Physical Chemistry - The ATOMS of Niels Bohr and Wave Particles Duality of Light and Matter_lecture4-6
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This course presents an introduction to quantum mechanics. It begins with an examination of the historical development of quantum theory, properties of particles and waves, wave mechanics and applications to simple systems — the particle in a box, the harmonic oscillator, the rigid rotor and the ...
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
ΔE
emission of a photon of frequency ν where ν = .
h
En1
hν hν
or
Absorption 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.
h
(3) Angular momentum is quantized: ! = n" where " =
2π
Angular momentum
! ! ! !
L=r×p "= L
L! !
For circular motion:
L
! ! !
L is constant if r and p are constant !
r
l = mrv is a constant of the motion ! !
p = mv
Other useful properties
1
, 5.61 Fall 2007 Lecture #4 page 2
v!
!
( )
= 2π r ⋅ ν rot = r ω rot
! !
velocity
(m/s) circumference frequency angular
(m/cycle) (cycles/s) frequency
(rad/s)
⇒ " = mvr = mr 2ω rot
Recall the moment of inertia I = ∑ mi ri2
i
2
∴ For our system I = mr
⇒ ! = I ω rot
Note: Linear motion vs. Circular motion
mass m ↔ I moment of inertia
velocity v ↔ ω rot angular velocity
momentum p = mv ↔ ! = Iω angular momentum
Kinetic energy is often written in terms of momentum:
Introduce Bohr’s quantization into the Rutherford’s planetary model.
For a 1-electron atom with
r e-
a nucleus of charge +Ze
+Ze
Ze2 n2 !2
r=
4πε 0 mv 2
⇒ r=
Z
(
4πε 0
me2
) The radius is quantized!!
!2
( 4πε ) 0
me2
≡ a0 the Bohr radius
2
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