Download Lecture 11 Atomic Structure Earlier in the semester, you read about

Lecture 11
Atomic Structure
Earlier in the semester, you read about the discoveries that lead to the proposal of the
nuclear atom, an atom of atomic number Z, composed of a positively charged nucleus
surrounded by Z electrons. When Rutherford proposed the nuclear atom, he expected to
be able to use classical mechanics (Newton's laws of motion) to describe the properties
of the sub-atomic particles. It soon became apparent that classical mechanics could not
be used to study the electrons in atoms. New laws were developed in the early twentieth
century to explain this unexpected behavior of electrons. These laws are known as
quantum mechanics. This is a fascinating, but complex, area of chemistry. We will
take a superficial look at the general principles.
When salt solutions are heated, the flames have distinct and diagnostic colors. Shown
below are flame tests of Na+, K+ and Li+ salts. Other salts have similar, characteristic
colors. These experiments suggest that there are spectral (related to light) properties
associated with atoms.
When elemental gases are heated, they also produce radiation, with each element emitting
only certain colors. In order to understand the connection between these spectral
properties and the quantum mechanical structure of the atom, we first have to learn more
about the nature of light, or more generally, electromagnetic radiation.
The Wave Nature of Light
All waves have a characteristic wavelength. In the figures shown below, the wavelength
is the distance between the peaks or troughs of the wave.
1 second
This is given the symbol "λ" in chemistry. As the wavelengths get smaller, there are
more cycles of the wave in a given distance on the x-axis. The x-axis is a measure of
time. In the first wave, there are two cycles per second, in the second, four cycles per
second and in the third wave, eight cycles per second. The unit for frequency (cycles
per second, or s-1) is Hz (hertz) and frequency is given the symbol "ν". Frequency and
wavelength are obviously related; the shorter the wavelength, the larger the frequency.
They are related by the equation, u = λν, where u is the speed that the wave is traveling.
Energy travels through space as electromagnetic radiation. It is comprised of an
electrical field (pink) and magnetic field (purple) traveling perpendicular to each other
and to the direction of propagation. Electromagnetic radiation has wave properties.
(Image from Jim Hawkins Radio and Technology Page
http://www.hawkins.pari.com)
The speed of propagation, u, of electromagnetic radiation is the speed of light, 2.998 x
108 meters per second (ms-1). This value is given its own symbol, c, the speed of light.
The speed of light is equal to the wavelength of the radiation, times the frequency of the
radiation, c = λν. There are no upper or lower limits in the electromagnetic spectrum.
At the long wavelength, low frequency end are radio waves. They have wavelengths of
the order of 100 meters. Their frequency can be calculated because, c = λν.
2.998 x 108 ms-1 = 100 m x ν
ν ≈ 3 x 106 s-1
At the short wavelength, high frequency end are gamma rays. They have wavelengths
of the order of 10-12m, and corresponding frequencies of about 1020 s-1. In the middle of
the electromagnetic spectrum is the visible light spectrum, represented below. These are
the colors that we can detect visually. They range from red (low frequency, long
wavelength, 700 nm), to violet, (high frequency, short wavelength, 400 nm). The
drawing to the right of the spectrum illustrates how the wavelength and frequency change
in the visible region, in going from red, to yellow to green light.
λ = 700 nm
λ = 580 nm
λ = 530 nm
Let's try to tie this in with Chemistry. When elements are heated, they produce
electromagnetic radiation. However, each element emits light of only a few, specific
frequencies.
Quantum Physics
This phenomenon could not be explained by classical physics, which would have
predicted that a continuous spectrum would be emitted. These line emission spectra
were first explained by the German scientist, Max Planck. He proposed that matter can
absorb or emit radiation with energies that are some multiple of a frequency, ν. The
equation that he proposed is:
E = hν
where E is energy, ν is wavelength, and h is Planck's constant, 6.626 x 10-34 J s. E is
the smallest quantity of energy that can be emitted or absorbed, a quantum of energy.
Classical physics describes energy as continuous. Quantum physics, describes energy
as quantized. Going from energy level A to energy level B can be viewed two different
ways:
B
A
B
A
Classical physics
Quantum physics
In Quantum physics, energy can only change by certain discrete amounts (steps):
ΔE = nhν
The symbol, ΔE, means, final energy minus initial energy. This explains why only a
fixed number of emission lines are seen from the elements. These frequencies are
associated with allowed quanta of energy leaving the atoms. For example, the yellow
light emitted by sodium with a wavelength of 589 nm, corresponds to a frequency of:
c = λν = 2.998 x 108 m s-1
ν = 2.998 x 108 m s-1 ÷ 589 x 10-9 m = 5.09 x 1014 s-1
Substituting this value into Planck's equation gives:
E = hν = (6.626 x 10-34 J s)(5.09 x 1014 s-1) = 3.37 x 10-19 J
This means that sodium (with a line at 589 nm) can absorb (or emit) 3.37 x 10-19 joules
of energy, or 6.74 x 10-19 J, or 10.11 x 10-19 J of energy, but not 5.00 x 10-19 J of
energy. Planck couldn't explain why this was the case.
The Photoelectric effect
Five years after Planck proposed the quantum theory, Albert Einstein used this theory to
explain another mystery of physics, the photoelectric effect. It was known that electrons
are ejected from the surface of certain metals when they are exposed to light of a certain
minimum frequency, called the threshold frequency. If a solid potassium surface is
exposed to red light, no matter how intense, no electrons are ejected. But, if even a weak
beam of violet light is used, electrons are given off. The number of electrons emitted is
proportional to the intensity of the light, above the threshold frequency. The energy of
the electrons given off is independent of the intensity of the light, but dependent on the
frequency. What these results suggested to Einstein, is the light exists as particles. He
called them, photons, each with an energy equal to hν. In each collision with a metal
atom, the photon gives up its energy to an electron in the metal atom. There is a
minimum energy needed to pull the electron from its atom. That is the threshold
energy. So, no amount of red photons can dislodge electrons from potassium, since they
do not possess enough energy. Any violet photon does possess this minimal energy, and
will allow electrons to escape. The higher the intensity of the light, the more photons will
hit the metal surface, and the more electrons will be ejected. The proposal that light has
properties of a particle was a very startling proposition.
The Bohr atom
Niels Bohr, a Danish scientist, combined the ideas of quantized energy and the line
emission spectra in an atomic model of hydrogen, H. He proposed that energy of the
electron in the H atom is quantized, that it is limited to a discrete set of values. E1 is the
ground state energy. E2 is the 1st excited state energy. Hydrogen atoms radiate energy
only when the electron "jumps" from one allowed energy level to another. ΔE = hν. The
diagram below shows the correspondence between energy levels (n = 1,2,3, etc.) and the
observed emission lines of H.
For the transition between n = 2 and n = 1, the wavelength corresponds to a line in the
emission spectrum. A number of scientists (Balmer, Lyman, Rydberg) had developed
mathematical equations to describe the placement of the lines in the emission spectra.
Bohr applied their equations to the hydrogen atom and derived an equation to predict the
allowed energy levels in hydrogen.
E n = -RH ( Z2/n2)
where n = principal quantum number and RH = the Rydberg constant = 2.18 x 10-18 J
and Z is the nuclear charge. By using this equation, we should be able to predict the
wavelength of any of the electronic transitions in the H atom. For example, what
wavelength of light will be emitted when an electrons falls from the n = 2 (initial energy
level) to n = 1 (final energy level)? Hydrogen has an atomic charge of +1, so Z = 1. It
will be given by:
E1 - E2 = -RH (1/12 - 1/22) = - 1.63 x 10-18 J
This number is negative, because energy is being given off by the atom. We would
expect this, since the electron is moving from a higher energy state to a lower energy
state. This can be used to calculate the frequency of the energy being emitted.
. ΔE = hν = 1.63 x 10-18 J = (6.626 x 10-34 J s) ν
ν = 2.45 x 1015 s-1
c = λν = 2.998 x 108 m s-1
λ = 121 nm
This falls outside of the visible spectrum, into the ultraviolet region. The red, green and
blue lines in the hydrogen emission spectrum (shown above) correspond to the transitions
between n = 3, 4 and 5 and n = 2. The green line appears at about 490 nm.
This gives it a frequency of (2.998 x 108 m s-1)÷(490 x 10-9 m) = 6.12 x 1014 s-1.
The energy of this photon would be ΔE = (6.626 x 10-34 Js) (6.12 x 1014 s-1) = 4.05 x 10-19
J.
4.05 x 10-19 J = (2.18 x 10-18 J) (1/22 - 1/ni2)
0.186 = (1/22 - 1/ni2) = 0.25 - 1/ni2
1/ni2 = 0.0640
ni2 = 16
n=4
So, the green line corresponds to the transition between n = 4 and n = 2. (The energy
levels are expressed in electron volts (eV) instead of joules in the diagram).
Allowed energy levels in
hydrogen
Bohr's model was important because it intoduced the idea of quantized states in atoms. It
can only be applied to systems with a single electron, like H, He+, Li2+, etc. After the
introduction of this model, the dual nature of light became quite well accepted. Light
has both wavelike and particle like properties. What is not explained by this model, is
why electrons seem to rotate around the nucleus at certain fixed distances.
The dual nature of matter
The French scientist and prince, Louis de Broglie, reasoned, that if light waves can
behave like particles, then why shouldn't particles, like electrons, behave like light
waves? He proposed that all matter has a dual nature, possessing both wavelike and
particle like properties and that electrons should behave like "standing waves". A
particle of mass, m, traveling at a velocity, v, has linear momentum, p. p = mv. By
combining this with Einstein's theory of relativity, and definition of a quantum of energy,
De Broglie derived the expression:
p = h/λ
This ties together wavelike properties (λ, the wavelength) and particle like properties, p,
momentum. Since p = mv = h/λ
λ = h/mv
With this equation, it is possible to calculate the wavelength of any particle.
The pictures below illustrate the "standing waves" of different frequencies possible in a
vibrating string.
De Broglie envisioned a standing wave associated with an electron around the nucleus.
Only "quantized" values of the wave (i.e., n = 4, 5 or some other multiple of the
wavelength) could exist about the nucleus. Non-integer values, like 4 ½ would lead to
destructive interference.
It had already been shown that X-rays form a diffraction pattern when they are passed
through crystals (X-ray crystallography). De Broglie's model predicted that accelerated
electrons (with increased velocities, v), should have wavelengths very similar to X-rays,
and should also give diffraction patterns with crystals. There was no experimental basis
for making this very bold prediction. Within three years, groups observed exactly this
phenomenon, electron diffraction. Electrons were accelerated to about 6 x 106 m/s.
With this velocity and with the mass of an electron, his equation predicted that the
wavelength of accelerated electrons should be about 0.122 nm, in the same spectral range
as X-rays. Shown below are two diffraction patterns. On the left is the diffraction
patterns of X-rays passing through aluminum foil. On the right is the diffraction pattern
of a beam of electrons passing through the same target.
This proved the wavelike nature of electrons.
Does this apply to all matter? One can actually compute the wavelength of a 6.0 x 10-2
kg tennis ball, going 85 ms-1 (190 mph, a good tennis serve).
λ = h/mv
λ = (6.63 x 10-24 Js) ÷(6.0 x 10-2g)(62 ms-1)
λ = 1.8 x 10-34 m
This is smaller than any atom, and could never be measured with existing devices.