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Quantum Theory
and Atomic Models
The Electron
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· Streams of negatively charged particles were found to
emanate from cathode tubes.
· J. J. Thompson is credited with their discovery (1897).
Discovery and Properties of the Electron
It was found that these rays could be deflected by electric
or magnetic fields. By adjusting those fields the charge to
mass ratio of the unknown "ray" was found.
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Discovery and Properties of the Electron
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First, they found the velocity of the particle by adjusting the
magnetic field electric forces so that they cancelled out; the
"ray" traveled in a straight line.
ΣF= ma
FB - FE = 0
FE
qvB = qE
FB
v
v = E/B
Since they could measure
E and B, they could
calculate v.
Discovery and Properties of the Electron
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Then they turned off the electric field and the particle moved in
a circular path. They measured the radius of the circle, by
seeing where the particle struck the tube, and then determined
the charge to mass ratio: q/m.
ΣF= ma
FB = ma
qvB = mv2/r
v
FB
qB = mv/r
q/m = v/Br
r
q/m = 1.76 x 1011 C/kg
1
Which one of the following
is not true concerning
cathode rays?
A
They originate from the negative
electrode.
B
They travel in straight lines in the
absence of electric or magnetic
fields.
C
They impart a negative charge to
metals exposed to them.
D
They are made up of electrons.
E
The characteristics of cathode rays
depend on the material from which
they are emitted.
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Millikan Oil Drop Experiment
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Once the charge/mass ratio
of the electron was known,
determination of either the
charge or the mass of an
electron would yield the
other.
Discovery and Properties of the Electron
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The mass of each droplet was estimated by its size. The electric
field was adjusted so the drop fell with constant velocity. The
data showed that the charge was always an integral multiple of a
smallest charge, e. That must be the charge of one electron.
ΣF= ma
FE - mg = 0
qE = mg
2
q = mg/E
FE
q was always an integer
multiple of the same
number, which was
given the symbol "e"
mg
Which of these could be the
charge of an object?
A
0.80 x 10-19 C
B
2.0 x 10-19 C
C
3.2 x 10
D
4.0 x 10-19 C
-19
C
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3
The charge on an electron
was determined in the
__________.
A
cathode ray tube, by J. J. Thompson
B
C
Rutherford gold foil experiment
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Millikan oil drop experiment
D
Dalton atomic theory
E
atomic theory of matter
Blackbody Radiation
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All objects emit electromagnetic
radiation which depends on their
temperature: thermal radiation.
A black body absorbs all
electromagnetic radiation (light)
that falls on it. Because no light
is reflected or transmitted, the
object appears black when it is
cold. However, black bodies emit
a temperature-dependent
spectrum termed blackbody
radiation.
Blackbody Radiation
At normal temperatures, we are
not aware of this radiation.
But as objects become hotter, we
can feel the infrared radiation or
heat.
At even hotter temperatures,
objects glow red and at still
hotter temperatures, object can
glow white hot such at the
filament in a light bulb.
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Blackbody Radiation
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This figure shows
blackbody radiation
curves for three different
temperatures.
The wavelength at the
peak, #p, is related to the
temperature by:
#pT = 2.90 x 10-3 m-K
Classical physics couldn't
explain the shape of these
spectra.
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4
Which of the following colors would indicate the
hotest temperature?
A
Black
B
Red
C
Yellow
D
Blue
Planck’s Quantum Hypothesis
· The wave nature of light could not explain the way an
object glows depending on its temperature: its
spectrum.
· Max Planck explained it by assuming that atoms only
emit radiation in quantum amounts...in steps given by
the formula:
E = hf
where h is Planck’s constant and f is the frequency of
the light
h = 6.6 ´ 10-34 J-s
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Planck’s Quantum Hypothesis
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Planck didn't believe this was a
real...it just worked. It was like
working from the answers in the
book...getting something that works,
but having no idea why.
It didn't make sense that atoms could
only have steps of energy. Why
couldn't they have any energy?
Planck thought a "real" solution
would eventually be found...but this
one worked for some reason.
Which brings us to our next
mystery...
The Photoelectric Effect
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When light strikes a metal, electrons
sometimes fly off.
Classical physics couldn't explain
some specific features about how the
effect works.
So Einstein used Planck's idea to solve
it.
The Photoelectric Effect
If atoms can only emit light in
packets of specific sizes; maybe
light itself travels as packets of
energy given by Planck's formula.
E = hf
where h is Planck’s constant
h = 6.6 ´ 10-34 J-s
or
4.14 x 10-15 eV-s
He called these tiny packets,
or particles, of light, photons.
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5
What is the energy (in nJ) of a photon with a
frequency of 5 x 1022 Hz?
The Photoelectric Effect
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The maximum kinetic energy of
these photons can be measured
using a variable voltage source and
reversing the terminals so that the
electrode C is negative and P is
positive. If the voltage is
increased, there is a point when the
current reaches zero. This is called
the stopping voltage, V0, and it is
given by:
KEmax = eV0
The Photoelectric Effect
We said earlier that when light strikes a metal, electrons
sometimes fly off. Since electrons are held in the metal by
attractive forces, some minimum energy, W0, called the work
function, is required just to get an electron free from the metal.
The input energy of the photon will equal the kinetic energy of
the ejected electron plus the energy required to free the
electron.
hf = KE + W0
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The Compton Effect
A. H. Compton scattered short-wavelength light from various
materials and discovered that the scattered light had a slightly
lower frequency than the incident light, which indicated a loss
of energy. He applied the laws of conservation of momentum
and energy and found that the predicted results corresponded
with the experimental results.
A single photon wavelength, #,
strikes an electron in some
material, knocking it out of its
atom. The scattered photon
has less energy since it gave
some to the electron and thus
has a wavelength of , #'.
Electron
after
collision
Incident
photon (#)
#
Electron
at rest
#
Scattered
photon (#')
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The Compton Effect
Electron
after
collision
The momentum of a photon is given by:
p = E/c
Incident
photon (#)
Since E = hf,
Electron
at rest
p = hf/c = h/#
Using conservation of momentum:
#
#
Scattered
photon (#')
Where m0 is the rest mass of the electron.
Photon Interactions
1. The Compton Effect - the photon can be scattered by an
electron and lose energy in the process.
2. The photoelectric effect - a photon can knock an electron
out of an atom and disappear in the process.
3. The photon can knock and atomic electron to a higher
energy state if the energy of the photon is not sufficient to
knock it out of the atom.
4. Pair production - a photon can produce an electron and a
positron and disappear in the process. (The inverse of pair
production can occur if an electron collides with a positron.
This is called annihilation.)
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Photon Theory of Light
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This particle theory of light assumes that an electron absorbs
a single photon and made specific predictions that proved
true. For instance, the kinetic energy of escaping electrons
vs. frequency of light shown below:
This shows clear
agreement with the
photon theory, and not
with wave theory.
This shows that light
is made of particles,
photons; light is not a
wave.
Wave-Particle Duality; the Principle of
Complementarity
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Earlier we proved that light is a wave.
Now we've proven that light is a particle.
Which is it?
This question has no answer; we must accept the
dual wave-particle nature of light.
While we cannot imagine something that is a wave
and is a particle at the same time; that turns out to
be the case for light.
6
The ratio of energy to
frequency for a given
photon gives
A
its amplitude.
B
C
Planck's constant.
D
its work function.
its velocity.
E = hf
c = lf
h = 6.6 ´ 10-34 J-s
c = 3.00 ´ 108 m/s
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What is a photon?
7
A
an electron in an excited state
B
a small packet of electromagnetic
energy that has particle-like
properties
C
one form of a nucleon, one of the
particles that makes up the nucleus
D
an electron that has been made
electrically neutral
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E = hf
c = lf
h = 6.6 ´ 10-34 J-s
c = 3.00 ´ 108 m/s
8
The energy of a photon
depends on
A
its amplitude.
B
C
its frequency.
D
none of the given answers
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its velocity.
E = hf
c = lf
h = 6.6 ´ 10-34 J-s
c = 3.00 ´ 108 m/s
9
Which color of light has
the lowest energy
photons?
A
red
B
C
yellow
green
D
blue
E = hf
c = lf
h = 6.6 ´ 10-34 J-s
c = 3.00 ´ 108 m/s
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10 The photoelectric effect is
explainable assuming
A
Slide 31 / 83
that light has a wave nature.
B
C
that light has a particle nature.
that light has a wave nature and a
particle nature.
none of the above
D
E = hf
c = lf
h = 6.6 ´ 10-34 J-s
c = 3.00 ´ 108 m/s
11 The energy of a photon that
has a frequency 110 GHz is
A
1.1 × 10-20 J
B
C
1.4 × 10-22 J
D
1.3 × 10-25 J
7.3 × 10
-23
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J
E = hf
c = lf
h = 6.6 ´ 10-34 J-s
c = 3.00 ´ 108 m/s
12 The frequency of a photon
that has an energy of 3.7 x
10-18 J is
A
5.6 × 1015 Hz
B
C
1.8 × 10-16 Hz
2.5 × 10-15 J
D
5.4 × 10 J
E
-8
2.5 × 10
15
J
E = hf
c = lf
h = 6.6 ´ 10-34 J-s
c = 3.00 ´ 108 m/s
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13 The energy of a photon that
has a wavelength of 12.3
nm is
A
1.51 × 10
-17
J
B
C
4.42 × 10
-23
J
1.99 × 10-25 J
D
2.72 × 10
E
1.62 × 10
-50
-17
J
J
E = hf
c = lf
h = 6.6 ´ 10-34 J-s
c = 3.00 ´ 108 m/s
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14 If the wavelength of a
photon is halved, by what
factor does its energy
change?
A
4
B
C
1/4
D
1/2
2
E = hf
c = lf
h = 6.6 ´ 10-34 J-s
c = 3.00 ´ 108 m/s
Energy, Mass, and Momentum of a Photon
Clearly, a photon must travel at the speed of light, since it is light.
Special Relativity tell us two things from this:
The mass of a photon is zero.
The momentum of a photon depends on its wavelength.
m=0
p = hf / c
p = h/l
and since c = lf
This last equation turned out to have huge implications.
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Wave Nature of Matter
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de Broglie asked, "If light can behave like a wave or a
particle, can matter also behave like a wave?"
Amazingly, it does!
de Broglie combined p = h/l with p = mv to get
The wavelength of matter
l = h/(mv)
This wavelength is really small for normal objects, so it had
never been noticed before.
But it has a dramatic impact on the structure of atoms.
Wave Nature of Matter
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Electron wavelengths are often about 10-10 m, about the size
of an atom, so the wave character of electrons is important.
In fact, the two-slit experiment that showed that light was a
wave, has been replicated with electrons with the same
result...electrons are particles and waves.
Electrons fired one at a time towards two slits show the same
interference pattern when they land on a distant screen.
The "electron wave" must go through both slits at the same
time...which is something we can't imagine a single particle
doing...but it does.
The most amazing experiment ever!!!
These photos show electrons being fired one
at a time through two slits.
Each exposure was made after a slightly
longer time. The same pattern emerges as
was found by light.
Each individual electron must behave like a
wave and pass through both slits.
But each electron must be a particle when it
strikes the film, or its wouldn't make one dot
on the film, it would be spread out.
This one picture shows that matter acts like
both a wave and a particle.
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15 What is the wavelength of
a 0.25 kg ball traveling at
20 m/s?
l = h/(mv)
h = 6.6 ´ 10-34 J-s
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16 What is the wavelength of a
80 kg person running 4.0
m/s?
l = h/(mv)
h = 6.6 ´ 10-34 J-s
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17 What is the wavelength of
the matter wave associated
with an electron (me = 9.1 x
10-31kg) moving with a
speed of 2.5 × 107 m/s?
A
0.29 nm
B
0.36 nm
C
0.48 nm
D
0.56 nm
l = h/(mv)
h = 6.6 ´ 10-34 J-s
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18 What is the wavelength of
the matter wave associated
with an electron (me = 9.1 x
10-31kg) moving with a
speed of 1.5 × 106 m/s?
A
0.29 nm
B
0.36 nm
C
0.48 nm
D
0.56 nm
l = h/(mv)
h = 6.6 ´ 10-34 J-s
The Atom, circa 1900
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· The prevailing theory was that
of the “plum pudding” model,
put forward by Thompson.
· It featured a positive sphere of
matter with negative electrons
imbedded in it.
Discovery of the Nucleus
Ernest Rutherford shot a
particles at a thin sheet of
gold foil and observed the
pattern of scatter of the
particles.
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Discovery of the Nucleus
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While most particles went
straight through, as
expected, some bounced
back...which was totally
unexpected.
Early Models of the Atom
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The only way to
account for that was to
assume all the positive
charge was contained
within a tiny volume.
A small very dense
nucleus must lie within
a mostly empty atom.
Now we know that the
radius of the nucleus is
1 / 10,000 that of the
atom.
The Nuclear Atom
Since some particles were
deflected at large angles,
Thompson’s model could
not be correct.
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Early Models of the Atom
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Rutherford's experiment showed that the positively charged
nucleus must be very small compared to the rest of the atom.
Then I remember two or three days later Geiger coming
to me in great excitement and saying "We have been
able to get some of the alpha-particles coming
backward …" It was quite the most incredible event
that ever happened to me in my life. It was almost as
incredible as if you fired a 15-inch shell at a piece of
tissue paper and it came back and hit you.
- Rutherford
The Nuclear Atom
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· Rutherford postulated a very small, dense nucleus with
the electrons around the outside of the atom.
· Most of the volume of the atom is empty space.
The Nuclear Atom
If an atom were magnified to be the size of a gymnasium
(about 100 m across), the proton would be about the size of a
ping pong ball (1 cm across), the electrons would be too
small to see, and all the rest would be just emptiness...not
filled with air (like a gym), but nothing.
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19 The gold foil experiment
performed in Rutherford's
lab __________.
A
confirmed the plum-pudding model of
the atom
B
led to the discovery of the atomic
nucleus
C
was the basis for Thomson's model
of the atom
D
utilized the deflection of beta
particles by gold foil
E
proved the law of multiple
proportions
20 In the Rutherford nuclearatom model, __________.
A
the heavy subatomic particles reside
in the nucleus
B
the principal subatomic particles all
have essentially the same mass
C
the light subatomic particles reside
in the nucleus
mass is spread essentially uniformly
throughout the atom
D
Subatomic Particles
· Protons were discovered by Rutherford in 1919.
· Neutrons were discovered by James Chadwick in 1932.
· Protons and electrons are the only particles that have charge.
· Protons and neutrons have essentially the same mass.
· The mass of an electron is extremely small.
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The Problem with the Nuclear Atom
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· The nucleus of an atom is small, 1/10,000 the size of the atom.
· The electrons are outside the nucleus, moving freely within the
vast empty atom
· The nucleus is positive; the electron is negative
· There is an electric force, FE = kq1q2/r2, pulling the electron
towards the nucleus
· There is no other force acting on the electron; there is nothing to
support it; it experiences a net force towards the nucleus
· Why don't the electrons fall in...why doesn't the atom collapse
into its nucleus?
The Problem with the Nuclear Atom
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Based on the equations
#F = ma
and
x = x0 + vot +1/2at2
All atoms would collapse in about 10-10 s
Earth would collapse to less than a mile across in
less than a billionth of a second.
The universe as we know it would end.
The Problem with the Nuclear Atom
Perhaps electrons orbit the nucleus...like planets orbit the
sun.
But then they would constantly be accelerating as they travel
in a circle:
a = v2/r
But it was known that an accelerating charge radiates
electromagnetic energy...light.
All the kinetic energy would radiate away in about that same
billionth of a second...then it would fall into the nucleus.
All the atoms in the universe would still collapse.
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The Problem with the Nuclear Atom
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Classical physics failed to explain how atoms could exist.
A new approach was needed.
The next step led to the Bohr model of the atom, which
was a semi-classical explanation of atoms. It would be an
important transition to modern quantum theory.
An important clue was found in the spectra of gas
discharge tubes.
Atomic Spectra
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A very thin gas heated in a discharge tube emits light
only at characteristic frequencies.
Atomic Spectra
An atomic spectrum is a line spectrum – only certain
frequencies appear. If white light passes through such a gas,
it absorbs at those same frequencies.
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Atomic Spectra
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Why don't atoms radiate, or absorb, all frequencies of light?
Why do they radiate light at only very specific frequencies, and
not at others?
Atomic Spectra
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The wavelengths of electrons emitted from hydrogen
have a regular pattern:
Balmer series
Lyman series
Paschen series
Atomic Spectra: Key to the Structure of
the Atom
A portion of the complete spectrum of hydrogen is shown
here. The lines cannot be explained by the Rutherford theory.
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The Bohr Atom
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Bohr proposed that electrons could orbit the nucleus, like
planets orbit the sun...but only in certain specific orbits.
He then said that in these orbits, they wouldn't radiate energy,
as would be expected normally of an accelerating charge.
These stable orbits would somehow violate that rule.
Each orbit would correspond to a different energy level for the
electron.
The Bohr Atom
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These possible energy states for atomic electrons were
quantized – only certain values were possible. The spectrum
could be explained as transitions from one level to another.
Electrons would only radiate when they moved between orbits,
not when they stayed in one orbit.
The Bohr Atom
As long as an electron was in an orbit given by the
below formula, it would not emit electromagnetic
radiation.
The observed spectrum of the hydrogen atom is
predicted successfully by transitions between
these orbits.
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The Bohr Atom
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An electron is held in orbit by the Coulomb force:
The Bohr Atom
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Using the Coulomb force, we can calculate the
radii of the orbits. These matched the sizes of
known atoms very well.
The Bohr Atom
The radii of the orbits of a hydrogen atom are
given by the below formula, with the smallest
orbit,
rn = n2r1,
Z
(for hydrogen, Z = 1)
r1 = 0.53 x 10-10m.
n = 1, 2, 3, 4, ....
Notice that the orbits grow in size as the square of
n, so they get much larger as n increases.
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21 The radius of the orbit for
the third excited state (n=4)
of hydrogen is ______r1.
22 The radius of the orbit for
the fifth excited state (n=6)
of hydrogen is ____ x 10-10m.
r1 = 0.50 x 10-10m
The Bohr Atom
Using the Coulomb force, he calculated the
energy of each orbit. For hydrogen he arrived at
this result:
E=
-13.6 eV
n2
n = 1, 2, 3, 4, ....
Notice that the energy levels are all negative,
otherwise the electron would be free of the atom.
The levels get closer together, and closer to zero,
as n increases.
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The Electron Volt (eV)
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In atomic physics, energies are so low that it's
awkward to use Joules (J). A smaller unit of
energy is the electron-volt (eV).
It's value equals the potenial energy of an electron
in a region of space whose voltage (V) is 1.0 volt.
UE = qV
UE = (1.6 x 10-19C)(1.0V)
UE = 1.6 x 10-19J # 1.0 eV
1.0 eV = 1.6 x 10-19J
The Electron Volt (eV)
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It's also convenient to convert Plank's constant to
units of eV-s rather than J-s.
h = 6.63 x 10-34 J-s
h = (6.63 x 10-34 J-s)
-15
h = 4.14 x 10
eV
( 1.61.0x 10
J)
-19
eV-s
h = 4.14 x 10-15 eV-s
1.0 eV = 1.6 x 10-19J
23 What is the energy of the
second excited state (n=3)
of hydrogen?
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24 What is the energy of the
fifth excited state (n=6) of
hydrogen?
The Bohr Atom
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The lowest energy level is
called the ground state; the
others are excited states.
de Broglie’s Hypothesis Applied to
Atoms
Scientists didn't like the lack of an explanation for why
electrons didn't radiate when in those orbits.
But de Broglie's wave theory of matter explained it very well.
As long as the wavelength on an electron in orbit was the
same as the circumference of the orbit, it would not radiate.
This approach yields the same relation that Bohr had
proposed.
In addition, it makes more reasonable the fact that the
electrons do not radiate, as one would otherwise expect
from an accelerating charge.
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de Broglie’s Hypothesis Applied to
Atoms
Slide 79 / 83
These are circular
standing waves for n = 2,
3, and 5.
Quantum Physics
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While a big step forward, Bohr's model only worked for
atoms that had one electron, like hydrogen or certain
ionized atoms.
It failed for all atoms other than hydrogen.
The idea that the electron was a particle in orbit around
the nucleus, but with wavelike properties that only
allowed certain orbits, worked only for hydrogen.
Semi-classical explanations failed except for hydrogen.
It turned out that only a lucky chance let it work even in
that case.
Quantum Mechanics
Our goal was to explain why electrons in an atom don't fall into the
nucleus.
An electron, as a charged particle, would fall in because of Newton's
Second Law.
#F = ma
But electrons, in atoms, aren't particles, they're waves.
Waves don't follow Newton's Second Law.
Schrodinger had to invent a new equation for wave mechanics.
H# = E#
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Quantum Mechanics
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H# = E#
The simplicity of this equation is deceptive. Here's what it
looks like when expanded for one type of problem.
It's only solved for general cases in advanced university
courses.
However, computers have been used to exactly solve it for
many specific cases: atoms, molecules, etc.
Quantum Mechanics
Those solutions will allow us to understand how the
microscopic world works: atoms, the periodic table,
molecules, chemical bonds, etc.
Quantum mechanics is very different from classical physics–
you can predict what a lot of electrons will do on average,
but have no idea what any individual electron will do.
In Chemistry, you'll be using the solutions to Schrodinger's
Equation equations, and the physics you've learned this
year, to explored the nature of matter.
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