5.61 Fall 2007
Lecture #4
page 1
The ATOM of NIELS BOHR
Niels Bohr, a Danish physicist who established the Copenhagen school.
(a) Assumptions underlying the Bohr atom
(1)
Atoms can exist in stable “states” without radiating. The states have
discrete energies En, n = 1, 2, 3,..., where n = 1 is the lowest energy
state (the most negative, relative to the dissociated atom at zero
energy), n = 2 is the next lowest energy state, etc. The number “n” is
an integer, a quantum number, that labels the state.
(2)
Transitions between states can be made with the absorption or
emission of a photon of frequency ν where ν =
ΔE
.
h
En1
hν
Absorption
or
hν
Emission
En2
These two assumptions “explain” the discrete spectrum of atomic vapor
emission. Each line in the spectrum corresponds to a transition between two
particular levels. This is the birth of modern spectroscopy.
(3)
Angular momentum is quantized:
! = n"
where " =
L!
!
L
Angular momentum
! ! !
L=r×p
!
"= L
For circular motion:
!
!
!
L is constant if r and p are constant
h
2π
!
r
l = mrv is a constant of the motion
Other useful properties
1
!
!
p = mv
5.61 Fall 2007
Lecture #4
v!
velocity
(m/s)
(
" = mvr = mr 2ω rot
I = ∑ mi ri2
Recall the moment of inertia
For our system
⇒
! = I ω rot
Note:
)
= 2π r ⋅ ν rot = r ω rot
!
!
!
angular
circumference frequency
frequency
(m/cycle)
(cycles/s)
(rad/s)
⇒
∴
page 2
I = mr
i
2
Linear motion
vs.
mass
m
↔
I
moment of inertia
velocity
v
↔
ω rot
! = Iω
angular velocity
momentum p = mv
Circular motion
↔
angular momentum
Kinetic energy is often written in terms of momentum:
1 m2 r 2 v 2 ! 2
K.E. =
=
2 mr 2
2I
1 2 p2
K.E. = mv =
2
2m
Introduce Bohr’s quantization into the Rutherford’s planetary model.
For a 1-electron atom with
a nucleus of charge +Ze
r=
Ze2
4πε 0 mv 2
⇒
r=
+Ze
n2
!2
4πε 0
Z
me2
(
( 4πε )
0
)
r e-
The radius is quantized!!
!2
≡ a0 the Bohr radius
me2
2
5.61 Fall 2007
Lecture #4
For H atom with n = 1, r = a
0
page 3
= 5.29x10-11 m = 0.529 Å (1 Å = 10 -10 m)
Take Rutherford’s energy and put in r,
1 Ze2
E=−
2 4πε 0 r
1 Z 2 me4
En = − 2
n 8ε 02 h2
⇒
Energies are quantized!!!
For H atom, emission spectrum
(
ν cm
Rydberg formula !
with
-1
)
1⎞
me4 ⎛ 1
=
−
= 2 3 ⎜ 2 − 2⎟
hc
hc 8ε 0 h c ⎝ n1 n2 ⎠
En2
En1
me4
R = 2 3 = 109,737 cm -1
8ε 0 h c
Measured value is 109,678 cm-1 (Slight difference due to model that gives
nucleus no motion at all, i.e. infinite mass.)
3