Transcript The world of Atoms - University of California, Irvine

The world of Atoms
Quantum Mechanics
Theory that describes the physical properties of smallest
particles (atoms, protons, electrons, photons)
Max Planck
"A scientific truth does not triumph by convincing its opponents and
making them see the light, but rather because its opponents eventually
die and a new generation grows up that is familiar with it."
Erwin Schrödinger
"I don't like it and I'm sorry I ever had
anything to do with it."
Werner Heisenberg
"An expert is someone who knows some of the worst
mistakes that can be made in his subject, and how to
avoid them"
Max Born
"It is true that many scientists are not philosophically
minded and have hitherto shown much skill and
ingenuity but little wisdom."
The hydrogen atom
- electron orbits around the nucleus like a wave
- orbit is described by wavefunction
- wavefunction is discrete solution of wave equation
- only certain orbits are allowed
- orbits correspond to energy levels of atom
Niels Bohr (1885-1962)
The hydrogen atom
In the Bohr model of the atom, the hydrogen atom is like a
planetary system with the electron in certain allowed circular
orbits.
The Bohr model
does not work for
more complicated
systems!
6 (r)
Quantum numbers
Each orbital is characterized by a set of quantum numbers.
n,l,m l (r)
Principal quantum number (n): integral values
(1,2,3). Related to the size and energy of the orbital.
Angular
momentum quantum number (l): integral
values from 0 to (n-1) for each value of n.
Magnetic quantum number (ml): integral values from
- l to l for each value of n.
Quantum numbers
How many orbitals are there for each principle quantum
number n = 2 and n = 3?
For each n, there are n different l-levels and (2l+1)
different ml levels for each l.
n=2:
n = 2 different l-levels
l = 0, 1
(2l+1) = 2 x 0 + 1 = 1 ml-levels for l = 0
(2l+1) = 2 x 1 + 1 = 3 ml-levels for l = 1
Total:
1 + 3 = 4 levels for n = 2
Quantum numbers
How many orbitals are there for each principle quantum
number n = 2 and n = 3?
For each n, there are n different l-levels and (2l+1)
different ml levels for each l.
n=3:
n = 3 different l-levels
l = 0, 1,2
(2l+1) = 2 x 0 + 1 = 1 ml-levels for l = 0
(2l+1) = 2 x 1 + 1 = 3 ml-levels for l = 1
(2l+1) = 2 x 2 + 1 = 5 ml-levels for l = 2
Total: 1 + 3 + 5 = 9 levels
for n = 3
The total number of levels for each n is n2
Quantum numbers
Names of atomic orbitals are derived from value of l :
Quantum numbers
Quantum numbers for the first four levels in the hydrogen
atom.
What is the meaning of  ?
Wavefunction itself is not an
observable!

Square of wavefunction is
proportional to probability density
“I cannot but confess that I attach only a transitory importance to
this interpretation. I still believe in the possibility of a model of reality
- that is to say, of a theory which represents things themselves and
not merely the probability of their occurrence. On the other hand, it
seems to me certain that we must give up the idea of complete
localization of the particle in a theoretical model. This seems to me
the permanent upshot of Heisenberg's principle of uncertainty.
(Albert Einstein, on Quantum Theory, 1934”
Wavefunction and probability
2
n,l,ml
n,l,m l
‘function’
‘probability’



n,l,ml
r
Quantum numbers
A subshell is a set of orbitals with the same value of l. They
have a number for n and a letter indicating the value of l.
n,l,m l
2
l = 0 (s)
l = 3 (f)
l = 1 (p)
l = 2 (d)
l = 4 (g)
Orbital Shapes
Heisenberg uncertainty principle
Where’s the electron?
That’s quite
uncertain!
Life is uncertain!
Werner Heisenberg
Heisenberg uncertainty principle
It is not possible to know both the position and momentum of
an electron at the same time with infinite precision.
h
x  mv 
4
x is the uncertainty in position.
 (mv) is the uncertainty in momentum.
h is Planck’s constant.
Heisenberg
The s orbitals in hydrogen
The orbital is defined as the surface
that contains 90% or the
2
total electron probability ( n,l,m l ).
probability distributions

The higher energy orbitals have
nodes, or regions of zero
electron density.
s-orbitals have n-1 nodes.
The 1s orbital is the ground
state for hydrogen.
orbital surfaces
Pauli exclusion principle
How many electrons fit into 1 orbital?
2,1,0
2
ms = +1/2
2,1,0
2
ms = -1/2
 2 electrons fit into1 orbital: 1 spin up
Only
1 spin down
Pauli exclusion principle
Electrons are fermions. There are also bosons
As the temperature is lowered, bosons pack much closer
together, while fermions remain spread out.
Energy Levels
n =∞
n =5
n =4
n =3
n =2
E
n =1
Z 2 
E  RH  2 
n 
RH = 2.178 x 10-18 J
Z = atomic number
n = energy level
Energy Transitions
For the energy change when moving from one level to another:
Z 2 Z 2 
E  RH  2  2 
n f ni 
n =∞
n =5
n =4

n =3
n =2
E
transition
n =1
Lines and Colors
Change in energy corresponds to a photon of a certain
wavelength:
E  h 
Change in
energy

hc

Frequency of
emitted light
Wavelength of
light emitted
Lines and Colors
What is the wavelength of the photon that is emitted
when the hydrogen atom falls from n=3 into n=2?
Z 2 Z 2 
E  RH  2  2 
n f ni 
12 12 
19
18

3.03
10
J

E  2.17810 J 

2 2 32 



19
hc
3.0310 J 


6.62610 34  3.00 10 8

 656 nm
19
3.03 10
Light out of Molecules
hydrogen
n =∞
n =5
n =4
Rhodamine
n =3
n =2
E
transition
532 nm
570 nm
n =1
‘Fluorescence’
Degeneracy
Orbital energy levels for the
hydrogen atom.
Beyond hydrogen
Hydrogen is the simplest element of the periodic table.
Exact solutions to the wave equations for other
elements do not exist!
Polyelectric Atoms
What do the orbitals of non-hydrogen atoms look like?
Multiple electrons: electron correlation
Due to electron correlation, the orbitals in non-hydrogen
atoms have slightly different energies
Polyelectric Atoms
Screening: due to electron repulsion, electrons in different
orbits ‘feel’ a different attractive force from the nucleus
e-
e-
e-
11+
ee-
eee-
Sees a different
effective charge!
Screening changes the energy of the electron orbital; the
electron is less tightly bound.
Polyelectric Atoms
Penetration: within a subshell (n), the orbital with the lower
quantum number l will have higher probability closer to the
nucleus
n =2 orbital
n=3 orbital
Polyelectric Atoms
Hydrogen
Orbitals with the same quantum
number n are degenerate
Polyelectric atom
Degeneracy is gone:
Ens < Enp < End < Enf
Spectra of Polyelectric Atoms
Due to lifting of degeneracy, many more lines
are possible in the spectra of polyelectric atoms