Physics 1C

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Physics 1C

• Lecture 29A

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Atomic Physics

• The study of quantum mechanics led to amazing
  theories as to how the subatomic world worked.
• One of the first theories of how the atom was
  composed was the Plum Pudding Model by J.J.
  Thomson (incorrect!!!).

• In this model the atom was thought to be a large
  volume of positive charge with smaller electrons
  embedded throughout.
• Almost like a watermelon with seeds.

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Atomic Physics

• But then in 1911, Ernest Rutherford performed an
  experiment where he shot a beam of positively
  charged particles (alphas) against a thin metal
  foil.
• Most of the alpha particles passed directly
  through the foil.

• A few alpha particles were deflected from their
  original paths (some even reversed direction).

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Atomic Physics

• This thin foil experiment led Rutherford to
  believe that positive charge is concentrated in
  the center of the atom, which he called the
  nucleus.

• He then predicted that the electrons would orbit
  the nucleus like planets orbit the sun.
• Centripetal acceleration should keep them from
  spiraling in (like the Moon).
• Thus, his model was named the Planetary Model.

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Atomic Physics

• But there were a few problems with the Planetary
  model of the atom. Such as, what happens when a
  charged particle (like the electron) is
  accelerated?

• It gives off light of a particular frequency.
• As light is given, the electron will lose energy
  and its radius should decrease.
• The electron should eventually spiral into the
  nucleus.

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Emission Spectra

• Finally, the key to understanding atoms was to
  look at the light that was emitted from them.
• When a low-pressure gas is subjected to an
  electric discharge, it will emit light
  characteristic of the gas.
• When the emitted light is analyzed with a
  spectrometer, we observe a series of discrete
  lines.

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Emission Spectra

• This is known as emission spectra.
• Each line has a different wavelength (color).
• The elemental composition of the gas will tell
  what the resulting color lines will be.
• Note that in general elements with a higher
  atomic number will have more lines.

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Emission Spectra

• The easiest gas to analyze is hydrogen gas.
• Four prominent visible lines were observed, as
  well as several ultraviolet lines.

• In 1885, Johann Balmer, found a simple functional
  form to describe all of the observed wavelengths

• where RH is known as the Rydberg constant
  RH  1.0973x107m-1.
• n 3, 4, 5, ....

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Emission Spectra

• Every value of n led to a different line in the
  spectrum.
• For example, n  3 led to a ?3  656nm and n  4
  led to a ?4  486nm.
• The series of lines described by this equation is
  known as the Balmer Series.

• Note how the spacing between the lines gets
  closer and closer the smaller the wavelength gets.

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Absorption Spectra

• In addition to emission spectra (lines emitted
  from a gas), there is also absorption spectra
  (lines absorbed by a gas).
• An element can also absorb light at specific
  wavelengths.
• An absorption spectrum can be obtained by passing
  a continuous radiation spectrum through a cloud
  of gas.
• The elements in gas will absorb certain
  wavelengths.

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Absorption Spectra

• The absorption spectrum consists of a series of
  dark lines superimposed on an otherwise
  continuous spectrum.
• The dark lines of the absorption spectrum
  coincide with the bright lines of the emission
  spectrum.
• This is how the element of Helium was discovered.

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Hydrogen Atom

• In 1913, Neils Bohr explained atomic spectra by
  utilizing Rutherfords Planetary model and
  quantization.

• In Bohrs theory for the hydrogen atom, the
  electron moves in circular orbit around the
  proton.
• The Coulomb force provides the centripetal
  acceleration for continued motion.

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Hydrogen Atom

• Only certain electron orbits are stable.

• In these orbits the atom does not emit energy in
  the form of electromagnetic radiation.
• Radiation is only emitted by the atom when the
  electron jumps between stable orbits.

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Hydrogen Atom

• The electron will move from a more energetic
  initial state to less energetic final state.

• The frequency of the photon emitted in the jump
  is related to the change in the atoms energy

• If the electron is not jumping between allowed
  orbitals, then the energy of the atom remains
  constant.

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Hydrogen Atom

• Bohr then turned to conservation of energy of the
  atom in order to determine the allowed electron
  orbitals.
• The total energy of the atom will be

• But the electron is undergoing centripetal
  acceleration (Newtons second law)

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Angular Momentum

• Recall from classical mechanics that there was
  this variable known as angular momentum, L.
• Angular momentum, L, was defined as
  L  I ?
• where I was rotational inertia and ? was angular
  velocity.
• For an electron orbiting a nucleus we have that

• Giving us

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Hydrogen Atom

• Bohr postulated that the electrons orbital
  angular momentum must be quantized as well

• where h is defined to be h/2p.
• This gives us a velocity of

• Substituting into the last equation from two
  slides before

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Hydrogen Atom

• Solving for the radii of Bohrs orbits gives us

• The integer values of n 1, 2, 3, give you the
  quantized Bohr orbits.
• Electrons can only exist in certain allowed
  orbits determined by the integer n.
• When n 1, the orbit has the smallest radius,
  called the Bohr radius, ao.
• ao 0.0529nm

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Hydrogen Atom Bohrs Theory

• We know that the radii of the Bohr orbits in a
  hydrogen atom are quantized

• We also know that when n 1, the radius of that
  orbit is called the Bohr radius (ao 0.0529nm).
• So, in general we have rn  n2ao
• The total energy of the atom can be expressed as
  

(assuming the nucleus is at rest)
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Hydrogen Atom

• Plus, from centripetal acceleration we found
  that

• Putting this back into energy we get

• But we can go back to the result for the radius
  (rn  n2 ao) to get a numerical result.

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Hydrogen Atom

• This is the energy of any quantum state (orbit).
  Please note the negative sign in the equation.
• When n  1, the total energy is 13.6eV.
• This is the lowest energy state and it is called
  the ground state.
• The ionization energy is the energy needed to
  completely remove the electron from the atom.
• The ionization energy for hydrogen is 13.6eV.

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Hydrogen Atom

• So, a general expression for the radius of any
  orbit in a hydrogen atom is rn  n2ao
• The energy of any orbit is

• If you would like to completely remove the
  electron from the atom it requires 13.6eV of
  energy.

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Hydrogen Atom

• What are the first four energy levels for the
  hydrogen atom?
• When n  1 gt E1  13.6eV.
• When n  2 gt E2   13.6eV/22 3.40eV.
• When n  3 gt E3   13.6eV/32 1.51eV.
• When n  4 gt E4   13.6eV/42 0.850eV.
• Note that the energy levels get closer together
  as n increases (similar to how the wavelengths
  got closer in atomic spectra).
• When the atom releases a photon it will
  experience a transition from an initial higher
  energy level (ni) to a final lower energy level
  (nf).

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Hydrogen Atom

• The energies can be compiled in an energy level
  diagram.
• As the atom is in a higher energy state and
  moves to a lower energy state it will release
  energy (in the form of a photon).
• The wavelength of this photon will be determined
  by the starting and ending energy levels.

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Hydrogen Atom

• The photon will have a wavelength ? and a
  frequency f

• To find the wavelengths for an arbitrary
  transition from one orbit with nf to another
  orbit with ni, we can generalize Rydbergs
  formula

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Hydrogen Atom

• The wavelength will be represented by a
  different series depending on your final energy
  level (nf).
• For nf 1 it is called the Lyman series (ni
  2,3,4,...).
• For nf 2 it is called the Balmer series (ni
  3,4,5...).
• For nf 3 it is called the Paschen series (ni
  4,5,...).

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Concept Question

• When a cool gas is placed between a glowing wire
  filament source and a diffraction grating, the
  resultant spectrum from the grating is which one
  of the following?

• A) line emission.
• B) line absorption.
• C) continuous.
• D) monochromatic.
• E) de Broglie.

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Atomic Spectra

• Example
• What are the first four wavelengths for the
  Lyman, Balmer, and Paschen series for the
  hydrogen atom?

• Answer
• The final energy level for either series will be
  nf  1 (Lyman), nf  2 (Balmer), and nf  3
  (Paschen).

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Atomic Spectra

• Answer
• Turn to the generalized Rydberg equation

• For the Lyman series we have

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Atomic Spectra

• Answer
• Finally for the Lyman series

• For the Balmer series we have from before
• ?1  656nm, ?2  486nm, ?3  434nm, ?4  410nm.

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Atomic Spectra

• Answer
• For the Paschen series we have

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Atomic Spectra

• The only series that lies in the visible range
  (390 750nm) is the Balmer series.
• The Lyman series lies in the ultraviolet range
  and the Paschen series lies in the infrared
  range.
• We can extend the Bohr hydrogen atom to fully
  describe atoms that are close to hydrogen.
• These hydrogen-like atoms are those that only
  contain one electron. Examples He, Li, Be
• In those cases, when you have Z as the atomic
  number of the element (Z is the number of protons
  in the atom), you replace e2 with Ze2 in the
  hydrogen equations.

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Concept Question

• Consider a hydrogen atom, a singly-ionized helium
  atom, a doubly-ionized lithium atom, and a
  triply-ionized beryllium atom. Which atom has the
  lowest ionization energy?

• A) hydrogen
• B) helium
• C) lithium
• D) beryllium
• E) the ionization energy is the same for all four

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For Next Time (FNT)

• Finish reading Chapter 29