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The Evolution of the
Atomic Model:
Thomson’s plumpudding
Rutherford’s nuclear
model
An electron in orbit about
the nucleus must be
accelerating; an accelerating
electron give off
electromagnetic radiation.
So, an electron should lose
energy and spiral into the
nucleus.
Gases in sealed tubes
can be made to emit
electromagnetic radiation
with a potential difference.
The light emitted can be
split into wavelengths by
a spectroscope.
The result is a series of bright
lines or fringes called a line
spectrum. Similar series of
spectra have been found in
the invisible spectrum, at
both shorter and longer
wavelengths than the visible.
The group in the visible
region is called the Balmer
series. The Lyman series is
comprised of waveleghts
shorter than the visible
spectrum; and the
Paschen series of longer
wavelengths than the visible.
Empirical equations:
Lyman series:
1/λ = R(1/12 - 1/n2) n = 2,3,4,...
Balmer series:
2
2
1/ λ = R(1/2 - 1/n ) n = 3,4,5,...
Paschen series:
1/ λ = R(1/32 - 1/n2) n = 4,5,6,...
R is the Rydberg
constant,
R = 1.097 x
7
10
-1
m .
Ex 2 - Find (a) the
longest and (b) the
shortest wavelengths
of the Balmer series.
The equations
show where the
lines are found,
but they do not
explain why.
Neils Bohr’s model of the
atom explains the why.
Bohr hypothesized that
there are only certain
values for the total energy
(kinetic energy plus
potential energy) of an
orbiting electron.
These energy levels
correspond to different
electron orbits. Bohr also
assumed that in these orbits
an electron does not radiate
electromagnetic energy. They
are called stationary orbits or
stationary states.
Bohr knew that radiationless
orbits violated the laws of
physics, but he knew they must
exist. He utilized Einstein’s
photon concept by theorizing
that a photon is emitted only
when an electron moves from a
higher energy orbit to a lower
energy one.
Electrons get to the higher
energy level when they
pick up energy as atoms
collide. This occurs more
often when atoms are
heated up or stimulated
by a flow of electricity.
When an electron moves
to a lower energy level,
the energy of the photon
is Ei - Ef. A photon’s
energy is hf, so
Ei - Ef = hf. Bohr used
this equation to find λ.
The formula for the radii of
the Bohr orbits in meters
-11
2
is: rn = (5.29 x 10 )n /Z
n is the orbit number,
Z is the number of
protons.
If Z = 1 and n = 1, it
is the smallest Bohr
orbit of a hydrogen
-11
atom: 5.29 x 10 m.
This the Bohr radius.
The Bohr energy levels
in joules:
En = -(2.18 x
-18
10
2
2
J)Z /n
n is the orbit number,
Z is the number of
protons.
The Bohr energy levels in
electron volts:
-19
(1.6 x 10 J = 1 eV)
2
2
En = -(13.6 eV)Z /n
n is the orbit number,
Z is the number of protons.
Ex 3 - Find the
ionization energy that
is needed to remove
the remaining
2+
electron from Li .
Bohr’s calculations led to a
calculation of the Rydberg
constant. Bohr’s theory’s major
accomplishment was the
agreement between the
theoretical and experimental
values of the Rydberg constant.
The Lyman series is produced
when an electron’s transition is
from higher levels to the first
level. In the Balmer series,
electrons end up in level two.
Electrons reach level three from
higher levels in the Paschen
series.
There is a different
amount of energy
released in each case, so
the frequency and
wavelength of the photon
emitted is different in each
case.
When electrons
move to lower energy
levels, photons are
emitted, so these are
called emission lines.
If light with a range of
wavelengths passes
through a gas, some
photons are absorbed
as they excite electrons
to a higher energy
level.
These are called
absorption lines,
which appear as
gaps in the
continuous spectrum.
Bohr’s equation for
the angular
momentum of the
electron was:
Ln = mvnrn = nh/2π.
n = 1,2,3,...
de Broglie pointed
out that the electron
in its circular orbit
must be pictured as a
particle wave.
As with any wave,
resonance can form a
standing wave when
the distance traveled is
any integer number of
wavelengths.
The total distance around a
Bohr orbit of radius r is the
circumference or 2πr. The
condition for a standing particle
wave would be
2πr = nλ. n = 1,2,3,...
“n” is the number of whole
wavelengths that fit into the
circumference of the circle.
In a particle wave, the de Broglie
wavelength of the electron is
λ = h/p (p is the electron’s
momentum and is equal to mv).
2πr = nλ becomes 2πr = nh/mv.
Rearranged, this is mvr = nh/2π.
n = 1,2,3,... This is what Bohr
assumed for the angular
momentum of the electron.
De Broglie’s
explanation
emphasizes the fact
that particle waves play
an important role in the
structure of the atom.
While Bohr’s model uses a
single quantum number,
quantum mechanics
reveals that four different
quantum numbers are
needed to describe each
state of the hydrogen atom.
1. The principle quantum
number n. Total energy of
the electron, n = 1,2,3,...
2. The orbital quantum
number l. Angular momentum
of the electron,
l = 0,1,2,...(n-1) and
L = √l(l+1)•h/2π
3. The magnetic quantum
number ml. An external
magnetic field influences the
energy of the atom.
ml = -l,...,-2,-1,0,1,2,...,+l
and angular momentum in
the z direction is Lz = ml h/2π
4. The spin quantum
number ms. Describes
the electron’ s spin
angular momentum.
ms = +1/2 or ms = -1/2
As the principle quantum
number n increases, the
number of possible
combinations of the four
quantum numbers (and
therefore, the number of
possible states of the atom)
increases rapidly.
Ex 5 - Determine the
number of possible
states for the hydrogen
atom when the
principle quantum
number is (a) n = 1
and (b) n = 2.
While the Bohr model
predicts an exact orbit for the
electron, in reality this is just
the most probable location of
the electron.
Actually it is found
somewhere in an electron
cloud around the nucleus.
This cloud takes
different, more complex
shapes than spherical
as the possible values
of the orbital quantum
number increase.
The Bohr model predicts a specific
location for an electron’s orbit.
From this we can calculate a
specific velocity. This means
a specific momentum. But
remember, the Heisenberg
uncertainty principle states that
specific location and specific
momentum can’t both be known.
When calculated, the
possible locations of the
electron in hydrogen range
from r = 0 to twice the Bohr
radius!
The Bohr model, while useful,
does not represent the reality
of the atom.
In multiple-electron atoms,
electrons with the same
value of n are said to be in
the same shell.
n = 1, the K shell
n = 2, the L shell
n = 3, the M shell
If the electrons have the
same value of n and l, they
are in the same subshell. The
n = 1 shell has a single l = 0
subshell. The n = 2 has two
subshells: l = 0 and l = 1.
If n = 3, three subshells:
l = 0, l = 2, l = 3.
When at room
temperature, an atom’s
electrons spend most of
their time in the lowest
energy levels possible.
This lowest-energy state
is called the ground state.
However, when a multipleelectron atom is in its ground
state, not every electron is in
the n = 1 shell. The Pauli
Exclusion Principle states that
no two electrons in an atom can
have the same set of values for
the four quantum numbers: n, l,
ml, and ms.
Ex 8 - Determine which
of the energy levels are
occupied by electrons
in the ground state of
hydrogen, helium,
lithium, beryllium, and
boron.
The l = 0 subshell, s
The l = 1 subshell, p
The l = 2 subshell, d
The l = 3 subshell, f
The l = 4 subshell, g
The l = 5 subshell, h
A shorthand notation used
is as follows:
5
2p
2 in this case is n,
p is l,
5 is the number of
electrons in the subshell
This subshell filling
explains the orientation
of the periodic table. In
the transition metals
3d, 4d, 5d, and 6d are
filled; lanthinide: 5d and
4f; actinide: 5f and 6d.
Light
Amplification by
Stimulated
Emission of
Radiation
Single frequencymonochromatic
Constant phase relationship
Light flash into a ruby crystal
sends photons into the
crystal. This sends some
atoms into a higher energy
state, electrons drop into
lower energy levels
releasing more photons.
These photons excite
more atoms, etc.
This results in an
amplification of the
photon beam.
Since the crystal is of a
specific length, a standing
wave of single-frequency light
is produced. As intensity
increases, photons pass
through a partially reflective
end in intense pulses of
coherent red light.