Transcript Physical Properties - Winthrop University

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
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 (n>1) are in
EXCITED STATES
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.
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/ninitial2) = -Rhc (1/ nfinal2 - 1/ninitial2)
(does the last equation look familiar?)
Origin of Line Spectra
Balmer series
Atomic Line Spectra and
Niels Bohr
Niels Bohr
(1885-1962)
Bohr’s theory was a great
accomplishment.
Rec’d Nobel Prize, 1922
Problems with theory —
• theory only successful for H.
• introduced quantum idea
artificially.
• So, we go on to QUANTUM or
WAVE MECHANICS
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
Wave-Particle Duality
L. de Broglie
(1892-1987)
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 / 
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
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
Curvature of the
wavefunction
Potential Energy (for
charged particles it is the electrical
potential Energy)
• The SchrÖdinger equation calculates both
wavefunction AND 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.
2 1/ 2 nx 
n (x)    sin
 where n=1,2,3,…
L 
 L 

n is the quantum number
•It defines a state

Particle In a Box
• Now we know that the allowable
energies are :
n 2h 2
En 
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:
h2
En 
8mL2
Zero 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.