with a hydrogen atom An electron collides with a hydrogen atom
in its ground state and excites it to a state in its ground state and excites it to a state
of of nn =3. How much energy was given to =3. How much energy was given to
the hydrogen atom in this collision?the hydrogen atom in this collision?
A. A. -12.1 eV-12.1 eVB.B. 12.1 eV 12.1 eVC. C. -13.6 eV-13.6 eV
D.D. 13.6 eV 13.6 eVE.E. Non of the above Non of the above
Solution:
ANS: B, Modern Technical Physics, Beiser, Example
25.6, pg. 786
0
3 0 0 2 2
13.6eV
( 13.6eV) 12.1eV
3 3
E
E E E E
9696
PYQ 1.6 KSCP 2003/04PYQ 1.6 KSCP 2003/04
Which of the following transitions in a hydrogen Which of the following transitions in a hydrogen
atom emits the photon of lowest frequency?atom emits the photon of lowest frequency?
A. A. nn = 3 to = 3 to nn = 4 = 4 B.B. nn = 2 to = 2 to nn = 1 = 1
C. C. nn = 8 to = 8 to nn = 2 = 2D.D. nn = 6 to = 6 to nn = 2 = 2
E.E. Non of the above Non of the above
ANS: D, Modern Technical Physics, Beiser, Q40, pg. 802, ANS: D, Modern Technical Physics, Beiser, Q40, pg. 802,
modifiedmodified
0 0
2 2 2 2
2 2
Lowest frequency means lowest energy:
1 1
13.6eV
1 1
The pair { =6, =2} happens to give smallest
of 0.22.
i f
i f f i
i f
f i
E E
E E E
n n n n
n n
n n
PYQ 1.6 KSCP 2003/04PYQ 1.6 KSCP 2003/04
Which of the following transitions in a hydrogen Which of the following transitions in a hydrogen
atom emits the photon of lowest frequency?atom emits the photon of lowest frequency?
A. A. nn = 3 to = 3 to nn = 4 = 4 B.B. nn = 2 to = 2 to nn = 1 = 1
C. C. nn = 8 to = 8 to nn = 2 = 2D.D. nn = 6 to = 6 to nn = 2 = 2
E.E. Non of the above Non of the above
ANS: D, Modern Technical Physics, Beiser, Q40, pg. 802, ANS: D, Modern Technical Physics, Beiser, Q40, pg. 802,
modifiedmodified
0 0
2 2 2 2
2 2
Lowest frequency means lowest energy:
1 1
13.6eV
1 1
The pair { =6, =2} happens to give smallest
of 0.22.
i f
i f f i
i f
f i
E E
E E E
n n n n
n n
n n
9797
Frank-Hertz experimentFrank-Hertz experiment
The famous experiment that shows the excitation of atoms to The famous experiment that shows the excitation of atoms to
discrete energy levels and is consistent with the results discrete energy levels and is consistent with the results
suggested by line spectrasuggested by line spectra
Mercury vapour is bombarded with electron accelerated Mercury vapour is bombarded with electron accelerated
under the potential under the potential V V (between the grid and the filament)(between the grid and the filament)
A small potential A small potential VV
00 between the grid and collecting plate between the grid and collecting plate
prevents electrons having energies less than a certain prevents electrons having energies less than a certain
minimum from contributing to the current measured by minimum from contributing to the current measured by
ammeterammeter
Frank-Hertz experimentFrank-Hertz experiment
The famous experiment that shows the excitation of atoms to The famous experiment that shows the excitation of atoms to
discrete energy levels and is consistent with the results discrete energy levels and is consistent with the results
suggested by line spectrasuggested by line spectra
Mercury vapour is bombarded with electron accelerated Mercury vapour is bombarded with electron accelerated
under the potential under the potential V V (between the grid and the filament)(between the grid and the filament)
A small potential A small potential VV
00 between the grid and collecting plate between the grid and collecting plate
prevents electrons having energies less than a certain prevents electrons having energies less than a certain
minimum from contributing to the current measured by minimum from contributing to the current measured by
ammeterammeter
9898
The electrons that arrive at the The electrons that arrive at the
anode peaks at equal voltage anode peaks at equal voltage
intervals of 4.9 Vintervals of 4.9 V
As As VV increases, the increases, the
current measured current measured
also increasesalso increases
The measured current The measured current
drops at multiples of a drops at multiples of a
critical potential critical potential
VV = 4.9 V, 9.8V, = 4.9 V, 9.8V,
14.7V14.7V
The electrons that arrive at the The electrons that arrive at the
anode peaks at equal voltage anode peaks at equal voltage
intervals of 4.9 Vintervals of 4.9 V
As As VV increases, the increases, the
current measured current measured
also increasesalso increases
The measured current The measured current
drops at multiples of a drops at multiples of a
critical potential critical potential
VV = 4.9 V, 9.8V, = 4.9 V, 9.8V,
14.7V14.7V
9999
InterpretationInterpretation
As a result of inelastic collisions between the accelerated As a result of inelastic collisions between the accelerated
electrons of KE 4.9 eV with the the Hg atom, the Hg atoms electrons of KE 4.9 eV with the the Hg atom, the Hg atoms
are excited to an energy level above its ground stateare excited to an energy level above its ground state
At this critical point, the energy of the accelerating electron At this critical point, the energy of the accelerating electron
equals to that of the energy gap between the ground state and equals to that of the energy gap between the ground state and
the excited statethe excited state
This is a resonance phenomena, hence current increases This is a resonance phenomena, hence current increases
abruptlyabruptly
After inelastically exciting the atom, the original (the After inelastically exciting the atom, the original (the
bombarding) electron move off with too little energy to bombarding) electron move off with too little energy to
overcome the small retarding potential and reach the plateovercome the small retarding potential and reach the plate
As the accelerating potential is raised further, the plate current As the accelerating potential is raised further, the plate current
again increases, since the electrons now have enough energy again increases, since the electrons now have enough energy
to reach the plateto reach the plate
Eventually another sharp drop (at 9.8 V) in the current occurs Eventually another sharp drop (at 9.8 V) in the current occurs
because, again, the electron has collected just the same because, again, the electron has collected just the same
energy to excite the same energy level in the other atomsenergy to excite the same energy level in the other atoms
InterpretationInterpretation
As a result of inelastic collisions between the accelerated As a result of inelastic collisions between the accelerated
electrons of KE 4.9 eV with the the Hg atom, the Hg atoms electrons of KE 4.9 eV with the the Hg atom, the Hg atoms
are excited to an energy level above its ground stateare excited to an energy level above its ground state
At this critical point, the energy of the accelerating electron At this critical point, the energy of the accelerating electron
equals to that of the energy gap between the ground state and equals to that of the energy gap between the ground state and
the excited statethe excited state
This is a resonance phenomena, hence current increases This is a resonance phenomena, hence current increases
abruptlyabruptly
After inelastically exciting the atom, the original (the After inelastically exciting the atom, the original (the
bombarding) electron move off with too little energy to bombarding) electron move off with too little energy to
overcome the small retarding potential and reach the plateovercome the small retarding potential and reach the plate
As the accelerating potential is raised further, the plate current As the accelerating potential is raised further, the plate current
again increases, since the electrons now have enough energy again increases, since the electrons now have enough energy
to reach the plateto reach the plate
Eventually another sharp drop (at 9.8 V) in the current occurs Eventually another sharp drop (at 9.8 V) in the current occurs
because, again, the electron has collected just the same because, again, the electron has collected just the same
energy to excite the same energy level in the other atomsenergy to excite the same energy level in the other atoms
100100
V = 14.7V
K
e
= 4.9eV
K
e
= 0
First
resonance
at 4.9 eV
Hg
Plate C Plate P
K
e
= 0 after first
resonance
Electron continue to
be accelerated by the
external potential until
the second resonance
occurs
Hg
K
e
reaches 4.9 eV again
Hg
First excitation
energy of Hg
atom E
1 = 4.9eV
Hg
K
e
= 0 after second
resonance
Electron continue to
be accelerated by the
external potential until
the next (third)
resonance occurs
K
e
reaches 4.9 eV again
here
second resonance
initiated
Third
resonance
initiated
If bombared by electron with K
e = 4.9 eV excitation
of the Hg atom will occur. This is a resonance
phenomena
electron is
accelerated
under the
external
potential
V = 14.7V
K
e
= 4.9eV
K
e
= 0
First
resonance
at 4.9 eV
Hg
Plate C Plate P
K
e
= 0 after first
resonance
Electron continue to
be accelerated by the
external potential until
the second resonance
occurs
Hg
K
e
reaches 4.9 eV again
Hg
First excitation
energy of Hg
atom E
1 = 4.9eV
Hg
K
e
= 0 after second
resonance
Electron continue to
be accelerated by the
external potential until
the next (third)
resonance occurs
K
e
reaches 4.9 eV again
here
second resonance
initiated
Third
resonance
initiated
If bombared by electron with K
e = 4.9 eV excitation
of the Hg atom will occur. This is a resonance
phenomena
electron is
accelerated
under the
external
potential
101101
The higher critical potentials result from The higher critical potentials result from
two or more inelastic collisions and are two or more inelastic collisions and are
multiple of the lowest (4.9 V)multiple of the lowest (4.9 V)
The excited mercury atom will then de-The excited mercury atom will then de-
excite by radiating out a photon of exactly excite by radiating out a photon of exactly
the energy (4.9 eV) which is also detected the energy (4.9 eV) which is also detected
in the Frank-Hertz experimentin the Frank-Hertz experiment
The critical potential verifies the existence The critical potential verifies the existence
of atomic levelsof atomic levels
The higher critical potentials result from The higher critical potentials result from
two or more inelastic collisions and are two or more inelastic collisions and are
multiple of the lowest (4.9 V)multiple of the lowest (4.9 V)
The excited mercury atom will then de-The excited mercury atom will then de-
excite by radiating out a photon of exactly excite by radiating out a photon of exactly
the energy (4.9 eV) which is also detected the energy (4.9 eV) which is also detected
in the Frank-Hertz experimentin the Frank-Hertz experiment
The critical potential verifies the existence The critical potential verifies the existence
of atomic levelsof atomic levels
102102
Bohr’s correspondence principleBohr’s correspondence principle
The predictions of the quantum theory for the The predictions of the quantum theory for the
behaviour of any physical system must behaviour of any physical system must
correspond to the prediction of classical physics correspond to the prediction of classical physics
in the limit in which the quantum number in the limit in which the quantum number
specifying the state of the system becomes very specifying the state of the system becomes very
large:large:
quantum theory = classical theoryquantum theory = classical theory
At large At large nn limit, the Bohr model must reduce to a limit, the Bohr model must reduce to a
“classical atom” which obeys classical theory“classical atom” which obeys classical theory
n
lim
Bohr’s correspondence principleBohr’s correspondence principle
The predictions of the quantum theory for the The predictions of the quantum theory for the
behaviour of any physical system must behaviour of any physical system must
correspond to the prediction of classical physics correspond to the prediction of classical physics
in the limit in which the quantum number in the limit in which the quantum number
specifying the state of the system becomes very specifying the state of the system becomes very
large:large:
quantum theory = classical theoryquantum theory = classical theory
At large At large nn limit, the Bohr model must reduce to a limit, the Bohr model must reduce to a
“classical atom” which obeys classical theory“classical atom” which obeys classical theory
n
lim
103103
In other words…In other words…
The laws of quantum physics are valid in The laws of quantum physics are valid in
the atomic domain; while the laws of the atomic domain; while the laws of
classical physics is valid in the classical classical physics is valid in the classical
domain; where the two domains overlaps, domain; where the two domains overlaps,
both sets of laws must give the same both sets of laws must give the same
result. result.
In other words…In other words…
The laws of quantum physics are valid in The laws of quantum physics are valid in
the atomic domain; while the laws of the atomic domain; while the laws of
classical physics is valid in the classical classical physics is valid in the classical
domain; where the two domains overlaps, domain; where the two domains overlaps,
both sets of laws must give the same both sets of laws must give the same
result. result.
104104
PYQ 20 Test II 2003/04PYQ 20 Test II 2003/04
Which of the following statements are correct?Which of the following statements are correct?
II Frank-Hertz experiment shows that atoms are Frank-Hertz experiment shows that atoms are
excited excited to discrete energy levelsto discrete energy levels
IIII Frank-Hertz experimental result is consistent with the Frank-Hertz experimental result is consistent with the
results suggested by the line spectraresults suggested by the line spectra
III III The predictions of the quantum theory for the The predictions of the quantum theory for the
behaviour of any physical system must correspond to behaviour of any physical system must correspond to the the
prediction of classical physics in the limit in which prediction of classical physics in the limit in which the the
quantum number specifying the state of the quantum number specifying the state of the system system
becomes very largebecomes very large
IVIV The structure of atoms can be probed by using The structure of atoms can be probed by using
electromagnetic radiationelectromagnetic radiation
A.A. II,IIIII,IIIB.B. I, II,IVI, II,IVC. II, III, IV C. II, III, IV
D.D. I,II, III, IV I,II, III, IV E. Non of the aboveE. Non of the above
ANS:D, My own questionsANS:D, My own questions
PYQ 20 Test II 2003/04PYQ 20 Test II 2003/04
Which of the following statements are correct?Which of the following statements are correct?
II Frank-Hertz experiment shows that atoms are Frank-Hertz experiment shows that atoms are
excited excited to discrete energy levelsto discrete energy levels
IIII Frank-Hertz experimental result is consistent with the Frank-Hertz experimental result is consistent with the
results suggested by the line spectraresults suggested by the line spectra
III III The predictions of the quantum theory for the The predictions of the quantum theory for the
behaviour of any physical system must correspond to behaviour of any physical system must correspond to the the
prediction of classical physics in the limit in which prediction of classical physics in the limit in which the the
quantum number specifying the state of the quantum number specifying the state of the system system
becomes very largebecomes very large
IVIV The structure of atoms can be probed by using The structure of atoms can be probed by using
electromagnetic radiationelectromagnetic radiation
A.A. II,IIIII,IIIB.B. I, II,IVI, II,IVC. II, III, IV C. II, III, IV
D.D. I,II, III, IV I,II, III, IV E. Non of the aboveE. Non of the above
ANS:D, My own questionsANS:D, My own questions
105105
Example Example
(Read it yourself)(Read it yourself)
Classical EM predicts that an electron in a circular motion Classical EM predicts that an electron in a circular motion
will radiate EM wave at the same frequencywill radiate EM wave at the same frequency
According to the correspondence principle, According to the correspondence principle,
the Bohr model must also reproduce this result in the the Bohr model must also reproduce this result in the
large large nn limit limit
More quantitativelyMore quantitatively
In the limit, In the limit, nn = 10 = 10
33
- 10 - 10
44
, the Bohr atom would have a size , the Bohr atom would have a size
of 10of 10
-3-3
m m
This is a large quantum atom which This is a large quantum atom which is in classical domainis in classical domain
The prediction for the photon emitted during transition The prediction for the photon emitted during transition
around the around the nn = 10 = 10
33
- 10 - 10
44
states should equals to that states should equals to that
predicted by classical EM theory. predicted by classical EM theory.
Example Example
(Read it yourself)(Read it yourself)
Classical EM predicts that an electron in a circular motion Classical EM predicts that an electron in a circular motion
will radiate EM wave at the same frequencywill radiate EM wave at the same frequency
According to the correspondence principle, According to the correspondence principle,
the Bohr model must also reproduce this result in the the Bohr model must also reproduce this result in the
large large nn limit limit
More quantitativelyMore quantitatively
In the limit, In the limit, nn = 10 = 10
33
- 10 - 10
44
, the Bohr atom would have a size , the Bohr atom would have a size
of 10of 10
-3-3
m m
This is a large quantum atom which This is a large quantum atom which is in classical domainis in classical domain
The prediction for the photon emitted during transition The prediction for the photon emitted during transition
around the around the nn = 10 = 10
33
- 10 - 10
44
states should equals to that states should equals to that
predicted by classical EM theory. predicted by classical EM theory.
106106
n
(Bohr)
n large
=
(classical theory)
n
(Bohr)
n large
=
(classical theory)
107107
Classical physics calculationClassical physics calculation
The period of a circulating electron is
T = 2r/(2K/m)
1/2
= r(2m)
1/2
(8e
0
r)
1/2
/e
This result can be easily derived from the
mechanical stability of the atom as per
Substitute the quantised atomic radius r
n
= n
2
r
0
into T, we obtain the frequency as per
n
= 1/T =me
4
/32
3
2
3
n
3
r
vm
r
eZe
e
2
2
04
1
Classical physics calculationClassical physics calculation
The period of a circulating electron is
T = 2r/(2K/m)
1/2
= r(2m)
1/2
(8e
0
r)
1/2
/e
This result can be easily derived from the
mechanical stability of the atom as per
Substitute the quantised atomic radius r
n
= n
2
r
0
into T, we obtain the frequency as per
n
= 1/T =me
4
/32
3
2
3
n
3
r
vm
r
eZe
e
2
2
04
1
108108
Based on Bohr’s theoryBased on Bohr’s theory
Now, for an electron in the Bohr atom at energy
level n = 10
3
- 10
4
, the frequency of an radiated
photon when electron make a transition from the n
state to n–1 state is given by
n
= (me
4
/64
3
2
3
)[(n-1)
-2
- n
-2
]
= (me
4
/64
3
2
3
)[(2n-1)/n
2
(n-1)
2
]
In the limit of large n,
(me
4
/64
3
2
3
)[2n/n
4
]
=me
4
/32
3
2
3
)[1/n
3
]
Based on Bohr’s theoryBased on Bohr’s theory
Now, for an electron in the Bohr atom at energy
level n = 10
3
- 10
4
, the frequency of an radiated
photon when electron make a transition from the n
state to n–1 state is given by
n
= (me
4
/64
3
2
3
)[(n-1)
-2
- n
-2
]
= (me
4
/64
3
2
3
)[(2n-1)/n
2
(n-1)
2
]
In the limit of large n,
(me
4
/64
3
2
3
)[2n/n
4
]
=me
4
/32
3
2
3
)[1/n
3
]
109109
Classical result and Quantum Classical result and Quantum
calculation meets at calculation meets at nn
Hence, in the region of large Hence, in the region of large nn, where classical , where classical
and quantum physics overlap, the classical and quantum physics overlap, the classical
prediction and that of the quantum one is prediction and that of the quantum one is
identicalidentical
classical =
Bohr = (me
4
/32
3
2
3
)[1/n
3
]
Classical result and Quantum Classical result and Quantum
calculation meets at calculation meets at nn
Hence, in the region of large Hence, in the region of large nn, where classical , where classical
and quantum physics overlap, the classical and quantum physics overlap, the classical
prediction and that of the quantum one is prediction and that of the quantum one is
identicalidentical
classical =
Bohr = (me
4
/32
3
2
3
)[1/n
3
]
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