Download Lecture 1 (Bohr model of the atom)

PA 1140 Waves and Quanta Unit 4: Atoms and Nuclei
PA1140
Waves and Quanta
Unit 4: Atoms and Nuclei
Dr Matt Burleigh (S4)
Tipler, Chapters 36 & 40
PA 1140 Waves and Quanta Unit 4: Atoms and Nuclei
Unit 4
Atoms and Nuclei (chapters 36 & 40).
•
•
•
•
Bohr theory
Radioactivity, fission and fusion
Atomic size and shape
Mass and binding energy
Lecture course slides can be seen at:
•
http://www.star.le.ac.uk/mbu/lectures.html
PA 1140 Waves and Quanta Unit 4: Atoms and Nuclei
• The 100 odd different stable nuclei make up the matter around us
• Their physical and chemical properties are determined by laws which govern the
behaviour of the electrons surrounding the nucleus
• The arrangement of the electrons determines their emission and absorption line
characteristics
• The spacings and wavelengths of the lines are characteristic of each element
Emission lines
Absorption lines
This is an interacting binary star system called a cataclysmic variable. The
small white dwarf star is pulling matter off its brown dwarf companion,
down its magnetic field lines onto its surface.
The absorption lines in the spectrum (left) are H gas in the white dwarf
atmosphere. The H emission lines are from the bright spot where the
accreting matter hits the white dwarf.
PA 1140 Waves and Quanta Unit 4: Atoms and Nuclei
PA 1140 Waves and Quanta Unit 4: Atoms and Nuclei
Ch. 36
Atomic Spectra
In 1884 Balmer found the wavelengths of lines in the visible
spectrum of H can be represented by
(364.6nm)m 2
l=
(m 2 - 4)
Rydberg & Ritz gave a more
general expression applicable to the
spectra of other elements
æ1 1ö
=R ç 2 - 2 ÷
l è n2 n1 ø
1
Where m=3,4,5….
Where n1 & n2 are integers
and n1 > n2. R is the Rydberg
constant
PA 1140 Waves and Quanta Unit 4: Atoms and Nuclei
• The Rydberg-Ritz formula works mathematically, but why?
• In 1913 Niels Bohr proposed a pre-QM model to explain the spectra
emitted by H atoms
• In this theory electrons are considered to be point objects in orbit
around the nucleus
• It gives a 1st order explanation of the spectral lines, & remains a
useful treatment of electron behaviour
• Next year you will gain enough QM knowledge to see how the true
quantum description of such systems works
PA 1140 Waves and Quanta Unit 4: Atoms and Nuclei
Ch. 36
The Bohr atom
The centripetal force acting
on the orbiting electron is
the electrostatic force of
attraction (Coulomb force)
(1)
PA 1140 Waves and Quanta Unit 4: Atoms and Nuclei
• The energy of the orbiting electron is thus:
1 kZe2
E =2 r
• But classical EM theory says such an atom is unstable, bcse
the electron must accelerate when moving in a circle &
radiate EM energy
• Thus the orbit would quickly collapse as the electron
spiralled into the nucleus as energy is radiated away
• Bohr proposed a way out of this difficulty with a set of
postulates
•
He had no proof, just a starting point of assumptions!
(2)
PA 1140 Waves and Quanta Unit 4: Atoms and Nuclei
Bohr’s postulates
(1) Bohr proposed that certain “magical” circular orbits existed, called “stationary
states”, which did not radiate, and that electrons could only exist in these
states, with radiation occurring when they made the transition from one to the
other.
(2) He also postulated that the frequency of the radiation from spectral lines was
determined by energy conservation during transitions from one stationary
state to the other. i.e.
Ei - E f
From E=hf, where h is Planck’s constant
f =
h
Combining this with the expression for electron energy gives
Ei - E f 1 kZe2 æ 1 1 ö
f=
=
ç - ÷
h
2 h è r2 r1 ø
(3)
(3). Trial and error led Bohr to his third postulate, that angular momentum is
quantized, specifically that
nh
L = mvr =
=n ,
2p
n =1, 2,...
h
=
2p
n is the quantum number of the state
PA 1140 Waves and Quanta Unit 4: Atoms and Nuclei
n2 2
r=
mkZe2
Radius of orbit:
(4)
Frequency of line:
mk 2 e4 æ 1 1 ö
f =Z
- 2÷
3 ç 2
4p è n2 n1 ø
2
f =c/l
so
æ1 1ö
mk 2 e 4 æ 1 1 ö
=Z
- 2 ÷ = Rç 2 - 2 ÷
3ç 2
l
4p c è n2 n1 ø
è n2 n1 ø
1
2
Where the Rydberg constant, R, is:
mk 2e4
7
-1
R=
=1.096776
´10
m
4p c 3
Energy of
orbit n
&
mk 2 e4 Z 2
2 Eo
En = =
-Z
,
2
2
2
2
n
n
Z =1
n =1, 2,3,...