WAVES AND THE WAVE NATURE OF LIGHT
A wave can be defined as:
- a continuously repeating change or oscillation in matter or in a
physical field.
- vibrating disturbance which transmits energy.
A wave is characterized by:
1) Amplitude (intensity)
2) Velocity (speed)
3) Wavelength, λ (distance
between two identical adjacent points on the wave.)
4) Frequency, ν (time
for one
wavelength to pass)
The speed of a wave is the product of wavelength (λ)
and frequency (ν).
[ν
is a Greek letter pronounced "nu".]
λν = speed
Electromagnetic radiation is a form of energy described in terms of perpendicular wave-like electric and magnetic fields that change, at the same time and in phase with time. Examples are light, radio signals, microwave, and x-rays.
λν = c [c = Speed of light in a vacuum is 3.00 x 108 m/s ]
λ is the wavelength measured in m or
nm.
1 m = 1 x 109 nm
ν is frequency; SI unit is hertz, Hz.
1
Hz = 1 s-1
EXAMPLE 1
What is the wavelength of the radio station WAFR which broadcasts at
a frequency of 88.3 MHz ?
λν = c
λ = c / ν
= 3.00 x 108 m/s / (88.3 MHz)(1 s/1 x 10-6 MHz) =
3.40 m
EXAMPLE 2
What is the frequency of yellow light from a sodium lamp with a
wavelength
of 589 nm?
λν
= c
ν = c/λ
= 3.00 x 108 m/s / 5.89 x 10-7 m = 5.09 x 1014
s-1
Red
Orange
Yellow
WHITE LIGHT-------> [Prism]----> Green
Blue
Indigo
Violet
When the light produced by excited atoms of elements is directed onto a prism, the light is also broken up but not into a continuous spectrum but a line spectrum. The light emitted by these atoms is at certain wavelengths characteristic for that element. Such spectra are known as line spectra.
Red 656 nm
Hydrogen Lamp-------> [Prism]----> Green
486 nm
Blue 434 nm
Violet 410 nm
Yellow
Na vapor lamp-------> [Prism]---->
Blue
At the end of the nineteenth century, Johann Balmer found an
empirical
formula that accurately describes
the hydrogen spectrum in the visible region, 1/λ
= (1.097 x 107 m-1)(1/22 - 1/n2),
where
n is an integer
greater than 2.
Spectra for other atoms could also be represented by similar equations but no one could explain why.
Photons: Energy by the Quantum
Planck's Quantum Hypothesis
1. When solids are heated, they emit radiation over a wide range of
wavelengths. (Electric heater and
tungsten light bulb)
2. The amount of energy emitted depends on the wavelength.
3. In 1900, Planck solved the problem by stating that atoms and
molecules
could emit (absorb) energy
only in discrete quantities, like small packages
or bundles. Up to this point it was assumed that any
amount of energy could be emitted or absorb. The
name given to the smallest amount of energy
is
quantum. According to Planck, the atoms
of the solid oscillate, or vibrate, with a definite frequency,
depending on the solid.
4. The energy (E) of an emitted single quantum of energy is
proportional to the frequency of the radiation.
5. According to Planck's quantum theory, energy is always emitted in
multiples of h.
E = hν [h = 6.626 x 10-34 J.s]
Matter is quantized. Stairs are quantized. Cats give birth to an integral number of kittens.
The following diagram shows the relationship between
wavelength,
frequency, & energy on the
electromagnetic radiation spectrum.
λ (wavelength)
--------------------------------------------------------------------------------------------------------------------------->
g rays x rays UV V I B G Y O R Infrared Microwave Radio waves
ν
(frequency)
<--------------------------------------------------------------------------------------------------------------------------
Energy
<---------------------------------------------------------------------------------------------------------------------------
The Photoelectric Effect: Einstein and Photons
1. When visible light falls on the active metals of Group IA (alkali
metals; Li, Na, K, Rb, Cs) electrons
called photoelectrons, are emitted (given off) from
the metal.
2. Thermal or light energy is required to knock the electrons off.
They do not spontaneously escape.
3. Brighter the light (greater intensity) the more electrons knocked
off.
4. Kinetic energy (speed) is dependent on the color (specific frequency
and wavelength) not on the
brightness. This certain minimum frequency
is called the threshold frequency.
5. Each metal has a characteristic minimum energy required to remove
its electrons. For potassium the
dimmest violet but not the brightest red. A bright
violet light produces more electrons than dim violet
light but the maximum kinetic energy of the
electrons
is the same.
6. In 1905 Einstein suggested that light is not only wavelike but also
like a stream of particles
called photons. When a photon hits
the metal, its energy h is taken up by the electron. The photon
ceases to exist as a particle; it is said to be
absorbed. Using Planck's quantum theory of radiation, he
deduced that each photon must possess energy E given
by the equation:
E = nhν
n = number of
photons; h = Planck's constant ( 6.626 x 10-34 J.s); ν
= frequency of radiation
Photon: smallest packet of electromagnetic
radiation.
(specific term)
Quanta: groups of photons or quantum.
Quantum: smallest packet of energy. (general term)
Dual Nature of Light is explained by:
a. Diffraction
pattern (Wave property)
b. Photoelectric
effect. (Particle property)
EXAMPLE 3
Calculate the energy of a photon of yellow light with a frequency of
5.09 x 1014 s-1.
E = nhν = (1)(6.626 x 10-34 J.s)(5.09
x 1014 s-1) = 3.37 x 10 -19 J
EXAMPLE 4
Calculate the energy of a photon of wavelength 5.00 x 104
nm (infrared).
E = nhν = nhc/λ =
(6.626 x 10-34 J.s)(3.00
x 108 m/s)/(5.00 x 10-5 m) = 3.98 x 10-21
J
EXAMPLE 5
Calculate the energy of a mole of photons of yellow light with a
frequency
of 5.09 x 1014 s-1.
E = nhν = (6.022 x 1023)(6.626
x 10-34 J.s)(5.09 x
1014
s-1) = 2.03 x 105 J
EXAMPLE 6
Radiation in the microwave region of the electromagnetic spectrum is
not very energetic. However, it
is this radiation that cooks your food in a microwave oven. If
you bombard a cherry pie with 1 mole
of photons with a wavelength of 0.500 cm, how much energy is your pie
absorbing?
E = nhν = nhc/λ =
(6.022 x 1023)(6.626 x 10-34 J.s)(3.00
x 108 m/s) / (5.00 x 10-3
m) = 23.94 J
QUANTUM VIEW OF ATOMIC STRUCTURE
Bohr's Hydrogen Atom: A Planetary Model
Niels Bohr (1913/Danish/student of Rutherford)
1) Electron travels around the nucleus in a circular orbit. An
electron
can have only specific energy
values in an atom, which are called its energy
levels.
Therefore the atom can have only specific
total energy values.
E =-Rh/n2
n = 1,2,3,... The principal quantum number Rh= 2.179 x 10-18J
n Energy
Energy kJ/mol
1
-2.178 x 10-18 J -1312.0
2
-5.445 x 10-19 J -328.0
3
-2.420 x 10-19 J -145.8
4
-1.361 x 10-19 J
-82.01
5
-8.712 x 10-20 J
-52.49
6
-6.050 x 10-20 J
-36.45
infinity
0
0 [When
n = infinity the electron is totally removed from the atom.]
2) The electron moves between energy levels. When energy is put into
the atom and absorbed, the electron
makes a discrete jump (transition) to a higher or
larger orbit (i.e. QUANTUM LEAP).
3) When the electron falls back to the original state of lower
energy
then a photon of light is emitted with
a characteristic energy, and giving rise to the
line spectrum.
Energy of electron final (Efinal) and initial (Einitial)
Energy of emitted photon = hν = energy to move the electron from a lower to a higher.
Einitial + hν = Efinal
hν = Efinal - Einitial = ΔE
ΔE = -Rh/n2final - (-Rh/n2initial)
ΔE = (2.179 x 10-18J) (1/n2initial - 1/n2final)
4) The lowest energy level is related to the smallest orbit known as
the Bohr orbit. This lowest energy
level is also called the ground state.
When the electron is in any higher energy level the atom is said
to be in an excited state.
5) Discrete jumps and falls correspond to characteristic energy emission or absorption.
6) Orbits (n) are quantized. n = 1, 2, 3, 4, 5... Ground state n =
1.
represents the electron totally
removed from the atom.
OBSERVATIONS OF ELECTRON TRANSITIONS IN A BOHR ATOM
If an electron goes from a lower orbital to a higher (e.g. n = 1 to
n = 3):
A) energy is required for this transition and must
be put into the atom.
B) the electron moves to a greater distance from
the nucleus.
C) the final energy of the electron is greater than
the initial.
ΔE
= + (i.e. analogous to an endothermic process)
If an electron goes from higher orbital to a lower (e.g. n = 5 to n
= 2):
A) energy is given off in the form of a photon of
electromagnetic radiation.
B) the electron moves closer to the nucleus.
C) the final energy of the electron is less than
the initial.
ΔE
= - (i.e. analogous to an exothermic process)
Ground States and Excited States
Ground State when an atom has all its
electrons in the lowest energy level possible.
Excited State when an electron has been promoted to a higher energy level possible.
Wave Mechanics: Matter as Waves
To answer the question why the energy of the electron in the hydrogen
atom should be quantized,
DeBroglie suggested in 1924 that electrons have wave properties as
well as particle properties.
DeBroglie reasoned that the electron was "fastened" in the space around
the nucleus by the attraction
of the positive charge of the nucleus. He argued that the electron
behaved like a standing wave.
Traveling waves move from one place to another. Water
waves.
Standing waves do not move from
one place to another. A vibrating guitar string is an example of a
standing wave.
The wavelengths of a standing wave are quantized, that is they can
have
only certain values due to the
fact that the wave has fixed nodes on either end. The wavelengths of
the allowed vibrations must be
such that a whole number of half-wavelengths are equal to the length
of the guitar string. At these places
the amplitude is always zero.
DeBroglie derived the following equation. λ = h/mv
In 1928 C. J. Davisson & L. H. Germer at Bell Labs in the U.S.
showed
that electrons do indeed have
wave properties when they directed a beam of electrons on a nickel
crystal and obtained a diffraction
pattern like those similar to x-rays.
QUANTUM MECHANICS
Quantum mechanics or wave mechanics is the branch of physics that
mathematically
describes the wave
properties of submicroscopic particles.
QUANTUM NUMBERS AND ATOMIC ORBITALS
Bohr's model works only for the hydrogen atom. Large errors result
when applied to atoms with more
than one electron.
In the 1920's an new approach was initiated which is called quantum
mechanics. Quantum mechanics
views the electron not as a particle located at some point in the atom
but as a wave whose mass and
charge is spread out in a standing wave surrounding the nucleus.
Instead of a single value to describe the position and energy of the
electron as the Bohr model uses,
wave mechanics is associated with a set of 3 numbers called quantum
numbers. Any particular set of
quantum numbers is referred to as an atomic orbital.
An ORBITAL is the volume of space where an electron with a given particular energy is most likely to be found.
Each set of 3 quantum numbers maps the electron much like longitude and latitude. They can be thought of as an "electron zip-code" which gives the electron an address or region of probability in a 3-dimensional space within the atom.
QUANTUM NUMBERS
The three quantum numbers are:
1. n: Principal
Quantum
Number
2. l:
Azimuthal Quantum Number
3. ml:
Magnetic Quantum Number
Each number has only certain allowed values.
n: Principal Quantum Number
The allowed values for n are 1, 2, 3, 4... (only integral values)
The principal quantum number n, relates the energy of the electron
and
distance of the electron from nucleus. In multi-electron atoms, two or
more electrons may have the same n value. Electrons with the same
n value are said to be in the same electron shell. A shell
contains
electrons having approximately the same energy and which are located
approximately
at the same distance from the nucleus. All shells except the first
shell
are subdivided into smaller shells.
l: Azimuthal Quantum
Number
(or angular momentum quantum number)
Electrons with the same n value can be grouped into subshells.
Each subshell has a particular value of the azimuthal quantum number l.
This quantum number relates the general shape of the orbital or
electron
cloud. The larger the l value
the
more complex the shape.
The allowed values of l are 0, 1, 2, 3...n-1 (any integral value except zero)
If n = 1 then l has one allowed value, 0.
If n = 2 then l has two allowed values, 0 & 1.
If n = 3 then l has three allowed values, 0, 1, & 2.
There are n values of l.
There are letter designations assigned to l
values.
l
=
0 1 2
3
4
Letter s p
d f g
ml: Magnetic Quantum Number
Relates the directionality in space of the electron cloud (orbital)
surrounding the nucleus.
The allowed values of ml are: -l...0...+l
If l is 0 then ml can be 0
If l is 1 then ml can be -1,0,+1
If l is 2 then ml can be -2,-1,0,+1,+2
There are 2l + 1 values for ml.
Permissible Values of Quantum Numbers n, l, & ml for n = 1 through 4
n
l
subshell label
ml
1
0
1s
0
2
0
2s
0
1
2p
-1, 0, +1
3
0
3s
0
1
3p
-1, 0, +1
2
3d
-2, -1, 0, +1, +2
4
0
4s
0
1
4p
-1, 0, +1
2
4d
-2 ,-1 , 0, +1, +2
3
4f -3, -2,
-1, 0, +1, +2, +3
ELECTRON SPIN & the PAULI EXCLUSION PRINCIPLE
The quantum numbers derived from the Schrodinger equation explain a
great deal of experimental data
but they do not account for the fact that some atomic spectral lines
consist of two closely spaced lines.
The Austrian physicist Wolfgang Pauli suggested that the two lines
could
be explained by the electron
having two states available to it, either on of which it can occupy.
These states were later identified with
electron spin. An electron is pictured as spinning like a top about
its axis. Like a top it can only spin in
one of two directions: Clockwise or Counterclockwise.
A fourth quantum number, the spin quantum number, ms,
had to be added to the three quantum
numbers derived from the Schrodinger equation.
ms: Electron Spin Number
An electron is said to spin either clockwise or counterclockwise.
The allowed values of ms are -1/2 or +1/2
Pauli also proposed that no two electrons in an atom can have all
four
quantum numbers alike.
This proposal is known as the Pauli Exclusion Principle.
Therefore an orbital can contain only two electrons, each with
opposite
spins.
2. What is the wavelength (in nm) of radiation of frequency 2.20 x 109Hz?
3. A photon has a wavelength of 624 nm. Calculate the energy of the photon in joules.
4. A photon has a frequency of 6.0 x 104 Hz. Calculate the energy (in joules) of 1 mole of photons all with this frequency.
5. Calculate the wavelength (in nm) associated with
a
beam of neutrons moving at 4.00 x 103 cm/s.
(Mass of a neutron = 1.675 x 10-27kg.)
6. According to the Bohr model, is electromagnetic
radiation
emitted or absorbed when the electron in a hydrogen atom
undergoes each of the following transitions? (a) from
n = 1 to n = 2 (b) from the orbit with radius = 476.1 pm to the orbit
with radius = 211.6 pm (c) from n = 4 to n = 3.
7. An electron in a certain atom is in the n = 2 quantum level. List the possible values of l and ml that it can have.
8. For the following subshells, give the values of
the
quantum numbers (n, l, and ml) and the
number of orbitals in each
subshell: (a) 4p, (b) 3d, (c) 3s, (d) 5f.
9. Which of the following sets of quantum numbers (n,
l,
ml,
ms)
are unacceptable?
(a) (1,0,1/2,-1/2), (b) (3,0,0,+1/2), (c) (2,2,1,+1/2),
(d) (4,3,-2,+1/2), (e) (3,2,1,1).
10. How many electrons can occupy a 4f orbital?
11. How many electrons can occupy a 4f subshell?
Answers
1. 6.59 x 1014s-1
2. 1.36 x 108nm
3. 3.19 x 10-19J
4. 2.39 x 10-5J
5. 9.89 nm
6. a) absorbed b) emitted c) emitted
7. l can = 0 or 1. If l = 0 then ml = 0. If
l = 1 then ml= -1, 0, +1.
8. a) n = 4 l = 1 ml = -1, 0, 1
b) n = 3 l = 2 ml = -2, -1, 0, 1, 2
c) n = 3 l = 0 ml = 0
d) n = 5 l = 3 ml = -3, -2, -1, 0, 1, 2, 3
9. a) no b) possible c) not d) possible e) not
10. 2
11. 14