Bohr Model of Atom

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Bohr Model of Atom

• Bohr proposed a model of the atom in which the
  electrons orbited the nucleus like planets around
  the sun
• Classical physics did not agree with his model.
  Why?
• To overcome this objection, Bohr proposed that
  certain specific orbits corresponded to specific
  energy levels of the electron that would prevent
  them from falling into the protons
• As long as an electron had an ENERGY LEVEL that
  put it in one of these orbits, the atom was
  stable
• Bohr introduced Quantization into the model of
  the atom

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Bohr Model of Atom

By blending classical physics (laws of motion)
with quantization, Bohr derived an equation for
the energy possessed by the hydrogen electron in
the nth orbit.
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Bohr Model of the Atom

• The symbol n in Bohrs equation is the principle
  quantum number
• It has values of 1, 2, 3, 4,
• It defines the energies of the allowed orbits of
  the Hydrogen atom
• As n increases, the distance of the electron from
  the nucleus increases

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Atomic Spectra and Bohr
Energy of quantized state - Rhc/n2

• Only orbits where n some positive integer are
  permitted.
• The energy of an electron in an orbit has a
  negative value
• An atom with its electrons in the lowest possible
  energy level is at GROUND STATE
• Atoms with higher energies (ngt1) are in EXCITED
  STATES

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Energy absorption and electron excitation

• If e-s are in quantized energy states, then ?E
  of states can have only certain values. This
  explains sharp line spectra.

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Spectra of Excited Atoms

• To move and electron from the n1 to an excited
  state, the atom must absorb energy
• Depending on the amount of energy the atom
  absorbs, an electron may go from n1 to n2, 3, 4
  or higher
• When the electron goes back to the ground state,
  it releases energy corresponding to the
  difference in energy levels from final to initial
• ?E Efinal - Einitital
• E -Rhc/n2
• ?E -Rhc/nfinal2 - (-Rhc/ninitial2) -Rhc (1/
  nfinal2 - 1/ninitial2)
• (does the last equation look familiar?)

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Origin of Line Spectra
Balmer series
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Atomic Line Spectra and Niels Bohr

• Bohrs theory was a great accomplishment.
• Recd Nobel Prize, 1922
• Problems with theory
• theory only successful for H.
• introduced quantum idea artificially.
• So, we go on to QUANTUM or WAVE MECHANICS

Niels Bohr (1885-1962)
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Wave-Particle Duality
DeBroglie thought about how light, which is an
electromagnetic wave, could have the property of
a particle, but without mass. He postulated that
all particles should have wavelike
properties This was confirmed by x-ray
diffraction studies
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Wave-Particle Duality
de Broglie (1924) proposed that all moving
objects have wave properties. For light E
mc2 E h? hc / ? Therefore,
mc h / ? and for particles
(mass)(velocity) h / ?
L. de Broglie (1892-1987)
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Wave-Particle Duality

• Baseball (115 g) at 100 mph
• ? 1.3 x 10-32 cm
• e- with velocity
• 1.9 x 108 cm/sec
• ? 0.388 nm

• The mass times the velocity of the ball is very
  large, so the wavelength is very small for the
  baseball
• The deBroglie equation is only useful for
  particles of very small mass

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1.6 The Uncertainty Principle

• Wave-Particle Duality
• Represented a Paradigm shift for our
  understanding of reality!
• In the Particle Model of electromagnetic
  radiation, the intensity of the radiation is
  proportional to the of photons present _at_ each
  instant
• In the Wave Model of electromagnetic radiation,
  the intensity is proportional to the square of
  the amplitude of the wave
• Louis deBroglie proposed that the wavelength
  associated with a matter wave is inversely
  proportional to the particles mass

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deBroglie Relationship

• In Classical Mechanics, we caqn easily determine
  the trajectory of a particle
• A trajectory is the path on which the location
  and linear momentum of the particle can be known
  exactly at each instant
• With Wave-Particle Duality
• We cannot specify the precise location of a
  particle acting as a wave
• We may know its linear momentum and its
  wavelength with a high degree of precision
• But the location of a wave? Not so much.

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The Uncertainty Principle

• We may know the limits of where an electron will
  be around the nucleus (defined by the energy
  level), but where is the electron exactly?
• Even if we knew that, we could not say where it
  would be the next moment
• The Complementarity of location and momentum
• If we know one, we cannot know the other exactly.

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Heisenbergs Uncertainty Principle

• If the location of a particle is known to within
  an uncertainty ?x, then the linear momentum, p,
  parallel to the x-axis can be simultaneously
  known to within an uncertainty, ?p, where
• h/2? hbar
• 1.055x10-34 Js
• The product of the uncertainties cannot be less
  than a certain constant value. If the ?x
  (positional uncertainty) is very small, then the
  uncertainty in linear momentum, ?p, must be very
  large (and vice versa)

?
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Wavefunctions and Energy Levels

• Erwin SchrÖdinger introduced the central concept
  of quantum theory in 1927
• He replaced the particles trajectory with a
  wavefunction
• A wavefunction is a mathematical function whose
  values vary with position
• Max Born interpreted the mathematics as follows
• The probability of finding the particle in a
  region is proportional to the value of the
  probability density (?2) in that region.

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The Born Interpretation

• ?2 is a probabilty density
• The probability that the particle will be found
  in a small region multiplied by the volume of the
  region.
• In problems, you will be given the value of ?2
  and the value of the volume around the region.

?
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The Born Interpretation

• Whenever ?2 is large, the particle has a high
  probability density (and, therefore a HIGH
  probability of existing in the region chosen)
• Whenever ?2 is small, the particle has a low
  probability density (and, therefore a LOW
  probability of existing in the region chosen)
• Whenever ?, and therefore, ?2, is equal to zero,
  the particle has ZERO probability density.
• This happens at nodes.

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SchrÖdingers Equation

• Allows us to calculate the wavefunction for any
  particle
• The SchrÖdinger equation calculates both
  wavefunction AND energy

Potential Energy (for charged particles it is the
electrical potential Energy)
Curvature of the wavefunction
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Particle in a Box

• Working with SchrÖdingers equation
• Assume we have a single particle of mass m stuck
  in a one-dimensional box with a distance L
  between the walls.
• Only certain wavelengths can exist within the
  box.
• Same as a stretched string can only support
  certain wavelengths

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Standing Waves
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Particle in a Box

• The wavefunctions for the particle are identical
  to the displacements of a stretched string as it
  vibrates.

where n1,2,3,

• n is the quantum number
• It defines a state

?
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Particle In a Box

• Now we know that the allowable energies are

Where n1,2,3,

• This tells us that
• The energy levels for heavier particles are less
  than those of lighter particles.
• As the length b/w the walls decreases, the
  distance b/w energy levels increases.
• The energy levels are Quantized.

?
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Particle in a BoxEnergy Levels and Mass

• As the mass of the particle increases, the
  separation between energy levels decreases
• This is why no one observed quantization until
  Bohr looked at the smallest possible atom,
  hydrogen

m1 lt m2
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Zero Point Energy

• A particle in a container CANNOT have zero energy
• A container could be an atom, a box, etc.
• The lowest energy (when n1) is

Zero Point Energy

• This is in agreement with the Uncertainty
  Principle
• ?p and ?x are never zero, therefore the particle
  is always moving

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Wavefunctions and Probability Densities

• Examine the 2 lowest energy functions n1 and n2
• We see from the shading that when n1, ?2 is at a
  maximum _at_ the center of the box.
• When n2, we see that ?2 is at a maximum on
  either side of the center of the box

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Wavefunction Summary

• The probability density for a particle at a
  location is proportional to the square of the
  wavefunction at the point
• The wavefunction is found by solving the
  SchrÖdinger equation for the particle.
• When the equation is solved to the appropriate
  boundary conditons, it is found that the particle
  can only posses certain discrete energies.