Physics of the Atom
Ernest
Rutherford
Niels Bohr
W. Ubachs – Lectures MNW-1
The Nobel Prize in Chemistry 1908
"for his investigations into the
disintegration of the elements,
and the chemistry of radioactive substances"
The Nobel Prize in Physics 1922
"for his services in the investigation
of the structure of atoms and
of the radiation emanating from them"
Early models of the atom
atoms : electrically neutral
they can become charged
positive and negative charges are around
and some can be removed.
popular atomic model
“plum-pudding” model:
W. Ubachs – Lectures MNW-1
Rutherford scattering
Rutherford did an experiment that showed that the
positively charged nucleus must be extremely small
compared to the rest of the atom.
Result from
Rutherford
scattering
2 ⎞2
dσ ⎛⎜ 1 Zze ⎟
1
=⎜
dΩ ⎝ 4πε 0 4 K ⎟⎠ sin 4 (θ / 2 )
Applet for doing the experiment:
http://www.physics.upenn.edu/courses/gladney/phys351/classes/Scattering/Rutherford_Scattering.html
W. Ubachs – Lectures MNW-1
Rutherford scattering
the smallness of the nucleus
the radius of the
nucleus is 1/10,000
that of the atom.
the atom is mostly
empty space
Rutherford’s atomic model
W. Ubachs – Lectures MNW-1
Atomic Spectra:
Key to the Structure of the Atom
A very thin gas heated in a discharge tube emits
light only at characteristic frequencies.
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Atomic Spectra:
Key to the Structure of the Atom
Line spectra: absorption and emission
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The Balmer series in atomic hydrogen
The wavelengths of electrons emitted from
hydrogen have a regular pattern:
Johann Jakob Balmer
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Lyman, Paschen and Rydberg series
the Lyman series:
the Paschen series:
Janne Rydberg
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The Spectrum of the hydrogen Atom
A portion of the complete spectrum of hydrogen is
shown here. The lines cannot be explained by
classical atomic theory.
W. Ubachs – Lectures MNW-1
The Bohr Model
Solution to radiative instability of the atom:
1. atom exists in a discrete set of stationary
states
no radiation when atom is in such state
2. radiative transitions Æ quantum jumps
between levels
hν =
hc
λ
= Ei − E f
3. angular momentum is quantized
.
These are ad hoc hypotheses by Bohr, against intuitions of classical physics
W. Ubachs – Lectures MNW-1
The Bohr Model: derivation
An electron is held in orbit by the Coulomb force: (equals centripetal force)
Fcentr = FCoulomb
mv 2
Ze 2
=
rn
4πε 0 rn2
Solve this equations for r:
Impose the quantisation condition:
W. Ubachs – Lectures MNW-1
rn =
Ze 2
4πε 0 mv 2
L = mvrn = nh
The Bohr Model: derivation
Solve: the size of the orbit is quantized
(and we know the size of an atom !)
.
Size of an atom:
W. Ubachs – Lectures MNW-1
0.5 Å
The Bohr Model: derivation
The energy of a particle in orbit
En = Ekin + E pot
2
1 2
Ze 2
⎛ 1 ⎞ Ze
= mv −
= ⎜ − 1⎟
2
4πε 0 rn ⎝ 2 ⎠ 4πε 0 rn
So
2
⎛ Z 2 ⎞⎛ e 2 ⎞ m
1 Ze
⎟
En = −
= −⎜⎜ 2 ⎟⎟⎜⎜
⎟ 2h 2
πε
2 4πε 0 rn
4
n
0
⎝
⎠⎝
⎠
2
Define
⎛Z2 ⎞
En = −⎜⎜ 2 ⎟⎟ R∞
⎝n ⎠
With the Rydberg constant
W. Ubachs – Lectures MNW-1
2
⎛ e 2 ⎞ me
⎟
R∞ = ⎜⎜
2
⎟
⎝ 4πε 0 ⎠ 2h
EXTRA
Reduced mass in the old Bohr model Æ the one particle problem
Relative coordinates:
r r r
r = r1 − r2
Centre of Mass
r
r
mr1 + Mr2 = 0
Position vectors:
r
r1 =
M r
r
m+M
r
r2 = −
W. Ubachs – Lectures MNW-1
m r
r
m+M
Velocity vectors:
r
v1 =
M r
v
m+M
r
v2 = −
M r
v
m+M
Relative velocity
r
r dr
v=
dt
EXTRA
Reduced mass
μ=
mM
m+M
Centripetal force
m1v12 m2v22 μv 2
F=
=
=
r1
r2
r
Kinetic energy
1
1
1
K = m1v12 + m2v22 = μv 2
2
2
2
Angular momentum
L = m1v1r1 + m2v2 r2 = μvr
W. Ubachs – Lectures MNW-1
Quantisation of angular momentum:
L = μvr = n
h
= nh
2π
These are the equations for a
single (reduced) particle
EXTRA
Result for the one-particle problem
Rewrite:
Quantisation of radius in orbit:
rn =
Z2 ⎛ μ ⎞1 2
En = − 2 ⎜⎜ ⎟⎟ α me c 2
n ⎝ me ⎠ 2
n 4πε 0h
n me
a0
=
2
Z e μ
Z μ
2
2
2
dimensionless energy
Energy levels in the Bohr model:
α≈
Z2 μ
En = − 2
R∞
n me
μ=
Isotope effect
W. Ubachs – Lectures MNW-1
mM
m+M
1
137
For all atoms:
μ ≈ me
Calculate the isotope shift on Lyman-α:
Optical transitions in The Bohr Model
The lowest energy level
is called the ground
state; the others are
excited states.
Calculate wavelengths
and frequencies
λ=
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c
hc
=
f E2 − E1
Quantum states and kinetics
Hydrogen at 20°C.
Estimate the average kinetic energy of whole hydrogen atoms
(not just the electrons) at room temperature, and use the result to
explain why nearly all H atoms are in the ground state at room
temperature, and hence emit no light.
3
K = k BT = 6.2 × 10 − 21 J = 0.04eV
2
Gas at elevated temperature – probability exited state is populated:
Pn (T ) = e
− En / kT
Energy in gas:
E =
∑ En e
n
∑e
n
W. Ubachs – Lectures MNW-1
− En / kT
− En / kT
Another way of looking at the quantisation in the atom
de Broglie’s Hypothesis
Applied to Atoms
Electron of mass m has a wave nature
Electron in orbit: a standing wave
2πrn = nλ
Substitution gives the quantum condition
nh
L = mvrn =
2π
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de Broglie’s Hypothesis
Applied to Atoms
These are circular standing
waves for n = 2, 3, and 5.
Standing waves do not radiate;
Interpretation:
electron does not move
(no acceleration)
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