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The QuantumMechanical Model of
the Atom
Matter and Energy
By 1900, physicists thought that the nature
of energy and matter was well understood and
distinct.
Matter, a collection of particles, have mass
and a defined position in space. Radiant energy,
as waves, is massless and delocalized.
It was also believed that particles of matter
could absorb or emit any energy, without
restriction.
The Failure of Classical Physics
Observations of the behavior of sub-atomic
particles in the early 1900s could not be
predicted or explained using classical physics.
Very small particles such as electrons appear
to interact with electromagnetic radiation (light)
differently than object we can see and handle.
Electromagnetic Radiation
Early atomic scientists studied the
interaction of matter with electromagnetic radiation.
Electromagnetic radiation, or radiant energy,
includes visible light, infrared, micro and radio
waves, and X-rays and ultraviolet light.
Electromagnetic Radiation
Electromagnetic radiation travels in waves.
The waves of radiant energy have three
important characteristics:
1. Wavelength - λ - (lambda)
2. Frequency – ν – (nu)
3. Speed – c – the speed of light
Wavelength
Wavelength, λ, is the
distance between two
adjacent peaks or troughs
in a wave.
The units may range
from picometers to
kilometers depending
upon the energy of the
wave.
Frequency
Frequency, ν, is
the number of waves
(or cycles) that pass a
given point in space
per second.
The units are
cycles/s, s-1 or hertz
(Hz).
The Speed of Light
All electromagnetic radiation travels at the
same speed. The speed of light ( c ) is:
c = 2.9979 x 108 m/s
Wavelength and Frequency
Wavelength and
frequency are
inversely related.
That is, waves with a
low frequency have a
long wavelength.
Waves with a high
frequency have short
wavelengths.
Electromagnetic Radiation
The relationship between wavelength and
frequency is:
λν = c
Properties of Light - Amplitude
Diffraction
Waves of electromagnetic radiation are bent
or diffracted with they a passed through an
obstacle or a slit with a size comparable to their
wavelength.
Interference Patterns
Planck & Black Body Radiation
Max Planck (1858-1947) studied the
radiation emitted by objects heated until they
glowed. He found that the energy emitted was
not continuous, but instead was released in
multiples of hν.
∆E = nhν
where n=integer
ν = frequency
h = 6.626 x 10-34 J-s (Planck’s constant)
Planck & Black Body Radiation
∆E = nhν
Planck’s work showed that when matter and
energy interact, the energy is quantized, and can
occur only in discrete units or bundles with
energy of hν. Each packet or bundle of energy is
called a quantum. A fraction of a quantum is
never emitted.
Einstein – Photoelectric Effect
Albert Einstein (1879-1955) won a Nobel
Prize for his explanation of the photoelectric effect.
When light of sufficient energy strikes the
surface of a metal, electrons are emitted from
the metal surface. Each metal has a
characteristic minimum frequency, νo , called the
threshold frequency, needed for electrons to be
emitted.
The Photoelectric Effect
Observations
1. No electrons are emitted if the frequency of
light used is less than νo, regardless of the
intensity of the light.
2. For light with a frequency≥ νo , electrons are
emitted. The number of electrons increases
with the intensity of the light.
3. For light with a frequency > νo , the electrons
are emitted with greater kinetic energy.
Explanation
Einstein proposed that light is quantized,
consisting of a stream of “particles” called
photons.
If the photon has sufficient energy, it can
“knock off” an electron from the metal surface.
If the energy of the photon is greater than that
needed to eject an electron, the excess energy is
transferred to the electron as kinetic energy.
The Photoelectric Effect
Ephoton= hν = hc/λ
If incident radiation with a frequency νi is used:
KEelectron = hνi -hνo = ½ mv2
The kinetic energy of the electron equals the
energy of the incident radiation less the
minimum energy needed to eject an electron.
The Photoelectric Effect
The frequency hνo is the minimum energy
needed to eject an electron from a specific
metal. This energy is called the binding energy of
the emitted electron.
Particle-Wave Duality
Einstein’s work suggested that the incident
photon behaved like a particle. If it “hits” the
metal surface with sufficient energy (hνi), the
excess energy of the photon is transferred to the
ejected electron.
In the atomic scale, waves of radiant energy
have particle-like properties.
Particle-Wave Duality
Einstein also combined his equations:
E=mc2
with
Ephoton= hc/λ
to obtain:
hc/λ
E
m= 2 = c2
c
h
m= λc
Particle-Wave Duality
The apparent mass of radiant energy can be
calculated. Although a wave lacks any mass at
rest, at times, it behaves as if it has mass.
Einstein’s equation was confirmed by
experiments done by Arthur Compton in 1922.
Collisions between X-rays and electrons
confirmed the “mass” of the radiation.
Louis de Broglie
Einstein showed that waves can behave like
particles. In 1923, Louis de Broglie (1892-1987)
proposed that moving electrons have wave-like
properties.
Louis de Broglie
Using Einstein’s equation:
m=h/λv
where v is the velocity of the particle,
de Broglie rearranged the equation to calculate
the wavelength associated with any moving
object.
Louis de Broglie
λ=h/mv
de Broglie’s equation was tested using a
stream of electrons directed at a crystal. A
diffraction pattern, due to the interaction of
waves, resulted. The experiment showed that
electrons have wave-like properties.
Particle Beams
Wave-Like Nature of the Electron
Particle-Wave Duality
It is important to note that the wavelike
properties of moving particles are insignificant
in our everyday world. A moving object such as
a car or a tennis ball has an incredibly small
wavelength associated with it.
It is on the atomic scale that the dual nature
of particles and light become significant.
Emission Spectrum of Hydrogen
When atoms are
given extra energy,
or excited, they give
off the excess
energy as light as
they return to their
original energy, or
ground state.
H2
Hg
He
Emission Spectrum of Hydrogen
Scientists expected atoms to be able to
absorb and emit a continuous range of energies,
so that a continuous spectrum of wavelengths
would be emitted.
Emission Spectrum of Hydrogen
A continuous spectrum in the visible range,
would look like a rainbow, with all colors visible.
Instead, hydrogen, and other excited atoms emit
only specific wavelengths of light as they return
to the ground state. A line spectrum results.
Emission Spectrum of Hydrogen
Emission Spectrum of Hydrogen
Instead, only a few wavelengths of light are emitted,
creating a line spectrum. The spectrum of hydrogen
contains four very sharp lines in the visible range.
Emission Spectrum of Hydrogen
The discrete lines in the spectrum indicate that
the energy of the atom is quantized. Only
specific energies exist in the excited atom, so
only specific wavelengths of radiation are
emitted.
The Bohr Atomic Model
In 1913, Niels Bohr (1885-1962) proposed
that the electron of hydrogen circles the nucleus
in allowed orbits.
That is, the electron is in its ground state in
an orbit closest to the nucleus. As the atom
becomes excited, the electron is promoted to an
orbit further away from the nucleus.
The Bohr Atomic Model
Classical physics
dictates that an electron
in a circular orbit must
constantly lose energy
and emit radiation.
Bohr proposed a
quantum model, as the
spectrum showed that
only certain energies are
absorbed or emitted.
The Bohr Atomic Model
Bohr’s model of the hydrogen atom was
consistent with the emission spectrum, and
explained the distinct lines observed.
The Bohr Atomic Model
The Bohr Atomic Model
Bohr also developed an equation, using the
spectrum of hydrogen, that calculates the energy
levels an electron may have in the hydrogen
atom:
E=-2.178 x 10-18J(Z2/n2)
Where Z = atomic number
n = an integer
The Bohr Atomic Model
Bohr also calculated the radius of the lowest
energy orbit in the hydrogen atom. He
proposed that the lowest energy orbit had a
radius of 52.9 pm.
Although the concept of circular orbits is
incorrect, the value of the Bohr radius is
consistent with calculations based on quantum
mechanics.
The Bohr Atomic Model
The Bohr model didn’t work for atoms other
than hydrogen. Though limited, Bohr’s
approach did attempt to explain the quantized
energy levels of electrons.
Later developments showed that any attempt
to define the path of the electron is incorrect.
The Quantum Mechanical Model
The quantum mechanical atomic model was
developed based on the theories of Werner
Heisenberg (1901-1976), Louis de Broglie (18921987) and Erwin Schrödinger (1887-1961).
They focused on the wave-like nature of the
moving electron.
The Quantum Mechanical Model
The electron in an
atom was viewed as a
standing wave. For an
energy level to exist,
the wave must
reinforce itself via
constructive interference.
The Quantum Mechanical Model
Schrödinger developed complex equations
called wave functions ( Ψ). The wave functions
can be used to calculate the energy of electrons,
not only in hydrogen, but in other atoms.
The Quantum Mechanical Model
The wave functions also describe
various volumes or spaces where electrons
of a specific energy are likely to be found.
These spaces are called orbitals.
The Quantum Mechanical Model
Orbitals are not orbits.
The wave functions provide no information
about the path of the electron. Instead, it
provides the space in which there is a high
probability (90%) of finding an electron with a
specific energy.
The Heisenberg Uncertainty
Principle
Werner Heisenberg showed that, due to the
wave nature of the electron,
It is impossible to know both the precise position and the
momentum of the electron at the same time.
This is known as the Heisenberg Uncertainty
Principle.
The Heisenberg Uncertainty
Principle
It is impossible to know both the precise position and
the momentum of the electron at the same time.
In mathematical terms, the principle is:
(Δx) (Δmv) ≥ (h/4π)
The Heisenberg Uncertainty
Principle
It is impossible to know both the precise position and
the momentum of the electron at the same time.
The Heisenberg Uncertainty
Principle
(Δx) (Δmv) ≥ (h/4π)
There is a limit to how well we can
determine position (x), if mass and velocity are
known precisely.
For large particles, the uncertainty is
insignificant. However, on the atomic scale, we
cannot know the exact motion of an electron.
The Heisenberg Uncertainty
Principle
(Δx) (Δmv) ≥ (h/4π)
For an electron in a hydrogen atom, the
uncertainty in the position of the electron is
similar in size to the entire hydrogen atom.
Thus the location of the electron cannot be
determined.
Orbitals
The Schrödinger equation is used to describe
the space in which it is likely to find an electron
with a specific energy.
The equation provides us with a probability
distribution, or an electron density map. It is
important to remember that the resulting shape
does not give us any information about the path
of the electrons.
Orbitals
Each orbital described by the Schrodinger
equations is associated with three interrelated
quantum numbers which relate to the energy of
electrons in the orbital and the probability of
finding the electron within a particular volume.
Quantum Numbers
The principal quantum number, n, determines
the overall size and energy of an orbital. It is an
integer with values of 1, 2, 3, etc.
The angular momentum quantum number, l,
determines the shape of the orbital. It is related
to the more familiar designations of s, p, d and f.
Orbitals
The orbital of lowest energy is the 1s orbital.
The probability density, or probability of finding
an electron per unit volume, shows electron
density in all directions, creating a spherical
shape.
The probability density decreases with
greater distance from the nucleus.
Orbitals
Orbitals
Radial Distribution Function
The radial distribution function is a graphical
representation of the probability of finding an
electron in a thin spherical shell a specific
distance from the nucleus.
It shows that there is zero probability that
the electron will be at the nucleus, and also
indicates the most probable distance the
electron will have from the nucleus.
Radial Distribution Function
The maximum at 52.9
pm is consistent with
Bohr’s radius for the
hydrogen atom. It
more correctly
indicates the most
probable distance
between the electron
and nucleus.
Orbitals
The first energy level of hydrogen (n=1)
consists of a 1s orbital.
The second energy level of hydrogen (n=2)
consists of a 2s orbital and 2p orbitals.
The third energy level of hydrogen (n=3)
consists of a 3s orbital, 3p orbitals, and 3d
orbitals.
Orbitals
As the value of n
increases, the orbitals,
on average, become
larger, with more
electron density
farther from the
nucleus.
Orbitals
The “white rings”
in the drawings are
nodes. This is the
region where the
wave function goes
from a positive value
to a negative value.
The 2s and 3s
Orbitals
Orbitals
p orbitals are “dumbbell” shaped, with two
lobes. In one lobe, the wave function is
positive, in the other lobe, it is negative.
Orbitals
p orbitals come in sets of three, called a subshell.
The three orbitals are designated as px, py and pz,
because the electron density lies primarily along
either the x, y or z axis.
Orbitals
All three orbitals have the exact same energy.
Orbitals with the same energy are called
degenerate.
Orbital Phase
The drawings of orbitals is an attempt to
visualize three-dimensional waves. Waves can
undulate from positive to negative amplitudes.
The sign of the amplitude is known as its phase.
The phase of a sine wave fluctuates between
positive and negative.
Orbital Phase
Orbital Phase
The phase of the wave functions or orbitals
is quite important when atoms bond together.
The orbitals must be of the same phase to
overlap and form covalent bonds.
Orbitals
The
n=3 level
contains s,
p and d
orbitals.
The d
orbitals are
shown.
Orbitals
The n=4
level contains
s, p, d and f
orbitals. The f
orbitals are
shown.