Download 1. dia

THE STRUCTURE OF THE ATOMS/QUANTUM NUMBERS
September 2014
UP, Medical School, Dept. Biophysics
ATOMOS = INDIVISIBLE
‘All that exists are atoms and empty space;
everything else is merely thought to exist.’
Democritus, 415 B.C.
1
THOMSON MODEL (1902)
Joseph John Thomson
1897 - electron
„Plum pudding”
THE ATOM MODELS
Atoms are stable
Their chemical properties show periodicity (Mendeleev 1869)
After excitation they emit light, and their emission spectra is linear
Johann Jacob Balmer’s
empirical formula (1885):
1 1 
 R  2 

4 n 
1
n: 3,4,5……
R: Rydberg constant
(R = 10 973 731.6 m-1)
2
RUTHERFORD MODEL (1911)
Ernest Rutherford
RUTHERFORD’S CONCLUSIONS
1. The majority of matter is „empty space”!
2. The positive charge is concentrated into a tiny space (nucleus ~10 -15 m).
3. Electrons are revolving around the nucleus, like planets around the Sun.
3
BOHR’S MODEL
Niels Bohr
Bohr’s postulates:
1. Electrons in an atom can only have defined orbits. The formula defining
the radius of the allowed orbits is:
L  mrv  n
h
2
Stationary wave!
h
2r  n  n
mv
2.
When the electron jumps from one allowed orbit to another, the energy
difference of the two states is emitted as a photon with the energy of hν:
h  f  E2  E1
h  6.6 10 34 Js
Planck constant
THE CONCLUSIONS OF THE BOHR’S MODEL
1. Radius of the 1st orbit: r1 = 5.3 ·10-11 m (Bohr-radius)
r2 = 4r1, r3 = 9r1…..
rn = n 2 r1
2. Energy of the first orbit: E1 = -13.6 eV (because it is bound)
E2 
E1
4
En 
E3 
E1
n2
E1
9
4
THE FRANK-HERTZ EXPERIMENT
The proof of the Bohr’s model
Atoms can absorb only precisely given amounts of energy. The Hg
atoms e.g. 4,9 eV. The 4,9 eV is equals to the energy difference
between the ground state and the first excited state of a Hg atom.
QUANTUM MECHANICAL ATOM MODEL
Matter wave – wave
function ()
Described by the
Schrödinger’s equation
Probability of
occurrence of an
electron:  2
The position of the ground state electron of a hydrogen
atom, around the nucleus.
The density of the spots is proportional to the finding
probability of the electron.
The graph shows Ψ2 in the function of the distance
measured from the nucleus.
5
Heisenberg’s uncertainty principle (1927)
It is impossible to precisely determine the position and the momentum of the
particle at the same time. The multiplication of the uncertainty (error) of two
measurements at the same time is always higher than h / 4 :
x  p x 
h
4
The relation gives a limit of principle: the multiplication of
the measured uncertainty of the two quantities can not be
smaller than h / 4.
QUANTUM MECHANICAL ATOM MODEL
QUANTUM NUMBERS
Quantum numbers describe values of conserved quantities in the
dynamics of the quantum system. They often describe specifically the
energies of electrons in atoms, but other possibilities include angular
momentum, spin etc.
It is already known from the Bohr’s atom model that the energy of the
electrons is quantized so they can have only one value. The energy values
are determined by the n principal quantum number.
The quantum mechanics is proved that there are sublevels of the given
energy levels that is why the n principal quantum number is not enough and
more other quantum numbers are needed.
6
QUANTUM NUMBERS
The principal quantum number (n)
It is known that the principal quantum number defines the energy, and an
energy value belongs to every n value ( n → En ). The electrons with given n values
are forming shells which are named with K, L, M, etc. letters. There can be more other
states inside a shell which states are determined by the orbital quantum number.
Bohr had predicted the positions of orbits with amazing accuracy but did not take
count that this is not the only position of electron, this is the place where the
electron can be found with the highest probability.
En 
E1
n2
QUANTUM NUMBERS
The orbital quantum number (l)
It defines the magnitude of the angular momentum of an electron.
Angular momentum:
The angular momentum of a body which is revolving around an r radius orbital
with v speed is a vectored quantity.
Its value is L = mvr.
Its direction is perpendicular to the
plane of the velocity.
The angular momentum resulting from
the movements of the electrons on
their orbital can only be:
L  l (l  1)
h
2
where h is the Planck constant and l is the orbital quantum number, which
can be an integer between 0 and n-1.
Example: n = 2; l = 0 (2s state): L = 0
h
l = 1 (2p state): L  2 2
7
QUANTUM NUMBERS
The magnetic quantum number (m)
It defines the direction of the angular momentum of an electron. That is why
the angular momentum can be set only in given directions. The projection of
the angular momentum on the direction of the outer magnetic field can only be:
Lz  m
h
2
where m is the magnetic quantum number which values are whole
numbers between -l and +l. This determines the direction of the angular
momentum definitely.
How can it define the angular momentum:
Example: if n = 2; l = 0, 1; m = -1, 0, +1
ZEEMAN EFFECT I
When an atom turns from an
initial higher energy level to
a stationary level with lower
energy then the energy
difference can be emitted as
a photon. This may give a
line in the visible spectrum.
In the presence of an external magnetic field, these different states will
have different energies due to having different orientations of the magnetic
dipoles in the external field, so the atomic energy levels are split into a
larger number of levels and the spectral lines are also split. The rate of split
is proportional to the applied magnetic field. The new lines appear
symmetrically on the right and on the left side of the original line. This is
the so-called Zeeman effect (normal Zeeman effect).
8
QUANTUM NUMBERS
The spin quantum number (s)
It defines the value of the spin angular momentum of the electron. It is
imagined as the electron (like the Earth) not just revolving around its orbit
but it is spinning around its own axis. The electron’s own angular
momentums can only be:
h
S  s ( s  1)
2
where s is the spin quantum number. The spin quantum number can only be ½.
It does not defines other sublevels.
QUANTUM NUMBERS
The magnetic spin quantum number (ms)
It defines the direction of spin angular momentum of an electron.
The projection of the angular momentum on the direction of the outer
magnetic field (z) can only be:
h
S z  ms
2
where ms is the magnetic spin quantum number which is ½ or -½, so
the spin (owned angular momentum) can be set only in two directions.
9
THE STERN-GERLACH EXPERIMENT
The Stern-Gerlach experiment involves sending a beam of particles through an
inhomogeneous magnetic field and observing their deflection. The particles
passing through the Stern-Gerlach apparatus are deflected either up or down
by a specific amount. This result indicates that spin angular momentum is
quantized (it can only take on discrete values), so that there is not a continuous
distribution of possible angular momenta.
http://www.youtube.com/watch?v=rg4Fnag4V-E
THE STERN-GERLACH EXPERIMENT
1922
Conclusions:
1st The experiment proves that the angular momentum is quantized.
2nd Why is the beam deflected into two beams?
if l=0 => m=0 => no deflection
if l=1 => m=0,  1 => deflects into three beams
(that is why a two-beam deflection can not caused by direction of the angular
momentum)
Phipps and Taylor reproduced the effect using hydrogen atoms in their ground
state in 1927.
1925 – Uhlenbeck and Goudsmit formulated their hypothesis of the existence of
the electron spin.
10
THE EINSTEIN-DE HAAS EFFECT
A freely suspended body consisting of a ferromagnetic material
acquires a rotation when its magnetization changes.
Because of the change of the
external
magnetic
field
mechanical
rotation
of
the
ferromagnetic
material
is
happened associated with the
mechanical angular momentum,
which, by the law of conservation
of angular momentum, must be
compensated by an equally large
and oppositely directed angular
momentum
inside
the
ferromagnetic material.
spin angular momentum is indeed of the same nature as the angular
momentum of rotating bodies
QUANTUM NUMBERS
Quantum
number
Symbol
Quantized value
Values
Principle
n
Energy
1,2,3…
Orbital
l
Value of angular
momentum
0,1……n-1
Magnetic
m
Direction of angular
momentum
-l, -l+1…0…l-1,
l
Spin
s
Value of own angular
momentum
½
ms
Direction of own angular
momentum
Magnetic spin
–½, +½
11
QUANTUM NUMBERS
http://dilc.upd.edu.ph/images/lo/chem/Quantum/quantum.swf
12