Lecture 29B - UCSD Department of Physics

Physics 1C
Lecture 29B
"Nuclear powered vacuum cleaners will
probably be a reality within 10 years. "
--Alex Lewyt, 1955
http://chemwiki.ucdavis.edu/Textbook_Maps/Physical_Chemistry_Textbo
ok_Maps/Blinder's_%22Quantum_Chemistry%22
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Outline
Last time – hydrogen atom
atomic spectra
quantized orbitals and energy levels
This time - Bohr’s model continued
principal and other quantum numbers
periodic table
EPR and MRI
nuclear stability
Problems with Bohr’s Model
Bohr’s explanation of atomic spectra includes some
features of the currently accepted theory.
Bohr’s model includes both classical and nonclassical ideas.
He applied Planck’s ideas of quantized energy levels
to orbiting electrons and Einstein’s concept of the
photon to determine frequencies of transitions.
What would happen if atoms could be excited to
same higher energy level and then emission could be
induced?
Laser
An acronym for Light
Amplification by Stimulated
Emission of Radiation.
A laser is a light source that
produces a focused, collimated,
monochromatic beam of light.
The laser operates using the
principle of stimulated emission
of light.
In 1917, Albert Einstein
established the theoretic
foundations for the laser.
Problems with Bohr’s Model
Discrete orbitals and energies come from solving the
Schrödinger equation (wave equation)
Another fundamental property of motion is angular
momentum, from Bohr description angular momentum
always value (not true in quantum mechanics)
Improved spectroscopic techniques, however,
showed that many of the single spectral lines were
actually groups of closely spaced lines.
Single spectral lines could be split into three closely
spaced lines when the atom was placed in a magnetic
field.
Quantum Numbers
n is considered the principal quantum number.
n can range from 1 to infinity in integer steps.
But other quantum numbers were added later on to
explain other subatomic effects (such as elliptical
orbits; Arnold Sommerfeld (1868-1951)) and the idea of
3D orbitals.
The orbital quantum number, ℓ, was then
introduced. ℓ can range from 0 to n-1 in integer steps.
Historically, all states with the same principal
quantum number n are said to form a shell. Shells are
identified by letters: K, L, M, …
All states with given values of n and ℓ are said to form
a subshell. Subshells: s, p, d, f, g, h, …
Quantum Numbers
Largest angular momentum will be associated
with:
ℓ =n–1
ℓ = 0 corresponds to spherical symmetry with no
axis of rotation.
This corresponds to a circular orbit.
spherical orbits associated with each value of n
Describes shape
Quantum Numbers
Because angular momentum is a vector, its direction
also must be specified.
Suppose a weak magnetic
field is applied to an atom and
its direction coincides with z
axis.
Then direction of the
angular momentum vector
relative to the z axis is
quantized!
Lz = mℓ ħ
Note that L cannot be
aligned with the z axis.
space quantization
Quantum Numbers
States with quantum numbers
that violate the rules below cannot
exist.
They would not satisfy the
boundary conditions on the wave
function of the system.
Quantum Numbers
In addition, it becomes
convenient to think of the
electron as spinning as it
orbits the nucleus.
There are two directions for
this spin (up and down).
Another quantum number
accounts for this, it is called
the spin magnetic quantum
number, ms.
Spin up, ms = 1/2
Spin down, ms = –1/2.
Pauli Exclusion Principle
Wolgang Pauli proposed that the four quantum
numbers identify the electron
No two electrons in an atom can be in the same
quantum state
No two electrons can have the same set of
quantum numbers
This prevents all atoms from having all electrons
in the ground state
Explained the order of the periodic table.
Periodic Table
The atoms position reflects the n and l values of the
last electron, the chemical electron.
In a single column, the elements’ last electrons differ
by the principle quantum number n, but have the
same l values.
Wave Functions for Hydrogen
The simplest wave function for hydrogen is the one
that describes the 1s state:
 1s (r ) 
1
a
3
o
e
 r ao
All s states are spherically symmetric.
The probability density for the 1s state is:
 1s
2
 1  2r ao
  3 e

a
 o
Wave Functions for Hydrogen
The radial probability density for the 1s state:
 4r  2r ao
P1s (r )   3  e
 ao 
2
The peak at the Bohr radius indicates the most
probable location.
The atom has
no sharply
defined boundary.
The electron
charge is
extended through
an electron cloud.
Wave Functions for Hydrogen
The electron cloud model is quite different from the
Bohr model.
The electron cloud structure does not change with
time and remains the same on average.
The atom does not radiate when it is in one particular
quantum state.
This removes
the problem of the
Rutherford atom.
Radiative
transition causes
the structure to
change in time.
Wave Functions for Hydrogen
The next simplest wave function for hydrogen is for
the 2s state n = 2; ℓ = 0.
This radial probability
density for the 2s state
has two peaks.
In this case, the most
probable value is r  5a0.
The figure compares 1s
and 2s states.
3
 1  2
r  r 2ao
 2s ( r ) 
  2  e
ao 
4 2  ao  
1
Clicker Question 29B-2
Which of the following hydrogen atom series can
emit a photon with a wavelength in the infrared part
of the spectrum?
A) Lyman
B) Balmer
C) Paschen
D) All of the above series can emit a photon
with a wavelength in the infrared region
Problem
•
If a photon with energy 1.40 eV were absorbed by a
hydrogen atom. What is the minimum n that can be
ionized by the photon?
•
ANSWER:
•
Related to energy difference between energy
in initial orbital and that upon ionization
•
Free electron is unbound so has energy = 0
•
Thus, look for orbital that has its magnitude of
energy closest to that of the photon (1.40 eV)
Problem
2 

1
e
13.6 eV
E tot   ke 2 
2
2  n ao 
n
•
Use
•
and solve for n.
•
n2 = 13.6/1.40 = 9.71 => n = 3.12
•

So does n = 3 work?
•
For n = 3, ionization requires 1.51 eV
•
Energy is not sufficient. Can we ionize n = 2?
•
Thus, we can ionize n = 4 or electrons with
higher values of n
Electron Paramagnetic Resonance
Apply magnetic field to
remove energy
degeneracy of spin up
and spin down electrons
When “splitting” matches
photon energy, get
absorption
Seen as a signal
Such radicals are
relatively rare in nature
(Block & Purcell, 1952 Nobel Prize)
Nuclear Magnetic
Resonance Imaging
Nuclei also have quantum
numbers
Detect nuclear magnetic
moments similar to NMR
Nuclear Magnetic
Resonance Imaging
magnetic moments of some of these protons change
and align with the direction of the field
radio frequency transmitter is briefly turned on,
producing a further varying electromagnetic field
photons of this field have just the right energy, known
as the resonance frequency, to be absorbed and flip
the spin of the aligned protons
Nuclear Magnetic
Resonance Angiography
generates pictures of
the arteries
administration of a
paramagnetic contrast
agent (gadolinium) or
using a technique
known as "flow-related
enhancement"
For Next Time (FNT)
Read Chapter 30 through page
1032
Chapter 30 homework due
Wednesday