Download Chapter 2 – Atoms and Elements - U of L Class Index

Chapter 7 – Atomic Structure
Rutherford’s model of the atom was superior to its predecessors,
but it was still not entirely accurate. The next model of the atom
was based on discoveries made by studying elements and light.
Waves and Electromagnetic Radiation
Society tends to consider visible light, radio waves and x-rays as
different; however, they are all forms of electromagnetic
radiation and all belong to the electromagnetic spectrum:
Electromagnetic radiation is a form of energy that travels in
waves. These waves all travel at the same speed (c =
2.99792458 × 108 m·s-1), but they do not all have the same
energy. Waves with higher energy are ______________ while
waves with lower energy are ________________:
Waves are characterized by several interrelated properties:
• wavelength (λ): the distance between successive crests
or successive troughs
• frequency (ν): the number of waves passing through a
point in a given period of time
• amplitude (A): the height of a wave (from the node)
• speed = wavelength × frequency
Different colours of light have different wavelengths, allowing a
prism to separate white light into a continuous spectrum.
Similarly, a radio tuner can separate radio waves into different
frequencies (which are inversely related to their wavelengths).
e.g. The University of Lethbridge campus radio station is CKXU
88.3 FM, broadcasting radio waves with a frequency of
88.3 MHz. Calculate the wavelength of these radio waves,
and report it using the most appropriate SI unit.
We can also calculate the energy of electromagnetic radiation
using Planck’s equation:
in which Planck’s constant (h) is 6.6260693 × 10-34 J·s
Calculate the energy of CKXU’s radio waves.
Would you expect them to have higher or lower energy than
radio waves from the old U. of L. radio station, CKUL 99.7 FM?
Atomic Line Spectra
Light has many properties that are best explained by considering
it to behave as __________, but it also has many properties that
are best explained by considering it to behave as ____________.
A ‘particle’ of light is called a _____________.
Each ____________ has a quantity of energy but no mass.
If an atom is struck by a photon that has enough energy, it will
absorb the photon. This puts the atom into an ______________
___________. (An atom that has absorbed no energy from
external sources is said to be in its ______________________.)
When an atom in an excited state relaxes back to its ground state
(or a less energetic excited state), it releases energy as a photon.
When the atoms in a pure gaseous sample of an element are
excited, they emit photons of light at a limited number of
wavelengths (specific to the element in question). When this
light is separated by a prism, it gives a series of lines called a
line emission spectrum.
The line emission spectrum for hydrogen was first reported by
Anders Ångström (1817-1874) in 1853. Over the course of
approximately 50 years, line emission spectra for the remaining
known elements were obtained, but the relationship between the
lines emitted and atomic structure remained elusive.
In 1885, Swiss math teacher Johann Balmer (1825-1898)
showed that the wavelengths of visible light emitted by a
hydrogen atom could be described by the mathematical formula:
1
λ
= 1.0974 x 107 m-1
1 - 1
4
n2
Later, Johannes Rydberg (1854-1919) generalized this equation
so that it described all of the spectral lines emitted by hydrogen:
1
λ
= R
1 - 1
n12
n 22
where R is the Rydberg constant (1.0974 × 107 m-1), and n1 and
n2 are any two integers. This is the Rydberg equation, and the
series of wavelengths for which n1 = 2 is the Balmer series.
The Rydberg equation allows scientists to predict all
wavelengths of light emitted by an excited hydrogen atom. Not
all of these wavelengths fall in the visible region of the
electromagnetic spectrum. When n1 = 1 (the Lyman series), the
light emitted is in the ultraviolet region.
Is more or less energy released when an atom relaxes to the
n1 = 1 state or to the n1 = 2 state?
e.g. Calculate the wavelength of a photon emitted when a
hydrogen atom relaxes to the n = 3 state from the n = 5
state. What type of electromagnetic radiation is this?
The Bohr Model of the Hydrogen Atom
Recall Rutherford’s model of the atom (1911):
For a hydrogen atom, this implies an electron circling around a
proton. The problem with this model is that circular motion
corresponds to constant acceleration since the particle is
constantly changing direction. Acceleration of a charged
particle should result in the continuous release of energy as
either heat or electromagnetic radiation. As such, the electron
should constantly be losing energy and therefore slowing down.
The negatively charged electron is attracted to the positively
charged nucleus so, as it slows, the electron should spiral in
toward the nucleus. This does not happen! Why not?
In 1913, Neils Bohr (1885-1962) proposed a solution to this
problem that also explained the atomic line spectrum observed
for hydrogen. His solution was based on three postulates:
1. The orbital angular momentum of electrons in an atom is
quantized. Only those electrons whose orbitals correspond
to integer multiples of h/2π are “allowed”.
2. Electrons within an allowed orbital can move without
radiating (so that there is no loss of energy).
3. The emission or absorption of light occurs when electrons
‘jump’ from one orbital to another.
Using these assumptions and basic physical constants, Bohr
calculated the energy of an electron in orbital n of a hydrogen
atom:
En = -
Rhc
n2
where En is the energy of an electron in orbital n, R is the
Rydberg constant, h is Planck’s constant, and c is the speed of
light. Note that En is less than zero! It usually takes less energy
for an electron to be part of an atom than to exist freely. Also,
the closer the electron is to the nucleus, the lower the energy.
***n is the principal quantum number***
This formula can be extended to any atom containing only one
electron (H, He+, Li2+, etc.):
En = -2.179 x 10-18 J
Z2
n2
or
En = -1 Ry
Z2
n2
where a rydberg (Ry) is a unit of energy equal to 2.179 × 10-18 J,
and Z is the number of protons in the nucleus (_____________).
Bohr also calculated the radius of each orbital:
2
rn = a0 n
Z
where rn is the radius of orbital n, and a0 is the Bohr radius
(5.29177 × 10-11 m).
By calculating the energies of the allowed orbitals for hydrogen,
we can generate an energy level diagram:
(The scale on the y-axis lists the energies of
each orbital in rydbergs.)
Notice that all of the energies are negative,
and the top of the scale is 0 Ry. This is
called the ionization limit. An electron
with higher energy than this has too much
energy to remain part of the atom, so it
leaves and a cation (H+) is formed. H+ is
just a _______________.
As the orbitals get closer in energy to the
ionization limit, they get closer together in
energy. Since there is an infinite number of
possible orbitals, they become so close in
energy near the ionization limit that they
are essentially continuous.
Calculating Atomic Spectra Using the Bohr Model
An atom in its ground state has its electron(s) in the lowest
energy orbitals allowed. e.g. A ground state hydrogen atom has
its electron in the _______ orbital.
When an atom absorbs a photon of energy, an electron is excited
into a higher energy (i.e. less negative) orbital. This puts the
atom into an excited state.
***Note that atoms have states. Electrons are in orbitals.***
The energy of the photon absorbed must be ________________
to the energy difference between the two orbitals.
When an ____________________ falls back into a lower energy
orbital, the photon released will have the same energy as the
energy difference between the two orbitals.
Recall:
En = - 2.179 x
10-18
Z2
J
n2
Therefore,
En2 - En1 = - 2.179 x 10-18 J . Z 2
1 - 1
n12
n22
Which can also be written as:
∆E = 2.179 x 10-18 J . Z 2
1 - 1
n22
n12
This formula can be used for one-electron atoms/ions only!
We can use this formula to repeat our calculation of the
wavelength of a photon emitted when a hydrogen atom relaxes
to the n = 3 state from the n = 5 state. Do we get the same
answer as before?
The Bohr model of the atom pictures an electron as a particle
circling the nucleus of an atom in a fixed orbital similar to the
way that planets circle the sun (except that planets cannot ‘jump’
from one orbital to another – which is probably a good thing!).
The main flaw in the Bohr model of the atom is that it only
works for one-electron atoms and ions (__________________),
and very few of these species are actually observable.
The reason this formula doesn’t work for multi-electron atoms is
that the electrons interact. One electron will often shield another
from part of the nucleus’ positive charge. The electrons will
also repel each other since they all have negative charges.
Electrons as Waves
Planck proposed that light, normally considered to behave as
waves, sometimes behaves like small fast-moving particles.
In 1924, Louis be Broglie (1892-1987) proposed that small fastmoving particles (e.g. electrons) might sometimes behave as
waves. In 1927, C. J. Davisson (1881-1958) and L. H. Germer
(1896-1971) proved this hypothesis by diffracting electrons.
The de Broglie equation relates the wavelength (a wave
property) of a particle to its momentum (a particle property):
λparticle =
h
ρparticle
=
h
mparticle vparticle
where λ is wavelength, ρ is momentum, m is mass, v is velocity,
and h is Planck’s constant.
Since electrons in an atom move at ~10% of the speed of light,
de Broglie felt that they should have significant wave properties.
A wave-like electron in an orbital might look something like
this:
The electron must be a standing wave; otherwise, the
mismatched crests and troughs eventually cancel out, leaving a
wave of ‘nothingness’ and the electron would have no energy.
This means that only a specific set of wavelengths are allowed
(and, therefore, a specific set of velocities). Could this be the
reason why orbital energies are quantized?
e.g. What is the wavelength of an electron moving with a
velocity of 2.5 × 108 cm/s?
Heisenberg Uncertainty Principle
While it is impossible to show an electron behaving as a wave
and a particle at the same time, we accept that it has properties
of both. This idea is known as __________________________.
Werner Heisenberg (1901-1976) showed that it is impossible to
know the precise location and momentum of quantum particles
(e.g. electrons) at the same time.
Let’s say we want to measure the position of an electron in an
atom. We can’t physically look at it, but imagine we could.
(The logic is the same.) A photon of light would have to bounce
off the electron and into our eyes. The momentum of a photon
is high enough that, when it collides with the electron, it disturbs
it. (i.e. The electron bounces away too.) So, we would know
where the electron *was* when it was struck by the photon but
not its location when the photon reaches our eye.
If we are willing to accept more uncertainty about an electron’s
momentum, we can have more certainty in knowing its position
– and vice versa. This inverse relationship can be described
mathematically:
∆x . ∆ρ >
h
4π
or
∆x . ∆(mv) >
h
4π
where ∆x is the uncertainty about position (i.e. distance between
the farthest apart possible positions), ∆ρ is the uncertainty about
momentum (i.e. difference between maximum and minimum
possible momentum values), and h is Planck’s constant. Note
that, as the mass of a particle increases, its position and velocity
can be known with greater certainty.
e.g. An electron (mass = 9.109 × 10-31 kg) is traveling with a
velocity between 3.021 × 107 m/s and 3.044 × 107 m/s.
What is the smallest uncertainty you can have about its
position? Repeat this calculation for an atom of hydrogen
(mass = 1.674 × 10-27 kg).
The Schrödinger Model of the Hydrogen Atom
Erwin Schrödinger (1887-1961) calculated how electrons might
act in an atom if they behaved as waves instead of particles.
This resulted in the complicated and difficult-to-solve
Schrödinger equation:
∂ 2Ψ ∂ 2Ψ ∂ 2Ψ
8π 2m
+
+
= − 2 ( E − V )Ψ
∂ x2 ∂ y2 ∂ z2
b
operator
scalar
This equation belongs to a special class known as eigenvector
equations: an operator acts on a function (Ψ), generating a
scalar (i.e. a number) times the same function. These kind of
equations always have an infinite number of solutions.
Ψ is called the wavefunction of the electron. There is an
infinite number of wavefunctions permitted by the Schrödinger
equation – each with a different energy (E). (V is the potential
energy from attraction of the electron to the nucleus; it is
constant. π, m and b are also fundamental constants.)
Solving the Schrödinger equation (which we will not be doing in
this course!) gives exactly the same principal energy levels for
electrons in the hydrogen atom as Bohr’s formula did. The main
difference is that it treats electrons as 3D ‘matter waves’ instead
of particles.
Schrödinger’s model also introduces an additional two quantum
numbers to describe the shape and orientation of the wave.
This gives us three quantum numbers to describe each orbital:
1. Principal quantum number (n) defines the energy of the
wavefunction and the shell it occupies.
2. Angular momentum quantum number (l) defines the
shape of the wavefunction and the subshell it occupies.
3. Magnetic quantum number (ml) defines the orientation
of the wavefunction and the specific wavefunction.
The allowed values for l depend on n, and the allowed values for
ml depend on l:
total number of orbitals
n
l
ml
1
0
0
1 orbital
2
0
0
1 + 3 = 4 orbitals
1
-1, 0, +1
1 + 3 + 5 = 9 orbitals
0
3
0
-1, 0, +1
1
-2, -1, 0, +1, +2
2
1 + 3 + 5 + 7 = 16 orbitals
0
4
0
-1, 0, +1
1
-2, -1, 0, +1, +2
2
-3, -2, -1, 0, +1, +2, +3
3
How many different orbitals exist for n = 5? n = 6? n = 7?
How many different orbitals exist for n = 6, l = 4?
In summary:
n defines the shell an electron is in. A shell contains one or
more subshells.
l defines the subshell an electron is in. A subshell contains one
or more orbitals.
ml defines the orbital an electron is in.
The orbitals of a hydrogen atom according to the Schrödinger
model look like this:
0
E
N
E
R
G
Y
n=∞
etc
n=5 l=0
l=1
l=2
l=3
n=4 l=0
l=1
l=2
l=3
n=3 l=0
l=1
l=2
n=2 l=0
l=1
n=1
The energy levels of the orbitals of the hydrogen atom
Each horizontal row represents a shell, each cluster of dashes
represents a subshell, and each dash represents an orbital.
The energy of an orbital is given entirely by the principal
quantum number (n), but there are many more orbitals present
than the Bohr model predicts.
If a magnetic field is placed around an atom, it interacts
differently with different orbitals depending on their shape and
orientation (i.e. are they lined up with the field, or do they lie
across it?). This makes some of the orbitals higher in energy
than before and makes some lower in energy:
0
E
N
E
R
G
Y
n=∞
etc
n=5 l=0
l=2
l=3
n=4 l=0
l=2
l=3
n=3 l=0
l=1
n=2 l=0
l=1
n=1
l=2
The energy levels of the orbitals of the hydrogen atom
under the influence of a magnetic field
Due to the magnetic field, not all orbitals in the same shell have
the same energy anymore. This leads to a more complicated
atomic line spectrum since there are more energy levels and
therefore more energy gaps (i.e. more ∆E values).
This is known as the Zeeman effect. It can be was one of the
key pieces of evidence for accepting the Schrödinger model of
the atom over the Bohr model.
An orbital is a wavefunction. The pictures we draw of
orbitals show regions in which an electron is most likely to
be found. They do not show its path around the nucleus.
Ψ2 (the amplitude of the wavelength squared) expresses the
probability of finding an electron within a given region of space.
This probability is called electron density.
Because wavefunctions describe probability distributions, they
do not have precise sizes, but they do have a maximum value.
For each orbital, there is a distance from the nucleus at which
the probability of an electron is highest. For a hydrogen atom,
these maximum values are very close to the Bohr radii.
i.e. The Bohr radius for the n = 1 orbital is very close to the
highest probability distance for an n = 1 electron calculated
using the Schrödinger equation.
Shapes of Atomic Orbitals
A hydrogen atom is spherical, and all of the orbitals predicted by
the Schrödinger equation fit into this sphere.
Just as 2-dimensional waves have nodes, so can 3-dimensional
waves. Each atomic orbital has _____ nodes. ____ of these
nodes are planar. The rest are radial.
s Orbitals (l = 0)
An s orbital is spherical. It has ___ planes through the nucleus.
The simplest orbital has quantum numbers n = 1, l = 0, ml = 0
{abbreviated as (1,0,0)}. This orbital is called Ψ1,0,0 or 1s.
The following diagrams all describe the shape of the 1s orbital:
An electron ‘dot’
picture is like a timelapse photo of an
electron’s position if
it is considered to act
as a particle for each
‘snapshot’.
A plot of the wave’s
amplitude
versus
distance from the
nucleus. (Note that
this is not a plot of
electron density.)
A probability surface
has a ‘skin’ drawn
over
the
wave,
encompassing some
percentage
(often
90% or 95%) of the
electron density.
The middle diagram shows that the amplitude of
the wave decreases as distance from the nucleus
increases. This does not, however, mean that the
electron is most likely to be found at the nucleus!
As the radius (r) of a sphere increases, its surface
area increases as r2. (____________) If we want
to know the most probable distance from the
nucleus to an electron, we plot the wavefunction
squared (Ψ2) multiplied by the surface area for a
given radius (4πr2) versus radius (r):
4πr2Ψ2 vs. r.
This type of plot is called a radial probability density plot
(also surface density plot or radial distribution plot). It describes
the relationship between electron density and radius. It has a
maximum at the Bohr radius.
The 1s orbital has no nodes; however, the ‘higher’ s orbitals
have radial nodes, as can be seen from their radial probability
density plots:
‘Dot’ diagrams
Wave amplitude plots
Radial probability density plots
Note that the maximum probability density is at
a higher radius than all of the nodes.
Also, note the different phases on either side of a node.
p Orbitals (l = 1)
A p orbital has ___ nodal plane passing through the nucleus:
The three p orbitals pack together to make a perfect sphere.
They can do this because they are ________________ functions.
The orientation of the x, y and z axes is arbitrary unless it is
imposed by a magnetic field (hence “magnetic quantum
number” for ml), and there is no direct correlation between ml
and direction. We arbitrarily assign the ‘positive’ phase to be
the positive direction on an axis and the ‘negative’ phase to be a
negative direction.
The higher p orbitals (3p, 4p, etc.) also have radial nodes.
Imagine one or more dumbbells tucked concentrically inside the
outer dumbbell:
The images above are 90% probability ‘skins’. The wave
amplitude plots and radial probability density plots look much
the same as for an s orbital with the same number of nodes.
d Orbitals (l = 2)
A d orbital has ___ nodal surfaces passing through the nucleus.
For four of the d orbitals, both nodal surfaces are planes, giving
a ‘petal-shaped’ orbital. For the fifth d orbital (_____),the nodal
surfaces looks more like a pair of inverted cones. This gives an
orbital that looks a bit like a p orbital with a doughnut around it.
(Note the phases, though; they are different from a p orbital.)
The higher d orbitals (4d, 5d, etc.) also have radial nodes.
f Orbitals (l = 3)
An f orbital has ___ nodal surfaces passing through the nucleus.
The higher f orbitals (5f, 6f, etc.) also have radial nodes. At this
point, orbitals start looking quite strange and are difficult to
draw. For pictures, see http://www.orbitals.com/orb/
Important Concepts from Chapter 7
• properties of waves (wavelength, frequency, amplitude, speed)
• electromagnetic spectrum, speed of light
• Planck’s equation and Planck’s constant
• wave-particle duality (for light, electrons, etc.)
• atomic line spectra and relevant calculations
• ground vs. excited states
• Heisenberg uncertainty principle
• Bohr and Schrödinger models of the atom
• quantum numbers (n, l, ml)
• shells, subshells and orbitals
• different kinds of atomic orbitals (s, p, d, f) and nodes