CHAPTER 30 THE NATURE OF THE ATOM
Rutherford Scattering and the Nuclear Atom
Early Greek models of atoms described them as tiny indivisible
spheres. In fact atomos, the Greek word for indivisible, is the
origin of the word atom.
The plum pudding model was an improvement proposed after
electrons were discovered. The idea was that electrons were
embedded in a sphere of positive matter like plums in a
pudding.
This model was replaced with the planetary model proposed by
Ernest Rutherford. He directed a beam of alpha particles at a
very thin sheet of gold foil.
Most of the alpha particles passed through undeflected as
expected. Some, however, were deflected up to 180 degrees.
The only explanation was that the positive charge of each atom
must be concentrated in a relatively small volume which was
named the nucleus.
The amount of empty space in an atom is difficult to imagine.
It may help to imagine an atom the size of our solar system. If
the nucleus of an atom were about the size of the sun, the
radius of the Earth's orbit (representing an electron) would be
400 times what it is now. The Earth's orbit would be 10 times
the current radius of Pluto's orbit.
Later in the chapter we will address the shortcomings of the
planetary model.
Example
The nucleus of a hydrogen atom is a proton with a mass of 1.67
x 10-27 kg and a radius of 1.0 x 10-15 m. The electron, with a
mass of 9.11 x 10-31 kg orbits at a distance of 5.29 x 10-11 m.
Compare the density of the nucleus to the density of the atom.
Line Spectra
Electromagnetic waves are emitted by all phases of matter
when energy is added to them by heating, electric current, etc.
In the case of a solid, a continuous spectrum is produced
because of the interactions among the atoms and molecules
that make up the solid.
If atoms of an element can be separated from other atoms and
each other, they can be made to emit specific wavelengths
characteristic of the atomic structure of that element. A device
that contains atoms in this state is called a gas discharge tube.
If high voltage is impressed across the gaseous element in the
tube, it will emit only the wavelengths that are characteristic of
that element.
If we break the light up into individual wavelengths with a
grating spectroscope, we can see the individual wavelengths as
colors.
When we see a series of bright fringes, we call this a line
spectrum since each bright fringe has the appearance of a
vertical line. The spectra above include two line spectra and
one continuous spectrum.
The simplest element is hydrogen and its spectrum has been
studied extensively. The three groups of lines that are most
famous are the Lyman series, Balmer series, and Paschen
series.
Each series is composed of the wavelengths emitted when the
electron in a hydrogen atom falls from higher energy levels to a
lower energy level. The energy of a photon = hf where h is
Planck's constant and f is the frequency of the photon.
In the case of the Lyman series, the electron is falling to the
first energy level.
In the case of the Balmer series, the electron is falling to the
second energy level.
In the case of the Paschen series, the electron is falling to the
third energy level.
The equations used to calculate the wavelengths are:
Lyman series
1/λ = R(1/12 - 1/n2)
n = 2, 3, 4, …
Balmer series
1/λ = R(1/22 - 1/n2)
n = 3, 4, 5, …
Paschen series
1/λ = R(1/32 - 1/n2)
n = 4, 5, 6,…
In these equations, R is called the Rydberg constant and has a
value of 1.097 x 107 m-1.
Each series has a longest and shortest wavelength. They can be
calculated as in the following example.
Example
Find the longest and shortest wavelength in the Balmer series.
The longest wavelength occurs when the energy level jump is
the smallest, so we consider a jump from shell 3 to shell 2.
For the shortest wavelength we need the largest energy level
jump, so we consider a jump from infinity to shell 2.
The Bohr Model of the Hydrogen Atom
In 1913, Neils Bohr presented a new model of atomic structure
in an attempt to solve some of the problems associated with the
planetary model. Planck and Einstein had already theorized
that light consisted of discrete packets of energy called photons
and that energy associated with photons was quantized (only
certain values allowed).
Bohr's idea was to relate atomic structure to this quantization
of energy in electromagnetic radiation. Since only certain
energies could be emitted by an atom (E = hf), the electrons
must only be able to occupy certain energy levels. This would
prevent the electrons from radiating energy continuously and
spiraling into the nucleus.
These orbits were called stationary orbits or stationary states.
The electron could only occupy one of these stationary orbits
or be in transition between them.
When an electron changes from one orbit to another, the
energy difference is:
Ei - Ef = hf
Since f = c/λ,
Ei - Ef = hc/λ
and the wavelengths associated with each energy level change
could be calculated if the energies could be determined.
The total energy of an atom is:
E = KE + EPE
E = ½mv2 + (-kZe2/r)
The electrical potential energy is negative since we are looking
at the energy of an electron.
Since we are considering a circular orbit, a centripetal force
exists due to the electric charge, so:
mv2/r = kZe2/r2
mv2 = kZe2/r
E = ½ kZe2/r + (-kZe2/r)
E = -kZe2/2r
This gives us Bohr's equation for the total energy of the atom
due to the location and speed of the electron.
To determine r, the radius of the orbit, Bohr had to assume
that the angular momentum of the electron is quantized. This
gives us the expression:
Ln = mvnrn = nh/2π
n = 1, 2, 3, . . .
Solving for vn, we get:
vn = nh/2πmrn
By substituting into the equation for centripetal force, we get:
m(nh/2πmrn)2 = kZe2/rn
rn = (h2/4π2mke2)n2/Z
n = 1, 2, 3, . . .
Using the accepted values for h, k, m, and e, the expression
reduces to:
rn = (5.29 x 10-11 m)(n2/Z)
For hydrogen, Z = 1, the smallest Bohr orbit (n = 1) has a
radius of 5.29 x 10-11 m. This is called the Bohr Radius.
Example
Find the radius of the He+ ion when it is in the second excited
state.
Bohr energy levels in joules can be calculated with:
En = (-2.18 x 10-18 J)Z2/n2
n = 1, 2, 3, . . .
If the energies are expressed in electron volts, the equation is:
En = (- 13.6 eV)Z2/n2
n = 1, 2, 3, . . .
In a hydrogen atom, when the electron has an energy of -13.6
eV, we call this the ground state. This is the lowest energy level
possible in a hydrogen atom. If energy is added to the electron,
it can occupy a higher energy level until it emits a photon and
falls back to a lower energy level.
If the electron is given 13.6 eV, it leaves the atom (n = ∞) and a
hydrogen ion is formed. 13.6 eV is called the ionization energy
for hydrogen.
The energy differences associated with the Lyman, Balmer,
and Paschen series are shown in the energy level diagram
above. Notice how the higher energy levels are very close
together.
Bohr's equations only apply to an atom with one electron.
Interactions among multiple electrons were not considered.
A gas can emit photons to form a bright line or emission
spectrum when its electrons fall from higher energy levels to
lower ones. The reverse process can also happen. If white light
is passed through a sample of a gas, electrons can absorb
photons and jump up to an excited state.
The frequencies corresponding to the energies absorbed are
absent from the normal continuous spectrum of white light and
appear as dark lines called absorption lines. Absorption lines
appear in the spectrum of the Sun and are called Fraunhofer
lines in honor of their discoverer.
The Bohr model provides a fairly good description of the
hydrogen atom but it does not accurately describe any other
atoms with multiple electrons. Quantum mechanics and the
Schrödinger equation do a better job.
DeBroglie's Explanation
Bohr's assumption that angular momentum was quantized as
an integral multiple of Planck's constant divided by 2π was
explained by DeBroglie.
DeBroglie felt that particles like electrons existed both as a
particle and as a wave. In order for an electron to remain in a
stable state, the circumference of the orbit must be an integral
multiple of the electron's wavelength. This equation is:
2πr = nλ
n = 1, 2, 3, . . .
The DeBroglie wavelength of the electron would be:
λ = h/p = h/mv
If we substitute for λ and rearrange the equation, we get:
2πr = nh/mv
n = 1, 2, 3, . . .
mvr = nh/2π
n = 1, 2, 3, . . .
This is what Bohr assumed for the angular momentum of the
electron.
The Quantum Mechanical Picture of the Hydrogen Atom
The Bohr model of the atom uses a single number to identify
the state of an electron in the hydrogen atom. It is called a
quantum number since it can only have certain discrete values
such as 1, 2, 3, etc.
Electrons with slightly different states exist within those
principal energy levels. These states are specified with a total
of four quantum numbers per electron.
1. The principal quantum number specifies the energy level
(shell) where the electron is located. It can have integral values
beginning with 1(n = 1, 2, 3, . . .).
2. The orbital quantum number, ℓ determines the angular
momentum of the electron due to its orbital motion. ℓ can have
the values 0, 1, 2, . . . (n-1). The value 0 corresponds to s
orbitals, the value 1 corresponds to p orbitals, 2 corresponds to
d orbitals, etc.
The magnitude of the orbital momentum of the electron is:
L = [ℓ(ℓ + 1)]½h/2π
3. The magnetic quantum number, mℓ, describes the effect of
an externally applied magnetic field on the energy of the atom.
If there is no magnetic field, mℓ plays no role in determining
the energy. This is called the Zeeman Effect for its discoverer,
Pieter Zeeman. Values of mℓ depend on ℓ and range from -ℓ to
+ℓ. The component of the angular momentum in the Z
direction is found with:
Lz = mℓh/2π
The magnetic quantum number describes the orientation of the
orbital within the atom. In the case of ℓ = 0, there is only one
possible orbital orientation. In the case of ℓ = 1, there are three
possible orientations, in the case of ℓ = 2, there are 5 possible
orientations, etc.
4. The spin quantum number, ms, indicates the direction of
rotation of the electron itself. The two possibilities are up and
down or +½ and -½. This number becomes important as
electrons pair up in orbitals since, in stable orbitals ,opposite
spins always pair up.
Example
The principal quantum number for an electron in an atom is n
= 6. The magnetic quantum number is mℓ = 2. What are the
possible values for the orbital quantum number?
Another Example
A hydrogen atom is in the second excited state. Determine (a)
the total energy (in eV) of the atom, (b) the magnitude of the
maximum angular momentum the electron can have, and (c)
the maximum value that Lz can have.
In the Bohr model of the atom, each energy state corresponds
to a specific distance from the nucleus. Quantum mechanics
changes this to a probability distribution with the distance r
being the most probable distance. Furthermore, when p, d, and
f orbitals are considered, their shapes are not spherical and
orientations with respect to the X, Y, and Z axes must be
considered.
The upper diagram shows the s orbital in the second shell. The
lower diagram shows the p orbital oriented along the Z axis.
The Pauli Exclusion Principle and the Periodic Table
In a multiple-electron atom, the Bohr equation for the energy
of an atom does not work. The four quantum number
approach still gives reasonably good results.
When all the electrons in an atom are in their lowest possible
energy states, this is called the ground state. Most atoms spend
most of their time in the ground state. The lowest energy level
present in an atom is n = 1 but only two electrons can occupy
that shell. The Pauli Exclusion Principle states that no two
electrons in an atom can have the same set of the four quantum
numbers.
Since there are only two unique sets in the first energy level,
only two electrons can be found there. In the second shell there
are eight sets so eight electrons can be found there, etc.
As the principal quantum number and the orbital quantum
number increase, the energy of the electron in that state
increases. There are some places, however, where principal
quantum number states (shells) overlap.
On the Periodic Table the fourth energy level begins before the
third energy level is complete. The 4s orbital has less energy
than the 3d orbitals. This overlap occurs again with the 5s and
4d orbitals and in several other places. This is what gives the
Periodic Table its distinctive structure.
X-rays
X-rays were discovered by Wilhelm Roentgen, a Dutch
physicist. When electrons are given large kinetic energies by
moving through a large potential difference, they release this
energy when they collide with a metal target, usually
molybdenum or platinum. The energy is released in a spectrum
of wavelengths as the electrons are stopped. This is called
Bremsstrahlung (braking radiation).
Sometimes electrons in the K shell of the target are knocked
out of the shell. An electron from the L shell (n =2) or the M
shell (n = 3) will fall to the K shell and release characteristic X-
ray radiation. It is called characteristic since the wavelength
depends on the energy difference between the two shells.
The cutoff wavelength, λ0, is determined by the maximum
kinetic energy an electron has when it strikes the target
material. Since the energy due to the voltage is eV, and E = hf
= hc/λ, we can write:
eV = hc/λ
λ0 = hc/eV
Example
Find the cutoff wavelength for a potential difference of 45,000
volts.
Conventional X-ray photographs have been used for many
years as a diagnostic tool. More dense tissue such as bone
absorbs X-rays better than muscle, fat or organ tissue. This,
however, is a two dimensional picture and the depth at which
absorption occurs is difficult to determine.
A CAT scan or CT scan (computerized axial tomography)
takes multiple X-rays of a slice of the body from different
angles. The images are then combined into one more detailed
image of that slice.
P 970 Questions 1, 3, 6, 7, 8, 9, 10, 11
P 971 Problems 3, 5, 7, 12, 15, 22, 23, 28. 29, 35, 38