Bohr's model

BOHR’S ATOMIC
MODEL & ITS
APPLICATIONS
Chapter # 2
Atomic Structure
Prepared by: Sidra Javed

The Fundamental Particles
Particle
Charge
(Coulomb)
Relative
Charge
Mass
(Kg)
Mass
(a.m.u)
Found
in:
Proton
+ 1.602 x
10-19 +1
1.6727 x
10-27 1.0073 Nucleus
Neutron 0 0
1.6750 x
10-27 1.0087 Nucleus
Electron
-1.602 x
10-19 -1
9.1095 x
10-31
5.4858 x
10-4
Outside
Nucleus

After the Gold Foil Experiment
Rutherford’s proposed the
Planetary Model of Atom.
Just like the solar system, the
Nucleus lies in the center of the
atom and electron revolves
around it in their orbits

But!!
The Revolution of the electron in a circular
orbit is not expected to be stable.
+ --

Any particle in a circular orbit would undergo
acceleration.
During acceleration, charged particles would
radiate energy (hν).
+ --

-
+
Thus, the revolving electron would lose
energy and finally fall into the nucleus.
If this were so, the atom should be highly
unstable and hence matter would not exist
in the form that we know.

As the electron spiral inwards, their angular
velocities and frequency would change
continuously and so will the frequency of
the energy emitted.
Thus, they would emit a continuous
spectrum, in contrast to the line spectrum
actually observed.

Neil Bohr, a Danish Physicist studied
in Rutherford Laboratory since
1912.
He successfully explained the
spectrum of hydrogen atom
and presented Bohr’s
Atomic model.
He was awarded Nobel
Prize in 1922

Bohr’s Model of Atom
1- Electrons revolves
around the nucleus
in definite energy
levels called orbits
or shells in an atom
without radiating
energy.
2- As long as an
electron remain in a
shell it never gains
or losses energy.
+
-
-
-
n = 3
n = 2
n = 1

3- The gain or loss of energy occurs within
orbits only due to jumping of electrons
from one energy level to another energy
level.
+
-
-
e
+hν
(Energy )
e
-hν
(Energy )
n=1
n=2

4- The angular momentum (mvr) of an
electron is equal to nh/2π.
The angular momentum of an orbit
depends upon its quantum number
(n) and it is integral multiple of the
factor h/2π
i.e. mvr = nh/2 π
Where,
n = 1, 2, 3, 4,….

Applications Of Bohr’s
Atomic Model
• Derivation of Radius of an Orbit of an
atom
• Derivation of Energy of an Orbit
• Derivation of Wave Number (ū)

Derivation of Radius of an Orbit
of an Atom
Consider an atom having
an electron e- moving
around the nucleus
having charge Ze where
Z is the atomic number.
Let m be the mass, r the
radius of the orbit and v,
the velocity of the
revolving electron.
Ze
e
r
v
m

According to Coloumb’s law, the
electrostatic force of attraction b/w nucleus
and electron :
Where is the vacuum permittivity
constant ( = 8.84 x 10-12 C2/J.m)
22
21
4
.
4 r
eZe
r
qq
F
oo
c


2
2
4 r
Ze
o

o
o

Centrifugal force acting on the electron =
The two forces are equal and balance each
other
r
mv2
)2..(..........
4
)1.(..........
4
4
2
2
2
2
2
22
mv
Ze
r
r
Ze
mv
r
Ze
r
mv
o
o
o







According to Bohr’s postulate:
Put value in eq(2).
)4(..........
4
2
)3......(..........
2
222
22
2
rm
hn
v
mr
nh
v
nh
mvr







)5...(..........
1
1
4
4
2
22
222
22
2
22
2222
mZe
hn
r
hnmrZe
hn
mrZe
hn
rm
m
Ze
r
















For Hydrogen atom, Z = 1
anr
me
hn
r
me
hn
r
mZe
hn
r
o2
2
22
2
22
2
22
)6..(..........
)1(











Where is a constant quantity,
o
o
o
o
anr
a
mxa
me
h
a
.
529.0
10529.0
2
10
2
2









a
o

So,
Therefore radius of orbits having n = 1, 2,
3… are as follows:
When n=1:
When n=2:
When n=3:
When n=4:
ο
2
4 A4.8529.0)4( r
o
nr A529.02

ο
2
3 A75.4529.0)3( r
ο
2
2 A11.2529.0)2( r

A529.0529.0)1( 2
1 r

Derivation of Energy of an
Electron in an Orbit
The energy of an electron in an orbit is the
sum of its potential and kinetic energy
r
Ze
mvE
r
Ze
mvE
EPEKE
o
T
o
T
T


42
1
)7......(
42
1
..
2
2
2
2










From eq(1)
Putting value in eq (7)
r
Ze
mv 
4
2
2

)8.....(..........
8
1
2
1
4
48
44.2
2
2
22
22
r
Ze
E
r
Ze
E
r
Ze
r
Ze
E
r
Ze
r
Ze
E
n
n
n
T




















Now putting the value of r from eq(5) into
eq(8),
For Hydrogen atom; Z=1
)9.(..........
8
8
222
42
22
22
hn
meZ
E
hn
mZeZe
E
o
n
n

















222
4
222
4
1
8
8
nh
me
E
hn
me
E
o
n
o
n



But
The negative sign indicated Decrease in
energy of the electron.
18
2
2
18
18
22
4
10178.2where
..(10)J.........
1
10178.2
J10178.2
8













k
n
k
E
n
E
h
me
n
n
o

For 1 mol of electron, multiply by Avogadro’s
No.
This energy is associated with 1.008 gram-
atoms of hydrogen.
KJ/mol
1
315.1313
KJ/mol
1000
1002.6
J/mol1002.6
2
23
2
23
2






















n
nE
n
k
nE
n
k
nE

If n=1, 2, 3,…. then;
The first energy level when n=1 is called
Ground state of H atom. All others are called
Excited states.
1
2
2
1 315.1313
1
1
315.1313 






 kJmolE
1
2
2
2 32.328
2
1
315.1313 






 kJmolE
1
2
2
3 92.145
3
1
315.1313 






 kJmolE
1
2
2
4 08.82
4
1
315.1313 






 kJmolE
1
2
2
5 53.52
5
1
315.1313 






 kJmolE

The first energy level when n=1 is called
Ground state of H atom. All others are
called Excited states.
1
1 315.1313 
 kJmolE
1
2 32.328 
 kJmolE
1
3 92.145 
 kJmolE
1
4 08.82 
 kJmolE
1
5 53.52 
 kJmolE
n=1
n=4
n=5
n=2
n=3

Frequency of Radiation emitted
by an Electron
From eq(9)
Let,
E1=energy of orbit n1
E2=energy of orbit n2
)9.(..........
8 222
42
hn
meZ
E
o
n



22
2
2
42
2
8 hn
meZ
E
o


22
1
2
42
1
8 hn
meZ
E
o



For H atom, Z=1












 




2
2
2
1
22
42
22
1
2
42
22
2
2
42
12
11
8
88
nnh
meZ
E
hn
meZ
hn
meZ
E
EEE
o
oo


)11(..........
11
8 2
2
2
1
22
4







nnh
me
E
o

Here,
But according to Planck’s Quantum Theory,
)12.(..........
11
1018.2
J1018.2
8
2
2
2
1
18
18
22
4










nn
E
h
me
o









2
2
2
1
18 11
1018.2
nn
h
hE



Again
or,






 2
2
2
1
22
4
11
8 nnh
me
h
o

)13........(secCycles.orHz
11
8
1
2
2
2
1
32
4








nnh
me
o

Hz
11
1029.3 2
2
2
1
15







nn
x

Derivation of Wave number
Where c is the velocity of Light
Putting value of ν from eq(13) into eq(14)
)14....(.......... c
)15...(..........
11
8
11
8
2
2
2
1
32
4
2
2
2
1
32
4














nnch
me
nnh
me
c
o
o





Putting values of constants, we get a factor
called Rydberg’s Constant, R.
17
32
4
m1009678.1
8

 R
ch
me
o
)16...(..........
11
2
2
2
1







nn
R

Defects of Bohr’s Atomic Model
1- According to Bohr, the radiation results
when an electron jumps from one energy
orbit to another energy orbit, but he did
not explained how this radiation occurs.

2- Bohr’s theory explained the existence of
various lines in H-spectrum, but it
predicted that only a series of lines exist.
Later on it was realized that the spectral
lines that had been thought to be a
single line was actually a collection of
several lines very close to each other.

3- Bohr’s theory successfully explained
the observed spectra for H – atom and
similar ions (He+1 , Li+2 , Be+3 etc) but it
can not explained the spectra for poly
electron atoms.
Hydrogen
1p, 1e
Helium
ion
2p, 1e
+1
Beryllium
Ion
4p, 1e
+3
Lithium
Ion
3p, 1e
+2

4- If a substance which gives line emission
spectrum is placed in a magnetic field, the lines
of the spectrum get split up into a number of
closely spaced lines. This phenomenon is known
as Zeeman effect. Bohr’s theory has no
explanation for this effect.

5- If a substance which gives line
emission spectrum is placed in an external
electric field, the lines of the spectrum get
split up into a number of closely spaced
lines. This phenomenon is known as Stark
effect. Bohr’s theory has no explanation
for this effect as well.

6- Bohr suggested circular orbits of
electron around the nucleus of H – atom
but later it was proved that the motion of
electron is not in a single plane, but takes
place in three dimensional space.

7- Bohr’s assumes that an electron in an atom is located at
a definite distance from the nucleus and is revolving
round it with definite velocity i.e. it has a fixed
momentum.
This idea is not in agreement with Heisenberg’s
uncertainty principle which states that it is impossible
to determine the exact position and momentum of a
particle simultaneously with certainty.

The End

Bohr's model

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