Post on 26-Apr-2018
W. Ubachs – Lectures MNW-1
Physics of the Atom
The Nobel Prize in Physics 1922"for his services in the investigation
of the structure of atoms and of the radiation emanating from them"
Niels Bohr
The Nobel Prize in Chemistry 1908"for his investigations into the disintegration of the elements,
and the chemistry of radioactive substances"
ErnestRutherford
W. Ubachs – Lectures MNW-1
atoms : electrically neutralthey can become chargedpositive and negative charges are aroundand some can be removed.
popular atomic model “plum-pudding” model:
Early models of the atom
W. Ubachs – Lectures MNW-1
Result fromRutherfordscattering
Applet for doing the experiment:http://www.physics.upenn.edu/courses/gladney/phys351/classes/Scattering/Rutherford_Scattering.html
( )2/sin1
441
4
22
0 θπεσ
⎟⎟⎠
⎞⎜⎜⎝
⎛=
Ω KZze
dd
Rutherford did an experiment that showed that the positively charged nucleus must be extremely small compared to the rest of the atom.
Rutherford scattering
W. Ubachs – Lectures MNW-1
the radius of the nucleus is 1/10,000 that of the atom.
the atom is mostly empty space
Rutherford scatteringthe smallness of the nucleus
Rutherford’s atomic model
W. Ubachs – Lectures MNW-1
A very thin gas heated in a discharge tube emits light only at characteristic frequencies.
Atomic Spectra: Key to the Structure of the Atom
W. Ubachs – Lectures MNW-1
Line spectra: absorption and emission
Atomic Spectra: Key to the Structure of the Atom
W. Ubachs – Lectures MNW-1
The wavelengths of electrons emitted from hydrogen have a regular pattern:
The Balmer series in atomic hydrogen
Johann Jakob Balmer
W. Ubachs – Lectures MNW-1
the Lyman series:
the Paschen series:
Lyman, Paschen and Rydberg series
Janne Rydberg
W. Ubachs – Lectures MNW-1
A portion of the complete spectrum of hydrogen is shown here. The lines cannot be explained by classical atomic theory.
The Spectrum of the hydrogen Atom
W. Ubachs – Lectures MNW-1
The Bohr Model
Solution to radiative instability of the atom:1. atom exists in a discrete set of stationarystatesno radiation when atom is in such state
2. radiative transitions quantum jumpsbetween levels
3. angular momentum is quantized
fi EEhch −==λ
ν
These are ad hoc hypotheses by Bohr, against intuitions of classical physics
.
W. Ubachs – Lectures MNW-1
An electron is held in orbit by the Coulomb force: (equals centripetal force)
The Bohr Model: derivation
hnmvrL n ==
Coulombcentr FF =
20
22
4 nn rZe
rmv
πε=
Solve this equations for r:
Impose the quantisation condition:
20
2
4 mvZern πε
=
W. Ubachs – Lectures MNW-1
The Bohr Model: derivation
Solve: the size of the orbit is quantized(and we know the size of an atom !)
.
Size of an atom: 0.5 Å
W. Ubachs – Lectures MNW-1
The Bohr Model: derivation
2
2
0
2
2
2
0
2
24421
h
menZ
rZeE
nn ⎟
⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛−=−=
πεπε
nnpotkinn r
Zer
ZemvEEE0
2
0
22
41
21
421
πεπε⎟⎠⎞
⎜⎝⎛ −=−=+=
2
2
0
2
24 h
emeR ⎟⎟⎠
⎞⎜⎜⎝
⎛=∞ πε
The energy of a particle in orbit
∞⎟⎟⎠
⎞⎜⎜⎝
⎛−= R
nZEn 2
2
So
Define
With the Rydberg constant
W. Ubachs – Lectures MNW-1
Reduced mass in the old Bohr model the one particle problem
Relative coordinates:
21 rrr rrr −=
Centre of Mass
021 =+ rMrm rr
Position vectors:
rMm
Mr rr
+=1
rMm
mr rr
+−=2
Velocity vectors:
vMm
Mv rr
+=1
vMm
Mv rr
+−=2
Relative velocity
dtrdvr
r =
EXTRA
W. Ubachs – Lectures MNW-1
Kinetic energy
2222
211 2
121
21 vvmvmK μ=+=
Reduced mass
MmmM+
=μ
Angular momentum
vrrvmrvmL μ=+= 222111
Centripetal force
rv
rvm
rvmF
2
2
222
1
211 μ
===
EXTRA
Quantisation of angular momentum:
hnhnvrL ===π
μ2
These are the equations for asingle (reduced) particle
W. Ubachs – Lectures MNW-1
EXTRA
Quantisation of radius in orbit:
0
2
2
20
2 4 amZn
eZnr e
n μμπε
==h
Result for the one-particle problem
Energy levels in the Bohr model:
∞−= Rmn
ZEe
nμ
2
2
Rewrite:
222
2
21 cm
mnZE e
en αμ
⎟⎟⎠
⎞⎜⎜⎝
⎛−=
dimensionless energy
1371
≈α
Isotope effectem≈μFor all atoms:
Calculate the isotope shift on Lyman-α:
MmmM+
=μ
W. Ubachs – Lectures MNW-1
The lowest energy level is called the ground state; the others are excited states.
Optical transitions in The Bohr Model
Calculate wavelengthsand frequencies
12 EEhc
fc
−==λ
W. Ubachs – Lectures MNW-1
Quantum states and kinetics
eVJTkK B 04.0102.623 21 =×== −
kTEn
neTP /)( −=
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.
Gas at elevated temperature – probability exited state is populated:
Energy in gas:
∑
∑−
−
=
n
kTE
kTE
nn
n
n
e
eEE /
/
W. Ubachs – Lectures MNW-1
de Broglie’s Hypothesis Applied to Atoms
λπ nrn =2Electron of mass m has a wave nature Electron in orbit: a standing wave
Substitution gives the quantum condition
π2nhmvrL n ==
Another way of looking at the quantisation in the atom