Download 6 - Rutgers Physics

Honors Physics III
Lecture 6:
Bohr Model of the Atom
http://www.physics.rutgers.edu/ugrad/273a
Weida Wu
Announcements
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First Midterm Oct. 5th
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10-minute talk after Oct. 12’s class
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Review on Monday Oct. 3rd.
Details to come soon on the course website
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Given by Naval Officer Recruiter
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2
Graduate Student Physics Tutors
The physics graduate students in the list
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There is in addition free peer tutoring
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http://physics.rutgers.edu/descr/tutorpage2016.pdf
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Convenient units
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The electron’s rest energy E0=mc2 is
=(9.11×10-31 kg)(3.0×108 m/s)2 ≈ 8.2×10-14 J
It’s frequently more convenient to measure energy in units of
electron volts:
 The amount of energy gained by a single electron when
moved across an electric potential difference of one volt.
Using 1 eV=1.6×10-19 J, so E0=0.511 MeV can write electron
mass m=E0/c2 = 0.511 MeV/c2
This implies a new, useful unit of mass (MeV/c2) and
momentum (MeV/c)
Of course, we can also use eV/c, keV/c, GeV/c, etc...
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Line Spectra
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Unlike continuous spectra from blackbodies, chemical elements produced
discrete spectra.
19th century experiments would excite elements (by applying high voltage)
and study their optical spectra. (Recall demo from Monday)
Emitted light is passed through a diffraction grating with thousands of lines
per ruling and diffracted according to its wavelength λ by the equation:
where d is the distance of line separation and n is an integer called the order
number.
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Balmer Series
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In 1885, Johann Balmer found an empirical formula for
wavelength of the visible hydrogen line spectra in nm:
nm
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(where k = 3,4,5…)
In fact, the Balmer series was only the first of several series to be
discovered (first because it was the only series in the visible part
of the light spectrum).
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Rydberg Formula
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Johannes Rydberg and Walther Ritz found a more
general empirical equation for calculating the
wavelengths:
RH=1.097×107 m-1
n and k are integers (where k>n)
For each value of n,
the set of
wavelengths
constitutes a “series”
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In fact, what was being observed were transitions between
different energy levels. The Lyman series was the
transition to the lowest level (i.e. the ground state).
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Atomic theory of matter
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First proposed by Greek philosophers
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Supporting evidence from physics and chemistry
Evidence of substructure of atoms
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Atoms are building blocks of matter
Discovery of electrons (J.J. Thomson, 1897)
What is the structure of atoms?
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Models of Atoms
Atoms contain negatively charged electrons
2) Overall, atoms are electrically neutral
3) Electrons account for only a tiny fraction of an atom’s mass
 Thomson Model (1898):
Uniform positive charge with embedded electrons
1)
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Please try this at home:
http://phet.colorado.edu/en/simulation/rutherford-scattering
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Models of Atoms
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Geiger-Marsden Experiments (1911):
 Ernest Rutherford, with Hans
Geiger and Ernest Marsden (a
graduate student) scattered alpha
particles from a radioactive source
off of a thin gold foil.
 Measure scattering angles θ.
 Much larger deflections observed.
Too many values of θ, even 180o!
 This was impossible in
Thomson model.
http://hyperphysics.phy-astr.gsu.edu/Hbase/hframe.html
(Alpha-particle = 2 protons + 2 neutrons)
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Rutherford Model
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“It was quite the most incredible event that has 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. On consideration, I
realized that this scattering backwards must be the
result of a single collision, and when I made
calculations I saw that it was impossible to get anything
of that magnitude unless you took a system in which
the greater part of the mass of the atom was
concentrated in a minute nucleus. It was then that I had
the idea of an atom with a minute massive center carrying a
charge.”
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Lord Rutherford, 1936
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Models of Atoms
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Rutherford Model (1915):
 Positive charge is contained in a tiny by massive
nucleus.
 Electrons orbit the nucleus.
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Experiments agreed with Rutherford model.
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Composition of Atoms
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If matter is primarily composed of atoms, what are
atoms composed of?
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Michael Faraday (1833): Discovered the law of electrolysis
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J.J. Thomson (1897): Identification of cathode rays as
electrons and measurement of ratio (e/m) of these particles
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Electron is a constituent of all matter!
Humankind’s first glimpse into subatomic world!
Robert Millikan (1909): Precise measurement of electric
charge
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Mass ∝ (q)(atomic weight)/(valence #)
Showed that particles ~1000 times less massive than the hydrogen
atom exist
Rutherford, with Geiger & Marsden (1910): Established the
nuclear model of the atom
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Atom = compact positively charged nucleus surrounded by an
orbiting electron cloud
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The Hydrogen Atom
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Pre-Quantum era questions about the Hydrogen Atom
 Why does the Rydberg formula work?
 Why is the absorption spectrum the same as the
emission spectrum?
 Why is the ionization energy of Hydrogen 13.6 eV?
 Why is the atom stable in the first place?
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Classical Atomic Model
Recall what would happen if we apply classical
electromagnetism to an electron orbiting a proton.
Coulomb’s Law binds electron to proton, but the electron
is accelerating.
An accelerating electron emits electromagnetic
radiation (this is an unavoidable consequence of
Maxwell’s equations).
Eventually (10-9 seconds) the electron loses enough energy
through radiation and collides with the nucleus (proton).
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The Bohr Atom
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The idea of the nuclear atom (Rutherford’s
planetary model) raised many questions at the
next deeper level.
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How do the the electrons move around the nucleus
and how does their motion account for the observed
spectral lines?
In 1913, Niels Bohr published
a revolutionary three-part paper.
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Bohr Model of Hydrogen Atom
Assumptions:
1.
Electron can only be in circular orbits
that have orbital angular momenta:
2.
3.
(n is the principal quantum number)
Atom does not radiate while in such
states
Atom radiates when electron jumps
from one allowed orbit to another.
Emitted photon carries off difference
in energy between the orbits.
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Bohr Theory of Hydrogen Atom
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Circular Oribit - In Bohr model, Coulomb force
still provides centripetal acceleration:
But now, orbital angular momentum is
quantized:
Put lower equation into above equation …
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So the n allowed radii are:
Bohr radius
Bohr radius gives the minimum radius of the hydrogen atom, i.e. the smallest radius of
the atom, because it describes the radius of the ground state, or the most tightly bound
state of the atom.
We now have radii in terms of fundamental
constants!
 And radii are quantized!
e.g. r1=a0=0.053 nm,
r2=4a0=0.211 nm,
r3=9a0=0.475 nm
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Electron Speed in Hydrogen Atom
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So, now let’s try to solve for the speed. Since:
and
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Then speed is quantized, too.
where
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α is the called the “fine-structure constant”. Notice that
it is dimensionless.
The fine-structure constant will show up in various places
and has an important role in fundamental physics.
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Energy in Hydrogen Atom
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Let’s solve for the energy. Since the velocity was not
(quite) relativistic, we can perform a classical
calculation.
We can plug our previous expressions for the velocity
and radius into this equation, as well. We now get a
quantized energy.
The ground state of the H atom is for n=1
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So what does this mean?
We now have electron energy in each orbit in terms of fundamental
constants!
The negative energy means that we must add energy to hydrogen to
ionize it.
Energy is not only quantized, but it has a ground state (n=1) below
which the photon (emitted by the electron) cannot fall.
So we have shown that the Hydrogen atom is stable and it’s ground
state is the measured ionization energy.
There are also excited states (n>1).
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Hydrogen Line Spectrum:
Bohr Model
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Let’s compare to the Rydberg equation.
The energy of a photon when a Hydrogen atom deexcites from one state to the next is:
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Solving for the inverse wavelength, we get
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But RH=0.01096776 nm-1
That’s really close, but can we do even better?
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Reduced-mass Correction
In fact the electron and the nucleus
Nucleus
are revolve around a common mass
CM
(the nucleus is not infinitely massive,
although, compared to the electron, it’s close).
Electron
 Replace mass in the equation with the reduced mass:
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where M is the mass of the nucleus.
 For Hydrogen, µ=0.99456m
 Once you do that, you get precisely the Rydberg constant
(M=Mproton).
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Absorption of Light
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If photon wavelength
satisfies Rydberg formula,
an electron in orbit will
absorb the photon and
jump to a higher orbit
Therefore,
absorption spectrum =
emission spectrum
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To Summarize
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From three assumptions, we have recovered the
full atomic spectrum of hydrogen!
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The discovery of deuterium
(Urey, 1932)
Natural hydrogen is 1 part in 6000 deuterium (nucleus
has one proton and one neutron, i.e. a deuteron).
So the reduced mass of deuterium is a little bit
different than regular hydrogen.
The well known Balmer line (n=3 to n=2 transition)
has wavelength 656.5 nm. But for deuterium, it is
656.3 nm!
In natural hydrogen, the regular emissions lines are
accompanied by very faint lines from deuterium!
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The Hydrogen Atom
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Pre-Quantum era questions about the Hydrogen Atom
 Why does the Rydberg formula work?
 Why is the absorption the same as the emission
spectrum?
 Why is the ionization energy of Hydrogen 13.6 eV?
 Why is the atom stable in the first place?
Addressed by the Bohr model. But the Bohr model is
just that: a model. Why should the angular
momentum be quantized in the first place? The
answer will come with quantum mechanics.
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Bohr’s Atomic model explained …
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Explained the limited number of lines seen in
the absorptions spectrum of Hydrogen
compared to emission spectrum
Emission of x-rays from atoms
Chemical properties of atoms in terms of
electron-shell model
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Deficiencies of Bohr Theory
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Many of the energy levels in hydrogen are actually
doublets, i.e. two levels closely spaced in energy
(wavelength). Bohr theory cannot account for this.
Quantization of angular momentum is just assumed,
not explained or derived.
Cannot explain spectra of complex atoms (with more
electrons).
Notions of fixed radii and speeds are inconsistent with
uncertainty principle (more on this next week).
Bohr theory is non-relativistic.
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Not too bad since v/c=1/137, but it means theory can’t be
exactly right.
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Atomic excitation by electrons
Franck-Hertz experiment
Quantized absorption of
electron kinetic energy.
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Please try this at home:
http://phet.colorado.edu/en/simulation/hydrogen-atom
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Question 1
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In the Bohr model, we have:
A electron in an Hydrogen atom drops from the n=6
state to the n=2 state. What is true about the
relationship between E6 and E2?
A. E6=E2/9
B. E6=E2/4
C. E6=E2
D. E6=4E2
E. E6=9E2
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Question 1
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In the Bohr model, we have:
A electron in an Hydrogen atom drops from the n=6
state to the n=2 state. What is true about the
relationship between E6 and E2?
A. E6=E2/9
B. E6=E2/4
C. E6=E2
D. E6=4E2
E. E6=9E2
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Example
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A hydrogen atom goes from the n=4 state to the
ground state. Find the atom’s recoil speed.
By momentum conservation, patom=12.75 eV/c
Atom’s mass ≈ proton mass = 938 MeV/c2
So non-relativistic formula is acceptable….
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Atom’s kinetic energy for v=4m/s works out to
K=1/2 mv2 = 8.7 x 10-8 eV
So it was ok to ignore it in developing the Bohr model!
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Question 2
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A.
B.
C.
D.
Mimicking the experiment of Rutherford, Geiger and Marsden,
you fire alpha particles of about 5 MeV into an unknown target
and measure the scattering angle. Which of the following is likely
to affect the scattering angle of the alpha particles coming off of
the target?
The temperature of the target.
The number of electrons orbiting the nuclei of the target.
The Coulomb force between the alpha particle and the electrons
in the target.
The Coulomb force between the alpha particle and the nuclei of
the target.
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Question 2
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A.
B.
C.
D.
Mimicking the experiment of Rutherford, Geiger and Marsden,
you fire alpha particles of about 5 MeV into an unknown target
and measure the scattering angle. Which of the following is likely
to affect the scattering angle of the alpha particles coming off of
the target?
The temperature of the target.
The number of electrons orbiting the nuclei of the target.
The Coulomb force between the alpha particle and the
electrons in the target.
The Coulomb force between the alpha particle and the nuclei
of the target.
The Coulomb force dominates in scattering experiments at lower
energies, and within the Coulomb force only the massive, positive
nucleus of the target interacting with the alpha particle determines
the scattering angle.
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
A.
B.
C.
D.
Question 3
In attempting to describe the atom, Bohr made a set of general
assumptions. Which of the following statements is NOT a result or
closely aligned with those general assumptions (the assumptions might
not be quantum mechanically correct)?
The radius of a hydrogen atom can be calculated from a certain
combination of fundamental constants.
Electrons in the hydrogen atom transfer between quantized energy
states and can exist nowhere else in the atom but in these energy states.
The velocity of the electron around the nucleus is the same in all
orbits, although the shape of the orbit changes with higher values of n,
the principal quantum number
Stationary states are states where the electron accelerates around the
nucleus but does not emit electromagnetic radiation.
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
A.
B.
C.
D.
Question 3
In attempting to describe the atom, Bohr made a set of general
assumptions. Which of the following statements is NOT a result or
closely aligned with those general assumptions (the assumptions might
not be quantum mechanically correct)?
The radius of a hydrogen atom can be calculated from a certain
combination of fundamental constants.
Electrons in the hydrogen atom transfer between quantized energy
states and can exist nowhere else in the atom but in these energy
states.
The velocity of the electron around the nucleus is the same in all
orbits, although the shape of the orbit changes with higher values
of n, the principal quantum number
Stationary states are states where the electron accelerates around the
nucleus but does not emit electromagnetic radiation.
The velocity of the electron is not constant according to Bohr, but
instead changes with changing principal quantum number.
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