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

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

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

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

WHITE LIGHT-------> [Prism]----> Green

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 

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

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)

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)

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

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

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

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

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.


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 or wave mechanics is the branch of physics that mathematically describes the wave
properties of submicroscopic particles.

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.

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

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.

1. What is the frequency of light of wavelength 456 nm?

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?

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