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Bohr Model of Atom

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Erwin Schr dinger introduced the central concept of quantum theory in 1927: ... Same as a stretched string can only support certain wavelengths. Standing Waves ... – PowerPoint PPT presentation

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Title: Bohr Model of Atom


1
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


2
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.
3
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

4
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

5
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.

6
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?)

7
Origin of Line Spectra
Balmer series
8
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)
9
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
10
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)
11
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

12
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

13
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.

14
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.

15
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)

?
16
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.

17
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.

?
18
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.

19
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
20
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

21
Standing Waves
22
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

?
23
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.

?
24
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
25
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

26
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

27
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.
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