__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 10^{8} m/s ]

λ is the wavelength measured in m or
nm.
1 m = 1 x 10^{9} 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 10^{8} 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 10^{8} m/s / 5.89 x 10^{-7} m = 5.09 x 10^{14}
s^{-1}

When white light is directed onto a prism, the light is broken up into a continuous spectrum (i.e. rainbow).

**
R**ed

**
O**range

**
Y**ellow

WHITE LIGHT-------> **[**Prism**]**----> **G**reen

**
B**lue

**
I**ndigo

**
V**iolet

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**.

**
R**ed 656 nm

Hydrogen Lamp-------> **[**Prism**]**----> **G**reen
486 nm

**
B**lue 434 nm

**
V**iolet 410 nm

**
Y**ellow

Na vapor lamp-------> **[**Prism**]**---->

**
B**lue

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 10^{7} m^{-1})(1/2^{2} - 1/n^{2}),
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:

n = number of photons; h = Planck's constant ( 6.626 x 10

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 10^{14} s^{-1}.

E = nhν = (1)(6.626 x 10^{-34} J** ^{.}**s)(5.09
x 10

__EXAMPLE 4__

Calculate the energy of a photon of wavelength 5.00 x 10^{4}
nm (infrared).

E = nhν = nhc/λ =
(6.626 x 10^{-34} J** ^{.}**s)(3.00
x 10

__EXAMPLE 5__

Calculate the energy of a mole of photons of yellow light with a
frequency
of 5.09 x 10^{14} s^{-1}.

E = nhν = (6.022 x 10^{23})(6.626
x 10^{-34} J** ^{.}**s)(5.09 x
10

__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 10^{23})(6.626 x 10^{-34} J** ^{.}**s)(3.00
x 10

__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 =-R_{h}/n^{2}

n = 1,2,3,... The principal quantum
number
R_{h}= 2.179 x 10^{-18}J

** n Energy**

1 -2.178 x 10

2 -5.445 x 10

3 -2.420 x 10

4 -1.361 x 10

5 -8.712 x 10

6 -6.050 x 10

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 (E_{final})
and initial (E_{initial})

Energy of emitted photon = hν = energy to move the electron from a lower to a higher.

E_{initial} + hν = E_{final}

hν = E_{final}
- E_{initial} = ΔE

ΔE = -R_{h}/n^{2}_{final}
- (-R_{h}/n^{2}_{initial})

ΔE = (2.179 x 10^{-18}J)_{}
(1/n^{2}_{initial} - 1/n^{2}_{final})

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. m_{l}:
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

**m _{l}: Magnetic Quantum Number**

Relates the directionality in space of the electron cloud (orbital) surrounding the nucleus.

The allowed values of m_{l} are: -l...0...+l

If l is 0 then m_{l}
can be 0

If l is 1 then m_{l}
can be -1,0,+1

If l is 2 then m_{l}
can be -2,-1,0,+1,+2

There are 2l + 1 values for
m_{l}.

**Permissible Values of Quantum Numbers n, l,
& m _{l} for n = 1 through 4**

__ n__ __
l __
__subshell label __
__m _{l}__

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, m _{s}**,
had to be added to the three quantum

numbers derived from the Schrodinger equation.

m_{s}: Electron Spin Number

An electron is said to spin either clockwise or counterclockwise.

The allowed values of m_{s} 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 10^{9}Hz?

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 10^{4}
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 10^{3} cm/s.

(Mass of a neutron = 1.675 x 10^{-27}kg.)

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 *m _{l}*
that
it can have.

8. For the following subshells, give the values of
the
quantum numbers (*n*, *l*, and* m _{l}*) 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*,
*m*_{l},
*m*_{s})
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 10^{14}s^{-1}

2. 1.36 x 10^{8}nm

3. 3.19 x 10^{-19}J

4. 2.39 x 10^{-5}J

5. 9.89 nm

6. a) absorbed b) emitted c) emitted

7. l can = 0 or 1. If l = 0 then m_{l} = 0. If
l = 1 then m_{l}= -1, 0, +1.

8. a) n = 4 l = 1 m_{l} = -1, 0, 1

b) n = 3 l = 2 m_{l} = -2, -1, 0, 1, 2

c) n = 3 l = 0 m_{l} = 0

d) n = 5 l = 3 m_{l} = -3, -2, -1, 0, 1, 2, 3

9. a) no b) possible c) not d) possible e) not

10. 2

11. 14