QUANTUM THEORY OF THE ATOM

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CHAPTER 7
QUANTUM THEORY OF THE ATOM
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I. Rutherford's model of the atom
A) The nucleus is very small - positively charged
- with the electrons outside the nucleus.
B) A new question arises. If the electron is
negatively charged, won't the attraction for
electrons by the nucleus cause the electron to
fall into the nucleus and therefore atoms should
collapse.
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They don't. Why not?
II. Rutherford attempts to explain his
experimental results.
A) He knew about the solar system - the
attraction of the planets by the sun - universal
gravitation.Yet planets are not pulled into the
sun. They are in motion around the sun and this
motion prevents them from being pulled into the
sun.
So Rutherford puts the electron in motion around
the nucleus.
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B) There is a BIG difference between electrons
and the nucleus and the sun and the planets. WHAT
IS THAT???
C) An electron in an orbit experiences an
acceleration. According to the laws of
electricity, this moving negative charge in the
vicinity of a positive charge will radiate energy
- as it accelerates, it loses energy. What
happens then?____________
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Problem for scientists since putting the electron
in motion around the nucleus does not keep it
outside the nucleus according to Newtonian
Physics.
D) The results of Rutherford's experiment cause a
real dilemma for the scientists of the time who
thought they had all the big problems solved.
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Only newer instruments were thought to be needed
to get more precise measurements.
III. CLUES TO THE SOLUTION OF THE PROBLEM
A) Electromagnetic radiation
1) Sunlight is composed of visible
electromagnetic radiation of wavelengths between
400 and 750 nm and other radiation as well.
2) Scientists had argued about the nature of
light for years. Was it a______________ or a
____________?
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3) At the beginning of the twentieth century, the
wave theory predominated. This was the result of
a diffraction experiment.
Light waves add when they are in phase and
subtract when out of phase. Matter was not known
to subtract.
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4) Characteristics of Waves
? - wavelength - distance between consecutive
peaks - crests - measured in m, nm, angstroms.
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? frequency - (nu) - number of times per second
a crest passes a given point (cycles per second)
1 Hz 1 cycle per second 1/sec sec-1
u speed ? X ?
nm/wave X wave/sec nm/sec
for light - speed of electromagnetic radiation in
a vacuum is a constant - c - 2.998 X 108 m/sec
? X ? c for light
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nu is inversely proportional to the wavelength.
What does this mean?
The range of frequencies or wave lengths is
called the electromagnetic spectrum - it ranges
from gamma rays to TV, FM, AM radio waves.
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for visible light ? is approximately 100 nm
red 750 - 610 nm long ? - low ?
purple 450 - 400 nm short ? - high ?
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5) Visible light from an incandescent light bulb
can be separated by a prism which bends light of
different ?'s, different amounts. Shorter ?'s
are bent more than longer ?'s. A continuous
spectrum results from an incandescent light bulb.
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6) Pass a current through a gas. The gas glows.
When examined through a spectroscope, a line
spectrum is observed, not a continuous one as
seen with the light bulb.
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7) Scientists named Angstrom and Balmer
investigated these lines of hydrogen gas.
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8) Through trial and error a formula was
developed through which the wavelengths in the
visible spectrum of hydrogen could be reproduced.
9) When attempts were made to apply this formula
to the spectrum of atoms with more than one
electron, it did not work.
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10) The wave theory of light could not explain
the line spectra of excited gases. Classical
Newtonian Physics, which worked for moving
automobiles, planets, etc. couldn't generate the
Balmer Equation.
B) Max Planck - explains black body radiation -
i.e. radiation from heated solids. What solids in
your home? kitchen? ???? Stars?
The wavelength distribution depends on
temperature.
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Planck tries to understand the relationship
between intensity and wavelength of emitted light
from hot objects. The then prevailing laws of
physics could not account for what was observed.
Planck made a daring assumption. Energy can be
absorbed by atoms only in chunks of some minimum
size. He gave the name quantum to the smallest
quantity of energy that can be absorbed as
electromagnetic radiation.
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He proposed that each chunk of energy carries a
definite quantity
E h?
The energy of one quantum is equal to Planck's
constant times the speed of light divided by the
wavelength. It must be set up to give Joules.
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According to Planck, energy is emitted in
multiples of h?. E nh? where n is an integer.
This means that only whole numbers are allowed
for n. So you can get 1 h?, or 2 h?, etc. , but
never 0.5 h?. or .2645 h?.
He cannot explain why energies associated with
atoms should be fixed in this way.
This is unusual. K.E. associated with cars,
baseballs, trucks is continuous. In going from 0
to 45 miles per hour, you go through all speeds,
not only certain ones.
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This is the stuff of science fiction where what
happens to speeds?______________
C) Another mystery was solved by Einstein in 1905.
1) He explained the photoelectric effect.
When light hits a metal, electrons are ejected.
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They are only ejected when the frequency of the
light exceeds a certain threshold value for each
particular metal.
Violet light will cause potassium to eject
electrons, no amount of red light, no matter how
bright (intense) it is will have any effect.
Einstein was surprised that the threshold energy
was related to color rather than the intensity.
More electrons are ejected with brighter light of
a certain color, but the energy of each electron
is the same.
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At the atomic level it appears that there is a 1
to 1 relationship between energy and phenomenon,
not a continuum.
2) Einstein takes Planck's theory and extends it
to the structure of light itself. He doesn't
think of it as a continuous wave, but as chunks
of 1wavelength. He calls these chunks of light,
photons.
Each photon has an energy E to h?
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IV. Bohr - 1913 - gave his model of the H atom
Nobel Prize awarded in 1922.
A) A single electron moves in a circular orbit
around the nucleus. To keep the e- from spiraling
into the nucleus, Bohr says that only certain
orbits are allowed which are at various distances
from the nucleus.
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B) Electrons do not obey the laws of classical
physics, therefore they do not lose energy in an
orbit.
C) In each orbit, the energy of the electron is
restricted to a certain value - E - RH/n2
where RH is a constant and n is an integer.
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D) Energy of the electron is quantized. Only
certain energies are allowed to the electron. A
staircase as opposed to an___________.
E) Each orbit is characterized by an integer - 1,
2, 3, etc.
RH is a constant in energy units
2.179 X 10-18 J.
F) When an electron changes orbits it changes
energies.
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G) If energy is emitted
This is the energy of the emitted photon. Recall
the Balmer-Rydberg equation, this equation now
has a basis in Bohr's theory.
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H) Energy is emitted in the form of light
(electromagnetic radiation as the electron moves
from a higher orbit to a lower one (from a higher
energy level to a lower one).
I) Energy is absorbed as electricity or heat as
the electron moves from a lower to a higher orbit
(energy level).
J) Energy levels are closer together as we move
away from the nucleus. The difference between the
energy levels becomes less and this is verified
by the wavelengths observed.
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K) This leads to the birth of quantum mechanics
(only certain energies allowed) new theory for
particles which barely exist. Remember the mass
of the electron is about 1 X 10-28 g.
V. PROBLEMS WITH THE BOHR ATOM
A) It is only successful with the Hydrogen atom
B) It could not account for extra lines in the H
emission spectrum when a magnetic field was
applied to the gas.
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C) The idea of circular orbits and knowing where
the electron is located is impossible.
VI. PARTICLE-WAVE DUALISM
A) 1923-24 - The French physicist de Broglie says
that if light waves exhibit particle properties,
under certain circumstances, then particles of
matter should show wave characteristics under
certain circumstances.
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B) He postulated that a particle of mass m and
speed v has a
where h 6.63 X 10-34 kg m2/sec
if m is large and the speed is small, then ? is
so small as to be meaningless- 10-34 m.
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BUT, if mass is small, the mass of the electron
for example 1 X 10-28 g, and the speed is large,
then the ? becomes measurable in picometers,
10-12 m.
C) In 1927, Davisson and Germer in the U.S., at
Bell Laboratories, and G.P.Thomson (son of J.J.
who showed the electron was a particle)
demonstrate that electrons have a wavelike
nature. What did they have to do???
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Diffraction patterns were obtained by reflecting
electrons off crystals as well as passing
electrons through thin gold foil. These patterns
were similar to those obtained when passing
X-rays through crystals.
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It is interesting to note that J.J. Thomson
received a Nobel Prize for the showing that the
electron was a particle and his son G.P. Thomson
received a Nobel Prize for showing that the
electron can be considered a wave.
Schrödinger - 1926 -awarded Nobel Prize in 1933
A) He developed a more powerful model of the atom.
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B) He combined the equations for the behavior of
waves with the de Broglie equation to generate a
mathematical model for the distribution of
electrons in atoms.
C) The advantage of Schrödinger's approach is
that it consists of mathematical equations known
as wave functions that satisfy the requirements
placed on the behavior of electrons.
D) The disadvantage is that it is difficult to
imagine electrons at waves.
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E) Schrödinger's equation provides information
about an electron's location in terms of
probability of finding an electron which has a
certain energy in a given location.
Why not certainty?
F) In 1927 Werner Heisenberg showed from quantum
mechanics that it is impossible to know
simultaneously, with absolute precision, both the
position and the momentum of a particle such as
an electron. The Heisenberg Uncertainty
Principle!
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G) The Bohr model was a 1 dimensional model (the
radius) that used 1 quantum number to describe
the distribution of electrons in an atom. The
only important information was the size of the
orbit which was described by the quantum number n.
H) Schrödinger's model allows the electron to
occupy 3D space. It requires 3 coordinates or 3
quantum numbers to the describe the ORBITALS in
which the electrons can be found..
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I) The three coordinates that come from the wave
equation of Schrödinger are the principal quantum
number (n),the angular quantum number ( ?),and
the magnetic quantum number (m?). A fourth
quantum number arises out of relativistic effects
(Einstein's Theory of Relativity),the spin
quantum number,(ms).
J) The 4 quantum numbers are like a zip code for
the electron.
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K) They specify an atomic orbital, a region in
space where there is high probability of finding
an electron with a characteristic energy, and the
number of electrons which can occupy the orbital.
VIII. THE QUANTUM NUMBERS
A) The Principal Quantum Number - the modern
equivalent of n in the Bohr Theory. It describes
the main energy level. It can have the values of
the positive integers 1, 2, 3, 4, 5,...?.
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It is related to the average distance of the
electron from the nucleus. The energy of the
electron depends principally on n. Orbitals of
the same quantum number n, belong to the same
shell.
B) Angular momentum quantum number ? - azimuthal
or subsidiary quantum number - distinguishes
orbitals of a given n having different shapes.
Other synonyms are sublevel and subshell. There
are n different kinds of orbitals each with a
distinctive shape denoted by ? .
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? has values from 0 to n-1. (It is important to
remember that in this case 0 does not mean
nothing.)
When n 1, ? can only equal 0 - only one
subshell
When n 2, ? can equal 0 and 1 - two subshells
When n 3, ? can equal 0, 1, and 2 - three
subshells
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Associated with each value of ? is a letter
related to a shape which is a region of space
with an approximate 90 occupancy rate by an
electron of a specified energy.
When ? 0, the letter designation is s and the
shape is spherical.
When ? 1, the letter designation is p and the
shape is dumbbell shaped.
When ? 2, the letter designation is d and the
shape is a cloverleaf and another shape.
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designations used are 1s, 2s, 2p, 3s, 3p, 3d,...
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C) m ? is the magnetic quantum number which
distinguishes orbitals of given n and ?. It
specifies the orientation in space of the atomic
orbital.
The number of different orientations in space
depends on the subshell designated. The allowed
values are integers from - ? through 0 to ?
giving 2 ? 1 possibilities.
When n 1 ? 0 m ? 0 - only 1 orientation
possible - a sphere.
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When n 2 ? 0 m ? 0 - only 1 orientation
possible - a sphere.
When n 2 ? 1 m ? -1, 0, 1 - 3
orientations are possible one dumbbell along
each of the three axes, x, y and z.
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When n 3 ? 0 m ? 0 - only 1 orientation
possible - a sphere.
When n 3 ? 1 m ? -1, 0, 1 The 3
orientations which are possible are one dumbbell
along each of the three axes, x, y, z.
When n 3 ? 2 m ? -2 -1, 0, 1 2 There are
5 orientations possible four cloverleafs and 1
other shape. One is along the xy axes, three
between the axes, and the special one is along
the z axis.
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D) ms is the spin quantum number. An electron has
magnetic properties that correspond to a charged
particle spinning in its axis. Either of 2 spins
are possible???
2 values are possible - ½ and -½ for every set
of n, ? , m ? - this gives two as the number of
electrons which can occupy each orbital.
2 e's in the 1s orbital - "zip code" 1,0,0, ½
and 1,0,0,- ½ .
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2 e's in the 2s orbital - "zip code" 2,0,0, ½
and 2,0,0,- ½ .
2 e's in each 2p orbital - "zip code" 2,1,-1,½
2,1,-1,- ½ 2,1,0, ½ 2,1,0, -½ 2,1,1, ½
2,1,1- ½ .
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IX. CAPACITIES OF PRINCIPAL LEVELS, SUBLEVELS,
AND ORBITALS
A) Each principal level of quantum number n can
hold 2(n2) electrons.
level 1 can hold 2e (2 X 12)
level 2 can hold 8e (2 X 22)
level 5 can hold 50e (2 X 52)
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B) Each principal level of quantum number n can
contain a total of n sublevels.
level 1 has 1 sublevel - s
level 2 has 2 sublevels - s and p
level 3 has 3 sublevels - s, p, and d
level 4 has 4 sublevels - s, p, d and f
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C) Each sublevel of quantum number ? can contain
a total of 2 ? 1 orbitals.
? 0 there are (2 x 0 1) orbitals, 1 orbital
called s.
? 1 there are (2 x 1 1) orbitals, 3 orbitals
called p.
? 2 there are (2 x 2 1) orbitals, 5 orbitals
called d.
D) Each orbital can contain only 2 electrons.
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The End