Modern Physics: Boh’r model of atom
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Introduction
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March 21, 2021
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To study the phenomenon deep inside matter several models of atomic structure were suggested from time to time. Each model had its own merits and
shortcomings. No model of atom could explain all the experimental observations. As a result modifications were done to the existing model. Some of
the well known atomic models are:
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• • Thompson’s model: This model considered atoms as a sphere of
positive charge embedded with negative charges
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• • Rutherford’s model of atom Rutherford’s model of atom was
one of the most accurate models as it could explain several atomic
phenomenon.Rutherford’s model suggested that most of the part of the
atom is empty. The center of the atom is the densest part containing
almost all the mass and was called Nucleus.Rutherford’s model proved
that the nucleus was positively charged.
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Rutherford’s model continued to be the widely accepted model for along
time and still is an accepted model however the drawback of the model was
Rutherford’s model was not able to explain the stability of the
atom
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The atomic spectra and its explanation
Spectrum in ordinary sense is obtained when white light passes through a
prism and breaks into several colors or wavelengths. In a similar manner
when an atom is taken above its ground stae or lowest energy state to
an excited state or higher energy state then allowed to cool down
or return to its lowest energy state then electromagnetic radiation similar to
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Figure 1: Thompson’s model of atom
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Figure 2: Rutherford’s model of atom
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Figure 3: spectrum from a prism
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white light is emitted. When this electromagnetic radiation is allowed
to pass through a prism or diffraction grating then the radiation
breaks into several wavelengths and is called atomic spectra. The
most widely studied atomic spectra is the spectra of hydrogen atom. The
features of the hydrogen atom spectra are :
• • the electromagnetic spectra is emitted by free atoms
• • the electromagnetic spectrum is concentrated at a number of definite
or discrete wavelengths
• bullet Each of the wavelength component is called line. The reason is
that the slit which produces the image is in the shape of a line.
The explanation of the atomic spectra of Hydrogen required a model more
advanced than the existing Rutherford model of atom.
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Figure 4: atomic spectra of hydrogen
Bohr’s Postulates and a new model of atom
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In 1913 Bohr came up with a new model of atom that was based on the
experimental observations of the hydrogen atom spectra. Bohr gave some
postulates (assumptions based on logical reasoning). The postulates
that Bohr presented were :
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• Bohr’s first postulate:An electron revolves around the nucleus in
circular orbits under the action of Coulomb Force and obeys the
laws of classical mechanics
• Bohr’s second postulate: The orbits in which the electron moves
round the nucleus are such that the angular momentum of the orbiting
electron is quantized , this means the angular momentum of the electron has only definite values which are multiples of the Planck’s
constant 6 h
• Bohr’s Third Postulate: The total energy E which is the sumof
kinetic and potential energy remains constant when an electron moves
on a fixed orbit (fixed means the orbit where the angular momentum
of the electron is quantized)
• Bohr’s fourth postulate: Bohr’s fourth postulate is the condition of
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quantization of energy .This postulate states that if an electron in an
orbit has total energy Ei makes a transition to another orbit where its
total energy is Ef then a photon (packet of fixed energy )is emitted
or absorbed depending on whether Ef > Ei or Ef < Ei
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Significance and meaning of first postulate
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Let us see what information the Bohr’sPostulates are providing. To get
the information we need some mathematical description to relate the postulates.The mathematics is very simple but each step contains a lot of physical
information
Bohr’s first postulate is related to the nucleus. The centripetal force that
makes the electron to move around the nucleus in circular orbits is provided
by the electrostaic Coulomb force. Remember we are considering here hydrogen atom that is the simplest atom with one electron and one proton.The
equation of motion is
1 e2
mv 2
=
r
4π0 r2 —-(1)
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now according to the second postulate the angular momentum of the electron
in a given orbit is quantized which means the angular momentum can have
only fixed or discrete values mathematically this means
L = mvr = n 6 h—-(2)
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now from equation (2) we get one fundamental result from simple mathematics
n6h
r = mv
—(3)
This result is of great importance when we will discuss the concept of allowed and not allowed orbits.
Equation (3) is a very familiar equation if observed carefully!!!
from equation (2) the velocity of the electron in the orbit is
v=
n6h
—(4)
mr
putting this value of v from equation (4) in equation (1) we have
mn2 6h2
1 e2
=
3
4π0 r2 —(5)
mr
from equation (5)we get the expression of the radius ras follows
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n2 6h2 4π0
–(6)
me2
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r=
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Figure 5: Bohr Model of Hydrogen atom
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From the equation (6) the information we get is
for each value of n there is a fixed orbit (only n is variable others
are constants)
as the value of n increases the value or radius increases, so with increasing
the electron orbit radius increases and the electron moves away from the
nucleus. the values on n are always eigenvalues or integral values
n = 1, 2, 3, ......
using the value of r obtained in (6) we can find the value of velocity v of the
electron
1 e2
v = 4π
–(7)
0 n6h
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5.1
Total Energy of the electron in an orbit
Potential energy ofthe electron in an orbit
5.2
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Let us calculate the total energy of the electron in the given orbit. The potential energy is due to the Coulomb attraction between positively charged
nucleus and negatively charged electron . From basics of electrostatics we
know the potential energy is defined as the work done in bringing a
charge from infinity to a point at a distance r from a given charge.
If we consider that the electron was initially at infinite distance from the
nucleus is brought at a distance r from the nucleus. The potential energy is
obtained by integrating the work done in the process.
R∞ 2
e2
—(8)
V = − r 4πe r2 dr = − 4π
0r
0
Significance of the minus sign
Kinetic energy of the electron in an orbit
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The minus sign occurs due to the fact that the Coulomb force is attractive in
nature. The positively charged nucleus and the negatively charged electron
are bound to each other by attractive force.
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Since the electron is moving in an orbit its kinetic energy is given by
KE = 12 mv 2–(9)
we take the value of velocity from equation (7) and the kinetic energy is given
by
e2
—(10)
KE = 8π
0r
summing (8) and (11) we get the total energy Eof the electron in the orbit
as
e2
e2
e2
E = KE + P E = 8π
−
=
−
4π0 r
8π0 r —(12)
0r
putting the value of r from equation (6) the total energy E is given by
e2
E=
—(13)
4π0 n2 6h2
8π0 ×
me2
Finally equation (13) becomes
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E = 32π2me
—(14).
20 n2 6h2
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Important aspects of the total energy E of the electron in an orbit
• Total energy is negative the negative sign indicates that the electron
is bound to the nucleus.
• The total energy is inversely proportional to n. This means that when
increases E decreasesbut there is negative sign. So with increasing
n the total energy approaches towards zero.
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• series limit: The situation of n = ∞ is called series limit. In this
case the total energy from equation (14) becomes zero which means the
kinetic energy is equal to potential energy. At this stage the electron
is far away from nucleus and the Coulomb force is no longer strong
enough to hold the electron. The electron is free.
• radius,angular momentum and total energy for a given value of n specifies a particular state of the atom.
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• the state n = 1 is called the ground state or fundamental state.
Energy level diagram
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The quantization of angular momentum leads to
the quantization of energy
The quantity n is called the principle quantum
number and signifies the quantized quantities
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The pictorial or graphical representation of the quantized energy
levels given by equation (14) is called energy level diagram
The enrgy of each level is evaluated from the equation (14)
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—(14).
E = 32π2me
20 n2 6h2
The energy values are shown on the left and the corresponding values of n
on the right.The energy level diagram for the Hydrogen atom is shown below .
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Figure 6: energy level diagram of hydrogen atom
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Explanation of Hydrogen atom spectra
To understand the spectrum of Hydrogen on the basis of Bohr model we need
to to keep the following points in consideration.
• The normal state of an atom is the state when the atom is in ground
state which is n = 1.
• any state with n > 1 bis called excited state
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• when an electron moves from n = 1 to any higher state n > 1 then the
atom is said to be in an excited state . An atom goes to excited state
only by absorbing energy.
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• when atom in an excited state n > 1 moves down to some lower state
then the extra energy is emitted in the form of electromagnetic radiation.
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• The wavelength of the emitted radiation depends on the initial ni and
final states nf .
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suppose there is an atom which is excited and sent to an excited state with
n=7. The atom dexcites or comes down to a low energy state by the following
steps
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• the electron comes from n = 7ton = 4
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• the electron then jumps from n = 4 to n = 2
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• finally the electron jumps from n = 2 to n = 1
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Thus there are three transitions and three electromagnetic radiations of
different wavelength are emitted.as shown in the diagram below
7.1
Frequency of radiation emitted in a transition
According to Bohr’s Postulate the frequency of radiation emitted when an
electron moves from a state of higher energy (high n) to a state of lower
energy (n) then frequency of emitted radiation is given by
Ef −Ei
—(15)
h
Where Ef and Ei are the energies of the initial and final states. The
wavelength of the emitted radiation totally depends on the energy difference
ν=
10
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Figure 7: Transitions made by an electron in an excited atom to come down
to ground state.The electron in higher state n = 7 moves down to ground
state n = 1 via three transitions.Each transition from a higher state to a
lower state is accompanied by the emission of electromagnetic radiation
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=
me4
(1
64π 2 20 6h3 n2f
− n12 )——-(18)
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Ei −Ef
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ν=
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between the final and initial states.
Let us consider that the electron was in an initial excited state n1 , the energy
of the electron in this state is given by equation (14) as
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En1 = − 32π2me
—-(16)
2
0 6h2 n21
similarly the energy of the final state is given by
4
–(17)
En2 = − 32π2me
20 6h2 n22
The frequency of radiation is then given by equation (15) according to Bohr’s
postulate
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Thus equation (18)gives the frequency of the electromagnetic radiation
emitted when an electron moves from a final state to an initial state.
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However in spectroscopic notation it is convenient to express the equation
(18) in wave number . Wave number is defined as the number of waves per
unit length and has unit of length inverse.
wave number is denoted as κ = λ1 = νc In terms of wave number the equation
(18)
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1
1
κ = 64πme
3 2 h3 c ( n2 − n2 )—(19)
0
i
f
κ = R∞ ( n12 − n12 )—-(19)
i
f
The quantity R∞ is called Rydeberg constant . The infinity subscript is
used because we have considered that the nucleus has an infinite mass.
Depending on the initial and final states where the electron makes transition there are several series(a group of lines ) in the atomic spectra of
Hydrogen .
7.1.1
Lyman series
when the final state of the electron is n = 1 state and the initial states can be
n = 2, 3, 4... then the lines observed in the spectra form the Lyman Series.
So for Lyman series nf = 1 and ni = 2, 3, 4...The equation of wave number
for Lyman series is
ν̄ = R( 112 − n12 )—(20). The lines of Lyman series appear in the ultra
violet region
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BalmerSeries
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7.1.2
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Figure 8: balmer series of Hydrogen spactra
7.1.3
Paschen Series
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Balmer series occurs when the final state is nf = 2 and the initial states are
ni = 3, 4, 5....
The Balmer series occurs inthe visible region.The first line from n = 3
is called Hα line. The line from n = 4 is called Hβ line. The wave number
equation is given by
ν̄ = R( 212 − n12 ) n = 3, 4, 5...—(21)
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Brackett Series
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The Paschen series occurs when the final state is nf = 3 and the initial states
are ni = 4, 5, 6.. The lines of Paschen series lie in the infrared region.The
wave number equation is given by
ν̄ = R( 312 − n12 ) where n = 4, 5, 6..–(22)
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Brackett series appears when the final state nf = 4 and the initial states are
ni = 5, 6, 7... The wave number equation is
ν̄ = R( 412 − n12 ) n = 5, 6, 7..–(23)
7.1.5
Pfund series
Pfund series occurs when the final state is nf = 5 and the initial states are
ni = 6, 7, 8, ... the Pfund series also occurs in the infrared region. The
wave number equation is
ν̄ = R( 512 − n12 )n = 6, 7, 8.. –(24)
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Figure 10: Pfund series
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Figure 9: Brackett series
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Nuclear mass and the significance of Rydberg
constant
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Till now we have considered that only the electron makes transitions and
there are changes in the total energy. We assumed that the nucleus has an
infinite mass with respect to the electron and has almost no role play. This
is the reason that in the last section we defined the Rydberg constant as R∞ .
However if the nucleus has finite mass then there is an effect on the energy
levels and the wavelengths of the emitted radiations.
We consider a binary system of an electron of mass m and and nucleus of
finite M mass at a distance r from each other.The nucleus and the electron
move about a common centre of mass C. The center of mass C is at a distance r1 from the nucleus and the electron is at distance r2 from the center.
Thus the nucleus and the electron can be considered to be rotating in circular
paths of radii r1 and r2 .
From the equation of center of mass
M r1 = mr2 —(8.1)
r = r1 + r2 —(8.2)
r1 = r − r2 = r − Mmr1 —-(8.3)
r1 (1 + M
)—-(8.4)
m
r1 = ( Mm
r)—(8.5)
+m
similarly
)r...(8.6)
r2 = ( MM
+m
The total angular momentum of the nucleus and the electron about the center of mass is given by
L = M r12 ω + mr22 ω(8.7)
putting the values of r1 and r2 from (8.1)and (8.2)we have
L = ( MmM
)r1 ω–(8.8)
+m
L = µr2 ω—(8.9)
the quantity µ is called reduced mass
From Bohr’s second postulate the angular momentum must be quantized
L = µr2 ω = n 6 h—(8.10)
if the nuclear mass is not considered then
mr ω = n 6 h—(8.11)
The physics is that if we consider the nucleus as a body of finite mass
then in that case the electron mass is replaced by µ subsectionEffect of nuclear mass on the emitted radiation. How the wave number of the
emitted radiation is affected by the nuclear mass The quantized energy levels
when nuclear mass is considered is given by
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4
En = − 82µen2 h2 –(8.1.1)
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Now suppose an electron jumps from initial state ni to a final state nf then
the wave number equation is given by
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1
1
ν̄ = − 8µe
2 ch3 ( n2 − n2 )—(8.1.2)
0
i
f
from equation (8.1.2) the Rydberg Constant is given by
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MZ
me4
1
RZ = 8µZe
2 ch3 = 82 ch3 × M +m = R∞ × 1+ m –(8.1.3)
Z
0
0
MZ
The subscript Z denotes the nuclear with finite mass and atomic number Z
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and R − ∞ = 8me
2 ch3 —(8.1.4)
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The general equation of the wave number of an atom with finite nuclear
mass is
ν̄ = Z 2 RZ ( n12 − n12 ) (8.1.5)
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Experimental evidences that support Bohr’s
Theory
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Ratio of mass of proton and mass of proton
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9.1
MH
RHe =
R∞
1+ Mm
—(9.1.2)
He
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The first evidence of Bohr’s theory and the significance of The Rydberg constant was proved when the ratio of mass of proton and mass of electron was
found. Following the previous formalism we can compare the Rydeberg constant for Hydrogen and Helium
RH= R∞m –(9.1.1)
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MHe = 4MH –(9.1.3)
taking the ratio of (9.1.1) and (9.1.2) and using(9.1.3)
1+ Mm
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RHe
RH
=
H
(9.1.4)
m
1+ 4M
H
RHe + R4He MmH = RH + R4H MmH – (9.1.5)(9.1.5)
RHe − RH = MmH = (RH − R4He )—(9.1.6)
m
H
–(9.1.7)
= RHe −R
R
MH
RH − He
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from the spectroscopic data available
m
1
≈ 1837
MH
this value is in exact match with the experimentally observed value.
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Experimental verification of Bohr’s Theory
The experimental verification of Bohr’s theory of quantized energy levels was
done by Franck and hertz in 1914. The experimental set up used by Franck
and Hertz is shown in the figure below The experimental set up contains the
following components
• Gas of the element to be studied and mercury vapor inside the gas tube
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• Filament C produces electrons
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• Electrons are accelerated by applying a potential between F and G
(grid)
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• Potential difference is varied from 0 to 60 volts by a potentiometer .
• P is plate where the electrons are collected
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• Plate P is kept at a small negative potential so that it can collect
electrons.
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Observations of Franck Hertz experiment
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The electrons are emitted from the cathode by thermionic emission. The
accelerating voltage speeds up the electrons. The electrons move towards G
, the grid which is at positive potential. Some of the electrons are able to
pass through the Grid G which has holes. Only those electrons can reach A
which have enough kinetic energy to overcome the retarding potential V of
the anode A. The observations are the current and the accelerating voltage.
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The observations of the experiment can be enumerated as follows
• The current I first increases with voltage at low accelerating voltages
• The current drops abruptly at 4.9eV
• The current again increases with increasing the accelerating potential
and again drops at 9.8eV
• the profile continues
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Figure 11: Frank Hertz Experiment set up
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Figure 12: observation and explanation of Franck Hertz experiment
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Figure 13: explanation of observation
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10.0.2
Explanation of the observation
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The explanation of the observation obtained in the Franck-Hertz experiment
is based on the quantized energy states in atom. The gas used in the experiment was Mercury vapor. At low accelerating voltages the electrons gain
energy and are able to pass through the positive potential grid and reach the
anode. As the accelerating potential is increased and reaches a value 4.9eV
the electron loses all this energy in elastic collision with Hg atom. The Hg
atom absorbs this energy and is excited to the first excited state.
Thus the electron is left with no energy to overcome the retarding potential
and reach A. At this stage the current drops. However the current does not
become zero as there are a large number of electrons and some of the electrons
may overcome the retarding potential and reach A. As the accelerating
voltage is further increased to 9.8 eV the electrons again undergo
inelastic collision with Hg atom and thus the second Hg atom is
excited at 9.8eV thus again a dip in current is observed at 9.8eV. As more
and more Hg atoms are excited more dips in current are observed at several
voltages. The dips in plate current occur at fixed voltages which
are same as the excitation or critical potential of Hg atoms. Thus
Franck hertz experiment confirms the existence of quantized energy levels in
Hg atoms. Franck Hertz experiment has also been performed with Neon as
shown in the figure below Definition:
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• Critical Potential:The minimum energy to raise an atom from its initial
ground state to higher excited state is called critical potential
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• Ionization potential: The minimum energy required to take out the last
valence electron from an atom is called ionization potential
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All these potentials are expressed in electron volts
Sommerfield correction and the generalized
quantum theory
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Figure 14: Franck-Hertz experiment with neon
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