Download The Quantum-Mechanical Model of the Atom

The QuantumMechanical Model of
the Atom
Behavior on the Atomic Scale
In the early 1900s, scientist tried to apply
their understanding of physics to the structure of
the atom. An entirely new approach, called the
quantum-mechanical model, was developed to explain
some of the unusual behavior of electrons within
an atom.
The Quantum-Mechanical Model
In this approach, very small fast-moving
particles such as electrons also have wave-like
properties associated with them. Likewise, light,
which travels as waves, can also exhibit particlelike properties.
Electromagnetic Radiation
Early atomic scientists studied the
interaction of matter with electromagnetic radiation,
or light.
Electromagnetic radiation, or radiant energy,
includes visible light, infrared, micro and radio
waves, and X-rays and ultraviolet light.
Atomic Spectroscopy
The study of the light emitted or absorbed
by matter is a branch of chemistry called
spectroscopy.
Atomic spectroscopy allows scientists to
understand the nature of the electrons in atoms.
Molecular spectroscopy provides information
about the bonds in molecules.
Electromagnetic Radiation
Light consists of oscillating electric and
magnetic fields that travel through space at the
rate of 3 x 108m/s.
The oscillating fields interact with electrons
in the atom.
Electromagnetic Radiation
This drawing
represents a
“snapshot”
of an electromagnetic
wave at a
given instant.
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
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 objects we can see and handle.
Black Body Radiation
Physicists focused on interactions between
light (electromagnetic radiation) and matter to
try to better understand the nature of the atom.
When objects are heated, they emit light in
relation to their temperature. Iron rods glow
red, and will glow yellow at higher temperatures.
Black Body Radiation
Classical physics, when applied to black body
radiation, predicted that the intensity of the
radiation emitted would dramatically increase at
shorter and shorter wavelengths. The result was
that any hot body should emit intense UV
radiation, and even x-rays. Even a human body
at 37oC would glow in the dark. This
discrepancy between theory and observation is
called “The Ultraviolet Catastrophe.”
The Ultraviolet Catastrophe
The failure of
classical physics is
seen in the shorter
wavelength ultraviolet
region
Planck & Black Body Radiation
Max Planck (1858-1947) studied the
radiation emitted by objects heated until they
glowed. In order to explain the observations, he
proposed (in 1900) that the energy emitted was
not continuous, but instead was released in
multiples of hν. h is known as Planck’s constant.
Planck & Black Body Radiation
∆E = nhν
where n=integer
ν = frequency
h = 6.626 x 10-34 J-s
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.
Planck & Black Body Radiation
∆E = nhν
Each packet or bundle of energy is called a
quantum. A fraction of a quantum is never
emitted. A quantum is the smallest amount of
energy that can be emitted or absorbed in the
form of electromagnetic radiation.
Planck’s Law
Planck’s approach
shows good
agreement between
the observed
spectrum (in blue)
and the calculated
values (in red).
Planck’s Law
Planck’s law was based on empirical data. He
found a mathematical relationship that fits the
observations. It is important to note that Planck
did not explain the reason for the relationship.
The concept of energy being quantized
rather than continuous was quite revolutionary.
Planck & Black Body Radiation
Planck received the Nobel Prize for his work
in 1918 (at the age of 42).
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.
Binding energy is often expressed in electron
volts (eV), with 1 eV = 1.602 x 10–19 J.
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 the “mass” of a photon:
hc/λ
E
m= 2 = c2
c
h
m= λc
Albert Einstein
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.
Particle-Wave Duality
Arthur Compton attempted to study the
collision of a light quantum with an electron
moving freely through space. However, creating
a collision between a beam of light and a beam
of electrons isn’t feasible, since it would take an
extremely long time for such a collision to occur.
Arthur Compton
Compton solved this problem by using
extremely high energy x-rays to bombard small
atoms. Since the energy of the radiation was so
high, the electrons in the atoms were viewed as
“free” by comparison.
Compton viewed the collision as if between
two elastic spheres, and perfectly predicted the
scattering of the x-rays and the decrease in
frequency as a result of the collision.
Arthur Compton
Compton received the Nobel prize in 1927.
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, Neils 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
Bohr’s orbits existed at specific fixed
distances from the nucleus. Thus the energy of
each orbit was fixed or quantized. Bohr called
these stable orbits stationary states.
Electrons can transition from one orbit to
another, but they are never observed between
states.
The Balmer Series
The emissions of hydrogen in the visible
region (the Balmer Series) produces four lines
with the following frequencies:
ν1 =4.569 x 1014sec-1
ν2 =6.168 x 1014sec-1
ν3 =6.908 x 1014sec-1
ν4 =7.301 x 1014sec-1
The Balmer Series
The emission frequencies of hydrogen in the
visible region (the Balmer Series) can be
calculated using the formula:
νm,n =3.289 x 1015 [1/4 -1/m 2]sec-1
where m has the value of 3, 4, 5 or 6
The Rydberg Equation
Johannes Rydberg suggested a different form
of the equation that lead to future discovery.
1/λ α [1/22 – 1/n2] where n = 3,4,5,…
This equation was adapted for lines found in
the infrared and ultraviolet spectrum of
hydrogen.
The Rydberg Equation
The general form of the equation is:
ν = R [1/n12 – 1/n22] where n1 = 1,2,3,..
and n2 =n1+1, n1+2, ….
R is determined experimentally and is
3.29 x 1015 Hz
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. (1 pm = 10-12 m)
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. It also failed to explain the fine
splitting of the lines of the emission spectrum.
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.
Neils Bohr
“If quantum
mechanics hasn't
profoundly shocked
you, you haven't
understood it.”
Bohr won the
Nobel prize in 1922.
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
In 1924, de Broglie (1892-1987) came up
with an explanation of why only certain orbits
(and energy levels) for the electrons in an atom
exist. Not only does electromagnetic radiation
have particle-like properties, he proposed that
moving electrons have wave-like properties.
Louis de Broglie
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.
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
Wave-Like Nature of the Electron
The interference pattern is
not the result of different
electrons (as waves)
interfering with each other.
It results from a single
electron interfering with
itself. If the beam emits
single electrons at a very
slow rate, the interference
pattern persists.
Wave-Like Nature of the Electron
de Broglie concluded that the wave nature of
the electron is an inherent property of individual
electrons. The electron goes through both slits,
existing in two states simultaneously, and
interferes with itself.
Louis de Broglie
De Broglie was awarded the Nobel prize in 1929.
Particle-Wave Duality
It is important to note that the wave-like
properties of moving particles are insignificant
in our everyday world. A moving object such as
a car or a tennis ball has an insignificant
radiation component associated with it.
It is on the atomic scale that the dual nature
of particles and light become significant.
The Uncertainty Principle
The dual nature of the electron presents a
challenge. How can a single electron interfere
with itself to create a diffraction pattern?
An experiment was designed to observe the
electron as it travels through the slits. A laser
beam was placed across the paths the electron
could travel. When an electron crosses the laser
beam, a tiny flash is produced.
The Uncertainty Principle
When the laser beam is on, the flash comes
from either slit, but never both at the same time.
In addition, the interference pattern is no longer
seen. There are just two bright spots directly
opposite each slit, as if the electrons are behaving
like ordinary particles. No wave-like behavior is
observed.
The Uncertainty Principle
We can never see both the interference
pattern and simultaneously know which slit the
electron has passed through.
We cannot simultaneously observe both the wave nature
and the particle nature of the electron.
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.
The more precisely we know the position of
the electron, the less we know about its velocity.
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.
Werner Heisenberg
“The problems of
language here are
really serious. We
wish to speak in some
way about the
structure of the
atoms. But we cannot
speak about atoms in
ordinary language.”
Werner Heisenberg
Werner Heisenberg won the Nobel prize in
1932. During world war II, he lead the German
research team that was developing nuclear
fission.
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
Erwin 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, the
equations (Ψ2) provide the space in which there
is a high probability (90%) of finding an electron
with a specific energy.
Erwin Schrödinger
Schrodinger won the Nobel prize in 1933.
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.
The value of l is 0 for an s orbital, 1 for a p
orbital, 2 for a d orbital, and 3 for an f orbital.
Quantum Numbers
For a given value of n, l is an integer with
values from 0 up to n-1.
For n=1, l can only = 0 [a 1s orbital].
For n=2, l can be 0 or 1 [a 2s or 2p subshell].
For n=3, l can be 0, 1 or 2 [a 3s, 3p or 3d
subshell].
Quantum Numbers
The magnetic quantum number, ml ,
describes the spatial orientation of the orbital.
For a given value of l, ml may have the value of:
–l,…0,…+l
Thus, a p subshell consists of three p orbitals
(px, py, pz) with ml values of -1, 0 and +1.
Quantum Numbers
The orbital is described by quantum
numbers n, l, and ml . To describe the electrons
within an orbital, a fourth quantum number, the
electron spin quantum number, ms is needed.
The quantum number relates to the direction
of spin of an electron around its own axis, and it
has the values of either +½ or -½ .
Electron Spin
Each orbital, regardless of type, can contain
zero, one or two electrons. If two electrons
occupy the same orbital, they must spin in
opposite directions.
The spin is quantized, and can be expressed
using quantum numbers, or simply specifying
the spin as up or down or clockwise and
counter-clockwise.
The Pauli Exclusion Principle
Quantum mechanics dictates that no two
electrons in an atom can have the same four
quantum numbers. Another way of stating the
Pauli Exclusion Principle is that if electrons occupy
the same orbital, they must have opposite spins.
Multi-electron Atoms
Orbitals of any
type can be empty, or
have 1 or two
electrons.
Experimental data
indicate that if two
electrons are in the
same orbital, they will
spin in opposite
directions.
Energy Levels
In any atom or ion
with only 1 electron,
the principal quantum
number, n,
determines the energy
of the electron. For
n=2, the 2s and 2p
orbitals all have the
same energy.
Energy Levels
Likewise, the 3s,
3p and 3d orbitals are
all degenerate, with
the same energy.
Hydrogen Energy Levels
For the hydrogen
atom, each principal
quantum level (n value)
relates to observed
wavelengths of light
emitted in the atomic
spectrum.
Each line in the
spectrum results from an
electron transition.
Energy Levels
In a multi-electron atom, there is interaction
between electrons. As a result of this
interaction, the various subshells of a principal
quantum level will vary in energy.
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.
Orbitals
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.