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Atomic Physics
Chapter III : Bohr’s Model of Hydrogen Atom
Ch. # 42
1431 - 1432
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Lecture # 1
Ch# 42
All objects emit thermal radiation characterized by a continuous distribution of
wavelengths.
1- Atomic Spectra of Gases
A line spectra observed when a low-pressure gas is subject to an electric discharge, this
produce lines called EMISSION lines.
When white gas from a continuous source passes through a gas or dilute solution,
ABSPORPTION lines produces.
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Balmer series of hydrogen
Johann Jocob Balmer (1825-1898)
•
The empirical equation by Johannes
Rydberg (1854-1919):
 1 1 
 RH  2  2 

2 n 
1
n  3,4,5,...
– RH: Rydberg constant =
1.0973732 x 107 m-1.
The Balmer series of spectral
lines for atomic hydrogen.
– The measured spectral lines agree
with the empirical equation to
within 0.1%
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Balmer series
Where RH is Rydberg constant =109673732 m-1
Lyman
Pashen
Bracket
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2- Early Models of the Atom
Thomson’s model of the atom: negatively charged electrons in a
volume of continuous positive charge.
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2- Early Models of the Atom
The classical model of the nuclear
atom. Because the accelerating
electron radiates energy, the orbit
decays until the electron falls into the
nucleus.
Rutherford’s planetary model
of the atom.
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Difficulties with Rutherford’s planetary model
•
•
Cannot explain the phenomenon that
an atom emits (and absorbs) certain
characteristic
frequencies
of
electromagnetic radiation and no
others.
Predication of the ultimate collapse of
the atom as the electron plunges into
the nucleus.
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3-The Bohr Model of the Atom
Bohr proposed that the possible energy
states for atomic electrons were quantized
– only certain values were possible. Then
the spectrum could be explained as
transitions from one level to another.
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Bohr’s Quantum Model of the Atom
Q: What are Bohr’s postulates?
1. The electron moves in circular orbits around the proton under the
influence of the attractive Coulomb force.
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2. Only certain special orbits are stable (STATIONARY STATES). While in
one of these orbits, the electron does not radiate (emit) energy.  Its
total energy is constant. (electron will not spiraling into the nucleus).
3. Radiation (e.g., light) is emitted by the atom when the electron transits
“jumps” from a higher energy orbit or “state” to a lower energy orbit or
“state”.
the frequency of the emitted radiation is found to be
Ei  E f  h f
where
Ei = energy of the initial state
Ef = energy of the final state
Ei > Ef
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4. The size of the stable or “allowed” electron orbits is determined by a
“quantum condition”
h
(angular orbital momentum equal to an integral multiple of  
)
2
nh
mvr 
 n
2
n  1,2,3,
h

2
Using the Bohr’s four assumptions enable us to calculate
the allowed energy level and emission wavelength of the
hydrogen atom
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The electric potential energy
Where Ke is the Coulomb constant and the negative sign arises from the
charge -e.
The kinetic energy is given by
Thus the total energy of the atom is as follow
Form Newton’s 2nd low, the electric force
mass and its centripetal acceleration
then
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must equal to its
So the total energy can be given as
Q: Prove that the radii of the allowed orbits given as
n 2 2
rn 
mke e 2
where n  1,2,3,
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mvr  n
The following constant is called Bohr’s radius
And hence the quantization of orbit radii is given as
 rn  0.0529(n )
2
nm
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The quantization of orbit radii leads to energy quantization which is the
allowed values of energy of H atom (i.e. allowed energy levels of H
atom), is given as follow
The minimum energy required to ionize
the atom in its ground state is called the
IONIZATION ENERGRY
to completely remove an electron from
the proton’s influence = 13.6 eV for
hydrogen.
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Q: Show that the frequency and the wavelength of an emitted photon if
transits from higher (outer orbit) state f to lower (inner orbit) state i is
given by the following expressions
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Bohr extended his model for hydrogen to other elements in which all but
one electron had been removed
Z, is the atomic number of the element (the number of protons in the
nucleus)
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Q: If the electron in the hydrogen atom was 207 times heavier (a
muon), the Bohr radius would be
1.
207 Times Larger
2.
Same Size
3.
207 Times Smaller
n 2 2
rn 
mke e 2
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Q: A hydrogen atom is in its ground state. Many photons are
incident on the atom, each having an energy of 10.5 eV. The result
is that:
(a) the atom is excited to a higher allowed state
to completely remove an electron from the proton’s influence = 13.6 eV
for hydrogen.
(b) the atom is ionized
(c) the photons pass by the atom without interaction
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Electron A falls from energy level n=2 to energy level n=1 (ground
state), causing a photon to be emitted.
Electron B falls from energy level n=3 to energy level n=1 (ground
state), causing a photon to be emitted.
n=3
Which photon has more energy?
n=2
• Photon A
A
• Photon B
B
n=1
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Q: Calculate the wavelength of photon emitted when an electron
in the hydrogen atom drops from the n=2 state to the ground state
(n=1).
hf  E2  E1 3.4eV  (13.6eV)  10.2eV
Ephoton 
n=3
E2= -3.4 eV
n=2
E1= -13.6 eV
n=1
hc

hc
1240


 124nm
10.2eV 10.2
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Q: Compare the wavelength of a photon produced from a transition
from n=3 to n=2 with that of a photon produced from a transition
from n=2 to n=1.
(1)
32 < 21
(2)
32 = 21
(3)
32 > 21
E32 < E21
so
n=3
n=2
32 > 21
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n=1
. If the energy difference between the electronic states of hydrogen
atom is 214.68 kJ mol-1, what will be the frequency of light emitted when
the electron jumps from the higher to the lower energy state? (Planck's
constant = 39.79 x 10-14 kJ mol-1)
Solution
The frequency (v) of emitted light is related to the energy difference of
two levels (ΔE) as
E = 214.68 kJ mol-1, h =39.79 x 10-14 kJ mol-1
= 5.39 x 1014 s
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Lecture # 2
Ch. 42
According to quantum mechanics, each electron is described by four
quantum numbers:
1.
2.
3.
4.
Principal quantum number (n)
Angular momentum quantum number (l)
Magnetic quantum number (ml)
Spin quantum number (ms)
•The first three define the wave function for a particular electron. The
fourth quantum number refers to the magnetic property of electrons.
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A wave function for an electron in an atom is called an atomic orbital
(described by three quantum numbers—n, l, ml). It describes a region of
space with a definite shape where there is a high probability of finding the
electron.
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•Principal Quantum Number, n
•This quantum number, which refers to energy level or shell, is the one on
which the energy of an electron in an atom primarily depends. The smaller
the value of n, the lower the energy and the smaller the orbital.
•The principal quantum number can have any positive value: 1, 2, 3, . . .
•Orbitals with the same value for n are said to be in the same shell.
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•Shells are sometimes designated by uppercase letters:
Letter
n
K
1
L
2
M
3
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N
4
...
•Orbital Quantum Number, l
•Sometimes called the azimuthal quantum number, this quantum number
distinguishes orbitals within a given n (shell) having different shapes.
•It can have values from 0, 1, 2, 3, . . . to a maximum of (n – 1).
•For a given n, there will be n different values of l, or n types of subshells.
•Orbitals with the same values for n and l are said to be in the same shell
and subshell.
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•Subshells are sometimes designated by lowercase letters:
n=
1
2
3
4
l≤
Letter
0
s
1
p
2
d
3
f
...
Not every subshell type exists in every shell. The minimum value of n
for each type of subshell is shown above.
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•Magnetic Quantum Number, ml
•This quantum number distinguishes orbitals of a given n and l , that is,
of a given energy and shape but having different orientations.
•The magnetic quantum number depends on the value of l and can
have any integer value from –l to 0 to +l. Each different value
represents a different orbital. For a given subshell, there will be (2l + 1)
values and therefore (2l + 1) orbitals.
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•Let’s summarize:
•When n = 1, l has only one value, 0.
When l = 0, ml has only one value, 0.
So the first shell (n = 1) has one subshell, an s-subshell, 1s. That
subshell, in turn, has one orbital.
•When n = 2, l has two values, 0 and 1.
When l = 0, ml has only one value, 0. So there is a 2s subshell with one
orbital.
When l = 1, ml has only three values, -1, 0, 1. So there is a 2p subshell
with three orbitals.
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•When n = 3, l has three values, 0, 1, and 2.
When l = 0, ml has only one value, 0. So there is a 3s subshell with one
orbital.
When l = 1, ml has only three values, -1, 0, 1. So there is a 3p subshell
with three orbitals.
When l = 2, ml has only five values, -2, -1, 0, 1, 2. So there is a 3d
subshell with five orbitals.
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•Spin Quantum Number, ms
This quantum number refers to the two possible orientations of the spin
axis of an electron.
It may have a value of either +1/2 or -1/2.
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Q: For a hydrogen atom, determine the number of allowed states
corresponding to the principal quantum number n=2, and calculate the
energies of these states.
When n = 2
l can be 0 or 1
When l = 0 the only value that ml can have is 0,
for l = 1 , ml = can be -1,0,1
States:
one state designated as the 2s state, that is associated with the quantum numbers
n=2, l=0 and ml=0
three states, designated as the 2p state, that is associated with the quantum
numbers
n=2, l=1 , ml=-1
n=2, l=1 , ml=0
n=2, l=1 , ml=1
Because all four of these states have the same principal quantum number n=2, they
have the same energy
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Lecture # 3
Ch. 42
The Wave functions for Hydrogen
The 1s state of the hydrogen atom, ψ1s(r):
 1s ( r ) 
1
a03
• Ψ1s is spherically symmetric.
• This symmetry exists for all s states.
• The probability density for the 1s state:
 1s 
2
Radial probability density function P(r):
The probability per unit radial length of finding the
electron in a spherical shell at radius r:
P(r )dr   1s dV   1s 4r 2 dr
2
 P(r )  4r 2  1s
2
2
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1  2 r / a0
e
3
a0
e  r / a0
Radial probability density function for the hydrogen atom in the 1s state:
P(r )  4r  1s
2
2
4 r 2  2 r / a0
 3 e
a0
• The peak indicates the most probable location of the
electron.
• The peak occurs at the Bohr radius.
• The average value of r for the 1s state of hydrogen is
3/2 a0.
Electron cloud:
The charge of the electron is extended throughout a
diffuse region of space, commonly called an electron
cloud.
• This figure shows the probability density as a
function of position in the xy plane.
• The darkest area corresponds to the most probable
region.
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The 2s state of the hydrogen atom:
 2 s (r ) 
1
4 2
1
 
 a0 
3
2

r 
 2  e r / 2 a0
a0 

• ψ2s depends only on r and is
spherically symmetric.
• The radial probability density
for the 2s state has two
peaks.
• The highest value of P
corresponds to the most
probable value (r ≈5a0).
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Quantum numbers of the hydrogen atom
42.6 Physical interpretation of the quantum numbers
I. Principle quantum number n :
Determines the energy of an atomic state.
II. Orbital quantum number l :
1) In the Bohr model, the angular momentum of the electron is restricted to
L=mevr = n ħ.
2) According to quantum mechanics, an atom in a state with principle quantum
number n
can take on the following discrete orbital angular momentum:
L  l (l  1)  l  0,1,2,3,, n 1
L can equal zero, which causes great difficulty when attempting to apply classical
mechanics to this system.
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III. Orbital magnetic quantum number ml:
• The atom possesses an orbital angular momentum L. A magnetic moment m
exists due to this angular momentum. (m  IA=(-e/2m)L).
• There are distinct directions allowed for the magnetic moment vector m with
respect to the magnetic field vector B.
• Because the magnetic moment m of the atom is related to the angular
momentum vector L, the discrete direction of m translates into the fact that the
direction of L is quantized.
• Therefore, Lz, the projection of L along the z axis, can have only discrete
values. The orbital magnetic quantum number ml specifies the allowed values
of the z component of orbital angular momentum:
Lz  ml  ml  l ,l  1,, l
Space quantization: The quantization of the possible orientations of L with
respect to an external magnetic field.
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Vector model of space quantization:
• L can never be parallel or antiparallel to B. (Lz < L).
• L lies anywhere on the surface of a cone that makes an
angle θ with the z axis. The angle θ is also quantized:
cos  
ml
Lz

L
l (l  1)
Zeeman effect: the splitting of
spectral lines in a strong
magnetic field.
Figure: The upper level has l = 1
and splits into three different
levels corresponding to the three
different directions of m.
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IV. Spin magnetic quantum number ms:
The fourth quantum number, spin magnetic quantum number ms, does not come
from the Schrödinger equation.
Electron spin:
• Only two directions exist for electron spins. The
electron can have spin up (a) or spin down (b).
• In the presence of a magnetic field, the energy of the
electron is slightly different for the two spin directions.
This produces doublets in the spectra of some gases.
• The electron cannot be considered to be actually spinning. The experimental
evidence supports that the electron has some intrinsic angular momentum
that can be described by ms.
• Dirac showed the electron spin from the relativistic properties of the
electron.
• Stern-Gerlach experiment: A beam of silver atoms is split in two by a
nonuniform magnetic field.
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Spin angular momentum:
• Electron spin can be described by a single quantum number s, whose value
can only be s = 1/2. The magnitude of the spin angular momentum S is
S  s( s  1) 
3

2
• The spin angular momentum can have two
orientations relative to the z axis, specified by
the spin quantum number ms = ± 1/2:
ms = + 1/2 corresponds to the spin up case;
ms = - 1/2 corresponds to the spin down case.
The z component of spin angular momentum is
1
S z  ms    
2
The spin magnetic moment: μ spin  
e
S.
me
The z component of the spin magnetic moment:
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μspin,z  
e
.
2me
Lecture # 4
Ch. 42
Exclusion principle and the periodic table
42.7 The exclusion principle and the periodic table
The four quantum numbers, n, l, ml, ms can be used to describe all the electronic
states of an atom regardless of the number of electrons in its structure.
Question: How many electrons can be in a particular quantum states?
Pauli’s exclusion principle: No two electrons can ever be in the same quantum
state; Therefore, no two electrons in the same atom can have the same set of
quantum numbers.
Sequence of filling subshells: Once a subshell is filled, the next electron goes into
the lowest-energy vacant state.
Orbital: The atomic state characterized by the quantum numbers n, l and ml.
From the exclusion principle, only two electrons can be present in any orbital. One
electron will have spin up and one spin down.
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Q: Which of the following are permissible sets of quantum numbers?
A.
n = 4, l = 4, ml = 0, ms = ½
B.
n = 3, l = 2, ml = 1, ms = -½
C.
n = 2, l = 0, ml = 0, ms = ³/²
D.
n = 5, l = 3, ml = -3, ms = ½
(A)
Not permitted. When n = 4, the maximum value of l is 3.
(B)
Permitted.
(C)
Not permitted; ms can only be +½ or –½.
(B)
Permitted.
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Question: How are the electrons aligned in an orbital?
Hund’s rule: When an atom has orbitals of equal energy, the order in which they
are filled by electrons is such that a maximum number of electrons have unpaired
spins. (Exceptions exist).
Electronic configuration:
The filling of the electronic
states must obey both Pauli’s
exclusion principle and Hund’s
rule.
The periodic table:
An arrangement of the atomic
elements according to their
atomic masses and chemical
similarities.
The
chemical
behavior of an element
depends on the outermost
shell that contains electrons.
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