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WAVE NATURE OF LIGHT
A. Properties of waves as they relate to light
1. Wave
A constantly repeating change or oscillation in matter or in a physical field
2. Wavelength
a. Definition
The distance between identical points on successive
waves
b. Symbol
(
c. Units
(1) Usually nanometers
[(nm) 1 nm = 1 x 10(9 m]
(2) Sometimes meters (m)
3. Frequency
a. Definition
The number of waves that pass through a given
point in one unit of time ( usually one second
b. Symbol
(
c. Units
(1) Reciprocal seconds
s(1
(2) Hertz
Hz = 1 EMBED Equation = 1 EMBED Equation = 1 s(1
4. Speed
a. Definition
The distance a wave travels in one unit of time ( usually one second
b. Symbol
c
c. Speed of light
(1) Depends on the medium
(2) values
2.9979 x 108 m/s
2.9979 x 1017 nm/s
(3) Mathematical relationship
c = ((
B. Light as waves
1. Light is one form of electromagnetic radiation
which also includes
radio waves
microwaves
infrared radiation
visible light
ultraviolet radiation
X rays
gamma rays
2. Electromagnetic radiation
a. Has an electric field component and a magnetic field
component hence the term electromagnetic
b. The electric field component and the magnetic field
component
(1) Travel in mutually perpendicular planes
(2) Have the same wavelength, frequency, and
speed
c. Electromagnetic spectrum
The whole range of wavelengths or frequencies of
electromagnetic radiation
C. Examples
1. Finding wavelength
A laser used to weld detached retinas has a frequency of
4.69 x 1014 s(1. What is the wavelength of its light?
c = ((
(= c/(
( = EMBED Equation.3 EMBED Equation.3
= 639 nm
2. Finding frequency
The light given off by a sodium lamp has a wavelength of
589 nm. What is the frequency of this light?
c = ((
( = c/(
( = EMBED Equation.3 EMBED Equation.3
= 5.09 x 1014 s(1
FROM CLASSICAL PHYSICS TO QUANTUM THEORY
A. Plancks theory
1. The reason for its development
a. Classical physics assumed that all energy
changes were continuous.
(1) This means that there are no restrictions on the
amount of one form of energy which can be
converted to another form of energy.
Example:
A ball rolling downhill can change
any amount of potential energy into kinetic energy.
(2) This meant that atoms and molecules should be
able to emit or to absorb any arbitrary amount
of energy.
b. The spectrum of light emitted by a hot object
blackbody radiation could not be explained by
classical physics.
2. Planck proposals
a. That energy could only be released or absorbed in
chunks of some minimum size.
b. That atoms and molecules could only emit or absorb
energy in discrete quantities
c. Two comparisons between continuous and discrete
(1) Violin and piano
A violin can play every pitch between the two notes B and C ( this is continuous.
A piano can only play the pitch for the note B or the pitch for the note C ( this is discrete.
(2) Inclined plane and stairs
A ball can roll down in inclined plane and have any possible height on the plane between the top and the bottom ( this is continuous.
A ball rolling down a flight of stairs can only have certain heights corresponding to the height of the stair it is on at the time ( this is discrete.
3. Plancks quantum theory
a. The smallest possible increment of energy that can be
gained or lost was called a quantum (the plural is
quanta).
b. Radiant energy is always emitted or absorbed in whole
number multiples of a constant h times the frequency.
(E = h(, 2 h(, 3 h(,
c. h is known as Plancks constant and has the value of
6.6262 x 10(34 J(s
B. The photoelectric effect
1. The photoelectric effect could not be explained by classical
physics.
a. The photoelectric effect described
(1) The photoelectric effect is the ejection of
electrons from the surface of certain metals
when light of at least a certain minimum
frequency (called the threshold frequency)
was shined upon them.
(2) Whether or not current flowed depended on the
frequency of the light NOT its intensity.
(3) Increasing the intensity of the light increased the
amount of the current.
(a) Dim low frequency light
(
(
(
No flow of electrons
(b) Intense low frequency light
(
(
(
No flow of electrons
(c) Dim high frequency light
(
(
(
Small flow of electrons
(d) Intense high frequency light
(
(
(
Greater flow of electrons
b. Classical physics predicted that intensity should
determine the amount of current and that frequency
should be irrelevant.
2. Einsteins use of Plancks theory to explain the photoelectric
effect
a. Einsteins three assumptions
(1) He assumed that a beam of light is really a
stream of particles (now called photons).
(2) He assumed that each photon ejects one electron
when it strikes the metal.
(3) He also assumed that this photon must have at
least enough energy to free the electron from the
forces that hold it in the atom.
b. Using Plancks constant he calculated the energy of a
photon from its frequency.
E = h(
c. Einsteins explanation using these assumptions:
It does not matter how many photons strike the metal surface if none of them have enough energy to kick out an electron.
If a photon with at least the minimum energy strikes the metal, then that photon is absorbed by the electron.
A certain minimum amount of energy is needed to free the electron.
The excess energy, if any, goes into the kinetic
energy of the electron.
3. Examples
a. Calculating energy from frequency
What is the energy of a X-ray photon with a
frequency of 6.00 x 1018 s(1?
E = h(
= (6.6262 x 10(34 J(s)(6.00 x 1018 s(1)
= 3.98 x 10(15 J
b. Calculating the energy from wavelength
What is the energy of an infrared photon with a
wavelength of 5.00 x 104 nm?
E = h(
Substituting for (
EMBED Equation.3 gives us
E = EMBED Equation
Since c = 2.9979 x 1017 nm/s
E = EMBED Equation.3
E = 3.97 x 10(21 J
C. The emission spectrum of hydrogen
1. The prediction of classical physics and Rutherfords model of
the atom
a. Has the electrons orbiting around the nucleus where the
attractive force of the nucleus is exactly balanced by the
acceleration due to the circular motion of the electron.
b. The two observed problems with the Rutherford model
(1) The stability of the atom
A charged particle, such as the electron, moving around the nucleus should lose energy and spiral down into the nucleusin
about 10(10 s.
(2) The line spectrum of atoms
The electron should be able to lose energy in
any amounts
Which should produce a continuous spectrum
(all colors like a rainbow)
Rather than a line spectrum
(a set of lines of specific colors)
2. The new model of the atom would use quantum theory.
BOHRS THEORY OF THE HYDROGEN ATOM
A. Bohrs Postulates
1. The electron moves in a circular orbit around the proton.
a. These orbits can only have certain radii corresponding to
certain definite energies.
b. These energies are given by the equation
En = (RH EMBED Equation.2
Rydberg developed this from his study of the line spectra of many elements.
RH = 2.179 x 10(18 J
(the Rydberg constant)
n = 1, 2, 3,
(indicates the energy level of the electron)
The ( sign indicates that the energy of the
electron in the atom is lower than the energy of
a free electron.
c. As the electron gets closer to the nucleus, En increases in
absolute value, but becomes more negative.
Think about a ball rolling down a staircase.
When it reaches the lowest step (n = 1) it has its lowest potential energy and it is the most stable.
d. Ground state and excited state.
n = 1 is the ground state for the hydrogen electron
n = 2, 3, are the excited states for the hydrogen
electron
2. Transitions of the electron occurs between specific energy states
and involves the emission or absorption of a photon of a specific
energy and frequency.
B. Bohrs explanation of the emission spectrum of hydrogen.
A photon is emitted when an electron transitions from one energy
level to a lower one.
(E = Efinal ( Einitial
Efinal = (RH EMBED Equation.3 and Einitial = (RH EMBED Equation.3
(E = EMBED Equation.3 ( EMBED Equation.3
(E = ( EMBED Equation.3 + EMBED Equation.3
(E = h( = RH EMBED Equation.3
( = c/(
hc/( = RH EMBED Equation.3
1/( = EMBED Equation EMBED Equation.3
C. Bohrs model completely explained the observed emission spectra of
hydrogen, including the Paschen (IR), the Balmer (visible), and the
Lyman (UV) series.
D. Example
What is the wavelength of the photon emitted when a hydrogen
electron transitions from the n = 4 state to the n = 2 state?
RH = 2.179 x 10(18 J
h = 6.6262 x 10(34 J(s
c = 2.9979 x 1017 nm/s
ninitial = 4
nfinal = 2
1/( = EMBED Equation EMBED Equation.3
1/( = EMBED Equation.3 EMBED Equation.3
1/( = (2.0567 x 10(3 EMBED Equation
( = ( 486.2 nm
The negative sign comes from the calculation of the energy, indicating that light is emitted.
( = 486.2 nm
THE DUAL NATURE OF THE ELECTRON
A. The history of deBroglies proposal
1. Physicists accepted Bohrs model but were puzzled as to why
the energy level of the hydrogen atom should be quantized.
2. Einstein has shown that light has both wave properties and
particle properties.
3. deBroglie proposed that particles such as electrons can also
posses wave properties under the proper circumstances.
B. deBroglies two proposals
1. The first proposal was the standing wave model of the electron
a. An electron bound to the nucleus behaves like a standing
wave.
(1) A standing wave is similar to plucking the string
of a guitar.
(2) These waves get their name from the fact that
they are stationary they do not travel along
the string.
b. If the electron behaves like a standing wave then the
length of the wave must fit the circumference of the
circle (the orbit) exactly, otherwise it would partially
cancel itself out.
c. Since 2( r = the circumference of the orbit, then the
wavelengths that will fit are:
2( r = (
2( r = 2(
2( r = 3(
2( r = n(
2. The second proposal was that very small particles moving very
fast would exhibit wavelike properties particularly a
wavelength.
a. This would NOT be observable in the macroscopic world
due to the insignificant wavelength.
b. In the submicroscopic world the wavelength could be
calculated using:
( = EMBED Equation
h = Plancks constant = 6.6262 x 10(34 J(s
but since 1 J = EMBED Equation
1 J(s = EMBED Equation (s
h = 6.6262 x 10(34 EMBED Equation
m = mass in kg
v = velocity in m/s
3. Example
What is the wavelength associated with an electron with a
mass of 9.11 x 10(31 kg and a velocity of 4.19 x 106 m/s?
( = EMBED Equation = EMBED Equation.3
= 1.74 x 10(10 m
C. Heisenbergs Uncertainty Principle
1. Because an electron can behave like a wave it is difficult to
determine exactly where an electron is.
2. The precise location of a wave cannot be specified because a
wave extends out in space.
3. To describe this problem Heisenberg formulated his uncertainty
principle
a. It is impossible to know simultaneously both the exact
position and the exact momentum of a particle.
b. There is always a limit to how precisely we can know
both values at the same time.
c. (x(p ( EMBED Equation.2 EMBED Equation
QUANTUM MECHANICS
A. Formulated by Ernst Schroedinger for the hydrogen atom.
B. The Schroedinger equation incorporates both particle behavior and
wave behavior.
C. Solving the Schroedinger equation for a hydrogen atom
1. Requires advanced calculus even for so simple a system
2. Produces a wave function (
3. Each wave function
a. Specifies the possible energy states that an electron can
occupy in a hydrogen atom
b. Is characterized by a set of quantum numbers
D. Electron density
1. The square of the wave function (2 gives the probability of
finding the electron in a certain region of space at a given
instant.
2. Regions of high electron density are areas where there is a high
probability of finding the electron.
3. Regions of low electron density are areas where there is a low
probability of finding the electron.
E. Orbitals
1. The complete set of solutions to the Schroedinger equation
yields a set of wave functions and a corresponding set of
energies.
2. Each of these allowed wave functions is called an orbital.
3. Each orbital describes a specific distribution of electron density
in space.
4. An atomic orbital describes the region of space where there is a
high probability of finding the electron.
5. Each orbital has
a. A characteristic size
b. A characteristic shape
c. A characteristic orientation in space
F. The many-electron atom
1. The Schroedinger equation for them cannot be solved.
2. The energies and wave functions of the hydrogen atom are a
good approximation of the behavior of the electrons in more
complex atoms.
QUANTUM NUMBERS
A. The principal quantum number
1. Describes the size of the orbital
2. Is symbolized by n
3. n can have the value of any non-zero integer
n = 1, 2, 3,
4. These are sometimes referred to as shells.
5. Is the quantum number on which the energy of an electron
principally but NOT exclusively depends
6. The larger the principal quantum number
a. The greater the average distance of the electron from the
nucleus
b. The greater the energy of the electron (generally)
c. The less tightly the electron is bound to the nucleus
d. The less stable the condition of having the electron in
that orbital
B. The angular momentum quantum number
1. Gives the shape of the orbital
2. Is symbolized by l
3. l can have integer values from 0 to n ( 1
4. There are n different values for l .
5. Different values of l are usually denoted by letters.
These are from the old spectroscopic terminology
a. 0 = s (sharp, NOT spread out)
b. 1 = p (principal, very strong)
c. 2 = d (diffuse, rather spread out)
d. 3 = f (fundamental)
e. 4 = g
6. These are sometimes referred to as sublevels or as
subshells.
7. Subshells are designated as the n followed by the letter of the
subshell
Examples:
1s
4f
8. To whatever extent the energy of an orbital does not depend on
the principal quantum number it will depend on the angular
momentum quantum number.
C. The magnetic quantum number
1. Gives the orientation of the orbital of the same energy and shape
2. Is symbolized by ml .
3. ml can have integer values from ( l to + l .
4. There are 2l + 1 different values for ml .
5. These are referred to as orbitals.
D. The spin quantum number
1. Describes which of the two possible spin orientations of the spin
axis an electron applies
2. Is symbolized by ms
3. Can have the values + or (
4. Analogous to the electrons spinning on its axis like the earth
a. A charged particle spinning causes a circulating electric
charge.
b. A circulating electric charge generates a magnetic field.
c. The north pole can either be pointing up or pointing
down.
d. Half of the electrons in a group of hydrogen atoms are
pointing one way and half are pointing the other way.
E. Table of Levels, Sublevels, Type of Sublevels, Number of Orbitals,
and Number of Electrons
see the handout with this title
F. Examples
1. How many subshells are there for n = 4? What are their labels?
l can have n values, therefore there are 4 subshells.
They are the 4s, the 4p, the 4d, and the 4f.
2. How many orbitals are there for n = 4?
l can have values from 0 to n ( 1.
ml = 2l + 1
l = 0 ml = 1 value
l = 1 ml = 3 values
l = 2 ml = 5 values
l = 3 ml = 7 values
16 orbitals
3. Which of the following sets are correct designations for
subshells?
a. 7s:
s is l = 0; since n = 7 then 0 ( n ( 1; OK
b. 5p:
p is l = 1; since n = 5 then 1 ( n ( 1; OK
c. 2d:
d is l = 2; since n = 2 then 2 is not less than or
equal to n ( 1; NOT OK
4. Why is each of these sets of quantum numbers not correct?
a. n = 2, l = 1, ml = (1, ms = 1
l ( n ( 1 ? yes
( l ( ml ( + l ? yes
ms = ( ? NO
b. n = 3, l = 1, ml = (2, ms = (
l ( n ( 1? yes
( l ( ml ( + l ? NO
PAGE
PAGE 16
Topic 6 Quantum Theory of the Atom
2006 Lloyd Crosby
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