# Electronic Structure of Atoms

Urban and Civil

Nov 16, 2013 (4 years and 7 months ago)

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Electronic Structure of Atoms

In this chapter we will explore, the quantum
theory and its importance in the study of
chemistry. We will look more closely at the
nature of light and how our study of light has
changed the quantum theory to explain how
electrons are arranged in an atom.

The Wave Nature of Light

Much of our present understanding of the electronic structure of
atoms has come from analysis of light either emitted or absorbed
by substances.

Light we see, visible light, is an example of electromagnetic
energy through space.

waves to gamma rays, differ from each other but also share many
fundamental characteristics

Speed of light (c) in a vacuum is
3.00
X
10
8

m/s

Wavelength (
λ
)
is from one point on one wave to the same point
on the next wave expressed in meters.

Frequency (
ע
) is the number of complete wavelengths or cycles
that pass a certain point each second expressed as
cycles/second or Hertz.

The inverse relationship between the frequency and the
wavelength of electromagnetic radiation can be expressed by
the equation c=
λ
ע

Sample Exercise
6.2
pg
215

Electromagnetic Spectrum

Quantized Energy and Photons

When solids are heated, they emit radiation as seen in the red glow of
stove and the bright white light of a tungsten light bulb.

The wavelength distribution of the radiation depends on temperature.

In 1900, Max Planck assumed that energy can be either be released or
absorbed by atoms only in discrete “chunks” of some minimum size.
Planck gave the name quantum to the smallest quantity of energy that
can be emitted or absorbed as electromagnetic radiation.

He proposed that the energy, E, of a single quantum equals a constant
time the frequency,
ʋ,

ʋ

The constant H is called Planck’s constant and has a value of

6.626 X 10
-
34

Joule seconds.

According to Planck’s theory, matter is allowed to emit and absorb
energy only in whole number multiple of hʋ, such as 2hʋ, 3hʋ and 4hʋ.

This theory can be compared to walking up stairs. As you can only
step on individual stairs or multiples stairs but not between them so
your increase in potential energy is restricted to certain values and is
therefore quantized.

Shows the device built by NIST researchers to perform a high
-
precision measurement of Planck's constant, the
number which describes the bundle
-
like nature of matter and energy at the atomic and subatomic scales. The
electrical power associated with the mechanical motions of the system contains quantities proportional to Planck's
constant.

Photoelectric Effect and Photons

In 1905, Einstein used Planck’s quantum theory to explain the
photoelectric effect.

Experiments had shown that light shining on a clean metal surface
causes the surface to emit electrons. Each metal has a minimum
frequency of light below which no electrons are emitted.

Einstein assumed that the radiant energy striking the metal surface
does not behave like a wave but rather as if it were a stream of tiny
energy packets. Each energy packet, called a photon behaves like a
tiny particle.

Extending Planck’s quantum theory, Einstein deduced that each
photon must have an energy equal to Planck’s constant times the
frequency of the light.

Analysis of data from the photoelectric experiment showed that the
energy of the ejected electrons was proportional to the frequency of
the illuminating light. This showed that whatever was knocking the
electrons out had an energy proportional to light frequency. The
remarkable fact that the ejection energy was independent of the total
energy of illumination showed that the interaction must be like that of a
particle which gave all of its energy to the electron!

This fit in well with Planck's hypothesis that light in the blackbody
radiation experiment could exist only in discrete bundles with energy.

Sample Exercise 6.3 and Practice Exercise pg 217

Line Spectra

The work of Planck and Einstein paved the way for
understanding how electrons are arranged in atoms.

In 1913, Danish physicist Niels Bohr offered a theoretical
explanation of line spectra. Most common radiation sources,
including light bulbs and stars, produce radiation containing
many different wavelengths.

A spectrum is produced when radiation from such sources is
separated into its different wavelength components. A prism
spreads light from a white light source into its component
wavelengths of a continuous range. A rainbow occurs when
raindrops acts as a prisms to separate sunlight.

Not all radiation sources produce a continuous spectrum.

When a high voltage is applied to tubes that contain different
gases under reduced pressure, the gases emit different colors
of light.

When that light is passed through a prism only a few
wavelengths are present in the resultant spectra which is
called a line spectra.

When scientists first detected the line spectrum
of hydrogen in the mid 1800s, they were
fascinated by it simplicity. At that time, only the
four lines in the visible portion of the spectrum
were observed as shown in 6.13 pg 219.

These lines correspond to wavelengths of
410nm, 434 nm, 486nm and 656nm.

In 1885, a Swiss named Johann Balmer showed
that the wavelengths of these four visible lines of
hydrogen fit a simple formula.

Soon Balmer’s equation was extended to a more
general one, called the Rydberg equation.

Bohr’s Model

Rutherford’s discovery of the nuclear nature of the atom
suggests that the atom can be thought of as a “microscopic
solar system” in which electrons orbit the nucleus.

To explain the line spectrum of hydrogen, Bohr assumed
that electrons move in circular orbits around the nucleus.

According to classic physics though, an electrically charged
particle (such as an electron) that moves in a circular path
should continuously lose energy by emitting electromagnetic
radiation. As the electron loses energy, it should spiral into
the positively charged nucleus. But since hydrogen atoms
are stable this must not be happening. Borh based his model
therefore on three postulates to explain this contradiction.

Only orbits of certain radii, corresponding to certain definite energies,
are permitted for the electron in a hydrogen atom.

An electron in a permitted orbit has a specific energy and is in an
allowed energy state. An electron in an allowed energy state will not
radiate energy and therefore will not spiral into the nucleus.

Energy is emitted or absorbed by the electron only as the electron
changes from one allowed energy state to another. This energy is
emitted or absorbed as a photon, E=h
v

Energy States of Hydrogen Atom

Bohr calculated the energies corresponding to each allowed orbit for the electron
in the hydrogen atom using the following formula:

E=(
-
hc/R
H
)(1/n
2
)

In this equation, h is Planck’s constant, c is the speed of light and R
H

is the
Rydberg constant, which means they can all be multiplied together to simplify the
equation to:

E=(
-
2.18 X 10
-
18
J)(1/n
2
)

Where n is called the principle quantum number or energy level from 1 to infinity.
Each orbit has a different number for n increasing out from the nucleus with 1
being closest to the nucleus.

The energies of the electron of a hydrogen atom given by the above equation are
negative for all values of n. The lower (more negative) the energy is, the more
stable the atom will be. Since the energy is lowest for n=1, when the electron is
there, the atom is the most stable. This is why n=1 is called the ground state.

When the electron is in a higher energy orbit it is said to be in an excited state.

In his third postulate, Bohr assumed that the electron could “jump” from one
allowed energy state to another by either absorbing or emitting photons whose
radiant energy corresponds exactly to the energy difference between the two
states. Energy must be absorbed for an electron to go to a higher energy state
and be emitted to fall to a lower state, so only specific energies of light can be
absorbed and emitted by the electron in the hydrogen atom.
Animation

Therefore, the existence of discrete spectral lines are due to the quantized jumps
of electrons between energy levels.

Results of Bohr’s Model

Model worked great to explain the hydrogen
atom, but not for any other atom.

Avoided problem of why the electron didn’t fall
into the nucleus.

Became an important step to the excepted
model today, as two ideas are also incorporated
into our current model

Electrons exist only in certain discrete energy levels,
which are described by quantum numbers

Energy is involved in moving an electron from one
level to another.

Wave Behavior of Matter

In the years following Bohr’s model, the dual nature of light
became a familiar concept, which says that light appears to
have either wavelike or particle
-
like characteristics. It was
thought then that if light could behave as a stream of particles
then maybe matter could have properties of a wave.

Louis de Broglie suggested that as the electron moves about
the nucleus, it is associated with a particular wavelength, and
that the characteristic wavelength of the electron depends on
its mass, m and on its velocity, v, by the equation
λ
=h/mv.

Within a few years, the wave properties of the electron were
demonstrated experimentally. As electrons passed through a
crystal, they were diffracted by the crystal, just as x
-
rays are
diffracted. Thus, a stream of electrons exhibits the same kinds
of wave behavior as electromagnetic radiation and has both
particle and wavelike characteristics.

The technique of electron diffraction has been highly
developed to produce such things as the electron microscope
which can obtain images at the atomic level.

Spider

Pollen

Atoms

Uncertainty Principle

German physicist Werner
Heisenberg proposed that the
dual nature of matter places a
fundamental limitation on how
precisely we can know both the
location and the momentum of
any object.

This is only significant for
things as small as an electron
as a small change in its
position often due to the
measuring device is more
significant than a small change
in our positions.

When applied to electrons it is
impossible to know both the
momentum of the electron and its
exact location in space at the
same time.

Quantum Mechanics and Atomic
Orbitals

In 1926, Austrian physicist Erwin Schr
ö
ndinger proposed a
wave equation that incorporates the wavelike and particle
like behavior of the electron.

We wont go into the equation but look at the results he
obtained that led to the current view of the atom.

Just like when a guitar string is plucked and creates a fuzz
showing where the string is most likely located. An electron
that is moving around a nucleus at a rapid wavelike speed
forms a cloud where there is a high probability that the
electron can be found.

The solution to Schrödinger's equation for the atom yields a
set of wave functions and corresponding energies. These
wave functions are called orbitals and each orbital describes
a specific distribution of electron density in space.

There are three quantum numbers that describe the orbitals
where the electrons in an atom are found.

Greatest probability of finding the electron

Quantum Numbers

1. The principle quantum number, n, has numerical values
of 1
-
7. As n increases, the orbital (distribution of electron
density) becomes larger. And the electron spends more time
farther from the nucleus. An increase in n also mean that
the electron has a higher energy and is therefore less tightly
bound to the nucleus.

2. The angular momentum quantum number,
ʅ
, has
numerical values from 0 to (n
-
1) for each value of n. This
quantum number defines the shapes of the orbital and is
more often described by the letters s p d f instead of the
numerical values (s=0, p=1, d=2, f=3).

The magnetic quantum number, m
ʅ
, has numerical values
between
ʅ

and
-
ʅ

including 0. this quantum number describes
the orientation of the orbital in space.

The collection of orbitals with the same value of n is called
an electron shell. All orbitals that have n=3, for example, are
said to be in the third shell. Each subshell is designated by a
number (the value of n) and a letter (s, p, d or f).

Review all with Table 6.2 on page 227.

s

p

d

f

The principle quantum number also states the number of
subshells in the whole shell. For example, when n=1 then
there is only 1 subshell (1s) in the 1
st

shell and when n=2
then there are 2 subshells (2s and 2p) in the 2
nd

shell.

Every s subshell contains 1 orbital (only one orientation in
space), every p subshell contains 3 orbitals (3 orientations in
space x,y,z), every d subshell contains 5 orbitals (5
orientations in space xy, xz, yz,x
2
-
y
2
, z
2
), and every f
subshell contains 7 orbitals (7 orientations in space).

In hydrogen, no matter what subshell the electron is in within
one shell, it will have the same energy such as 3s, 3p or 3d.

In many electron atoms, however, the electron
-
electron
repulsion causes the different subshells to be at different
energies but the orbitals within a subshell to be equal.

Electron Spin

We know where the electron that hydrogen has resides in the
ground state (1s) but where do the electrons in the many electron
atoms reside.

When scientists studied the line spectra of many electron atoms in
great detail, they noticed that lines that were originally thought to
be single were actually closely packed pairs. This meant that there
were twice as many energy levels as there were “supposed” to be.

In 1925, George Uhlenbeck and Sam Goudsmit proposed that
electrons have an intrinsic property, called electron spin, that
causes each electron to behave as if it were a tiny sphere spinning
on its own axis.

This observation led to the fourth and final quantum number for the
electron, the spin magnetic number, ms. The two numerical values
for are +½ and

½, which was first interpreted as indicating the
two opposite directions in which the electron can spin.

A spinning charge produces a magnetic field so the two opposite
directions of spin therefore produce oppositely directed magnetic
field. These two opposite magnetic fields lead to the splitting of
spectral lines into closely spaced pairs.

Pauli Exclusion Principle

Electron spin is crucial for the electron structure of atoms.

In 1925, Wolfgang Pauli discovered the principle that governs the
arrangements of electrons in many electron atoms.

The Pauli Exclusion Principle states that no two electrons in an
atom can have the same four quantum numbers, n,
ʅ

, m
ʅ,

m
s
.

Following this principle, an orbital can hold a maximum of two
electrons and they must have opposite spins.

Now we can look at how electrons are arranged in the various
orbitals for many electron atoms, which is called the electron
configuration.

The most stable electron configuration of an atom

the ground
state
-
is that in which the electrons are in the lowest possible
energy states.

Thus orbitals are filled in order of increasing energy with no more
than two electrons in each orbital.

Finally Hund’s rule says that for orbitals with equal energies, the
lowest energy is attained when the number of electons with the
same spin is maximized (1 in each first then the second ones go in
with opposite spins).

Table to use for writing electron
configurations…