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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
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.
Bohr Model of the Atom
• The symbol n in Bohr’s 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
• Only orbits where n = some positive Only orbits where n = some positive integer are permitted.integer are permitted.
• The energy of an electron in an orbit has The energy of an electron in an orbit has a negative valuea negative value
• An atom with its electrons in the lowest An atom with its electrons in the lowest possible energy level is at possible energy level is at GROUND GROUND STATESTATE– Atoms with higher energies (n>1) are in Atoms with higher energies (n>1) are in
EXCITED STATESEXCITED STATES
Energy of quantized state = - Rhc/nEnergy of quantized state = - Rhc/n22
Atomic Spectra and Bohr
If e-’s are in quantized energy states, If e-’s are in quantized energy states, then ∆E of states can have only then ∆E of states can have only certain values. This explains sharp certain values. This explains sharp line spectra.line spectra.
Energy absorption and electron excitation
Spectra of Excited Atoms•To move and electron from the n=1 to an excited state, the atom must absorb energy
•Depending on the amount of energy the atom absorbs, an electron may go from n=1 to n=2, 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/ninitial
2) = -Rhc (1/ nfinal2 - 1/ninitial
2)(does the last equation look familiar?)
Origin of Line SpectraOrigin of Line Spectra
Balmer seriesBalmer series
Atomic Line Spectra and Atomic Line Spectra and Niels BohrNiels Bohr
Atomic Line Spectra and Atomic Line Spectra and Niels BohrNiels Bohr
Bohr’s theory was a great Bohr’s theory was a great accomplishment.accomplishment.
Rec’d Nobel Prize, 1922Rec’d Nobel Prize, 1922Problems with theory —Problems with theory —• theory only successful for H.theory only successful for H.• introduced quantum idea introduced quantum idea
artificially.artificially.• So, we go on to So, we go on to QUANTUMQUANTUM or or
WAVE MECHANICSWAVE MECHANICSNiels BohrNiels Bohr
(1885-1962)(1885-1962)
Wave-Particle DualityDeBroglie 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
Wave-Particle DualityWave-Particle DualityWave-Particle DualityWave-Particle Duality
L. de BroglieL. de Broglie(1892-1987)(1892-1987)
de Broglie (1924) proposed de Broglie (1924) proposed that all moving objects that all moving objects have wave properties. have wave properties. For light: E = mcFor light: E = mc22
E = hE = h = hc / = hc / Therefore, mc = h / Therefore, mc = h / and for particles:and for particles: (mass)(velocity) = h / (mass)(velocity) = h /
Baseball (115 g) at 100 mphBaseball (115 g) at 100 mph = 1.3 x 10= 1.3 x 10-32-32 cm cm
e- with velocity = e- with velocity = 1.9 x 101.9 x 1088 cm/sec cm/sec = 0.388 nm= 0.388 nm
Wave-Particle DualityWave-Particle DualityWave-Particle DualityWave-Particle Duality
•The mass times the velocity of the ball is very large, so the wavelength The mass times the velocity of the ball is very large, so the wavelength is very small for the baseballis very small for the baseball•The deBroglie equation is only useful for particles of very small massThe deBroglie equation is only useful for particles of very small mass
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 @ 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 particle’s mass
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.
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.
Heisenberg’s 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 J·s
• 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)
Wavefunctions and Energy Levels
• Erwin SchrÖdinger introduced the central concept of quantum theory in 1927:– He replaced the particle’s 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.
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.
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.
SchrÖdinger’s Equation
• Allows us to calculate the wavefunction for any particle
• The SchrÖdinger equation calculates both wavefunction AND energy
Curvature of the wavefunction
Potential Energy (for charged particles it is the electrical
potential Energy)
Particle in a Box
• Working with SchrÖdinger’s 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
Standing Waves
Particle in a Box
• The wavefunctions for the particle are identical to the displacements of a stretched string as it vibrates.
€
n (x) =2
L
⎛
⎝ ⎜
⎞
⎠ ⎟
1/ 2
sinnπx
L
⎛
⎝ ⎜
⎞
⎠ ⎟ where n=1,2,3,…
n is the quantum number•It defines a state
Particle In a Box
• Now we know that the allowable energies are :
€
En =n2h2
8mL2 Where n=1,2,3,…
• This tells us that:1. The energy levels for heavier particles are less than
those of lighter particles.2. As the length b/w the walls decreases, the ‘distance’
b/w energy levels increases.3. The energy levels are Quantized.
Particle in a Box:Energy 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 < m2
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 n=1) is:
€
En =h2
8mL2Zero Point Energy
•This is in agreement with the Uncertainty Principle:•∆p and ∆x are never zero, therefore the particle is always moving
Wavefunctions and Probability Densities
• Examine the 2 lowest energy functions n=1 and n=2
• We see from the shading that when n=1, 2 is at a maximum @ the center of the box.
• When n=2, we
see that 2 is at a maximum on either side of the center of the box
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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