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Ionization energy of hydrogen
To completely remove the electron from
a hydrogen atom is an example of
ionization.
The Bohr model allows us to predict the
ionization energy of an electron in a
hydrogen atom from the ground state ni =
1 to an unbound electron where nf = ∞
Chapter 8
Electrons in Atoms
Dr. Peter Warburton
[email protected]
http://www.chem.mun.ca/zcourses/1050.php
∆E=hν=R H
1
1
−
= R H 1−0 =R H
ni 2 nf 2
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Ionization energy of hydrogen-like ions
Ionization energy of hydrogen-like ions
− =
If we consider ions that have only one
electron like He+, Li2+, Be3+ 3 then we
can use the Bohr models to predict the
energy levels and the ionization energy of
the last electron
As the atomic number goes up, the ionization
energy increases because we are trying to
remove a negatively charged electron from a
nucleus with a total Z+ charge. The electron is
more tightly electrostatically bound to a nucleus
with larger positive charge!
− =
where Z is the atomic number for the ion
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Problem
Problem answer
Determine the wavelength (and color if it
falls in the visible spectrum) of light emitted
in an electron transition from n = 5 to n = 3
in a Be3+ ion.
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Problem
λ = 80.13 nm. This is actually ultraviolet
light.
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Problem answer
The frequency of the n = 3 to n = 2
transition for an unknown hydrogen-like ion
occurs at a frequency 16 times that for
hydrogen. What is the identity of the ion?
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Since there is a Z2 dependence for
hydrogen-like ions, Z must be 4 for a 16
times increase in frequency (and energy
of transition). This means the ion is Be3+.
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Inadequacies of the Bohr model
Building a new theory
We only talk about the Bohr model for hydrogen
and hydrogen-like ions because the model
cannot handle more than one electron!
Two or more electrons in an atom, molecule or
ion will interact with each other, changing the
energy levels in comparison to a single electron
interacting with a nucleus.
A new theory was required!
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Wave-particle duality
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Wave-particle duality
Starting from Einstein’s equation telling us
that matter and energy are interconvertible
This means
hν = mc2
E = mc2
which we can rearrange to
de Broglie related this to the energy of a
photon
hν/c = mc = p
where p is the momentum (mass
times velocity) of the photon
E = hν
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Wave-particle duality - we saw in the
photoelectric effect that Einstein treated
light (which we think of waves) as
particles called photons.
Louis de Broglie proposed that matter and
energy actually have “dual” nature where
“Small particles of matter may at times
display wave-like properties.”
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Wave-particle duality
de Broglie wavelength
Since
h/p = h/(mu) = λ
λν = c or ν/c = 1/λ
Any wave can be described as a particle in
motion, and any particle in motion can be
described with a wavelength.
However, we see from the equation, that
since h is very small, then the wavelength
for a particle must also be very small –
UNLESS it’s mass is very small as well!
then
p = h/λ = mu
where u is the speed of any
particle
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de Broglie wavelength
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de Broglie wavelength
We also saw on slide 50 that the Bohr
radius of an electron in the ground state of
a hydrogen atom is 53 pm!
If moving electrons have wavelengths in
the picometer range, we should be able to
use them to probe the structure of matter.
h/p = h/(mu) = λ
Consider the problem we did on slides 40
and 41 where an electron was ejected
from a potassium atom with a speed of
1.68 x 107 m s-1. The moving electron can
be treated as a wave with
How? We’ll see soon!
λ = h/(meu) = 43.3 pm
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X-rays can be in the pm range!
X-ray diffraction
Boats and waves
Electron beam
diffraction
To observe an object requires the waves
interacting with the object to be similar in size, or
smaller than the object. If the waves are too big,
the small object gets “overwhelmed” by the wave.
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More SEM
Scanning electron microscopy (SEM)
Using the wave-like
properties of a beam of
electrons, we can get
very detailed images of
small objects that we
cannot get using lenses.
Here we have a white
blood cell (pink) infected
by HIV viruses (blue) at
13400x magnification!
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5
Building a new theory
Heisenberg uncertainty principle
Heisenberg uncertainty principle - the idea
that very small particles have wave-like
qualities means we can never be
absolutely certain about it’s behavior, since
a particle is seen as discrete and localized
and a wave is variable and “spread out”
(delocalized)
If we try and describe a particle by two
variables (usually it’s position x and
momentum p), it turns out that if we are
dealing with a small enough particle, then
we cannot accurately measure both things
at the same time. The uncertainty of the
measurements ∆x and ∆p are related!
∆x∆p ≤ h/4π
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Wave packet
Analogy with camera shutter speed
Is the car moving? Short
time frame of the picture
means we can’t be
absolutely sure. We
know x but aren’t sure
about p.
Here, the car is definitely
moving. But the long
time frame of the picture
means we don’t know
exactly where it was.
We know p but not x.
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To “localize” a small
particle, we must
describe it as a group
of waves - a wave
packet. The more
waves we have, the
better we know x, but
the harder it is to know
p, and vice versa (less
waves to get p, but
don’t know x)
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Types of waves
Types of waves
Traveling waves – like
ocean waves or sound
waves, the crests and
troughs move
through space
Standing waves – like
a plucked guitar string,
the crests and
troughs DO NOT
move through space
n=1
λ = 2L/n
n=2
Standing waves also
have nodes where the
displacement is zero.
The number of such
nodes is n + 1.
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Standing waves in a Bohr orbit
Imagine a guitar
string of length 2L
wrapped around to
form a circle of
circumference 2L.
Standing waves in
the string must still
obey
λ = 2L/n!
Standing waves have
a wavelength that is an
integer fraction of twice
the total length
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Standing waves in a Bohr orbit
If the standing wave
doesn’t obey the
relationship, it will
destructively interfere
with itself, cancelling
out to zero. The
standing wave is not
allowed!
Allowed standing
wave
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n=3
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Disallowed
standing wave
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de Broglie vs Bohr
3D description of an electron
The electrons in an
atom set up standing
waves to achieve the
allowed stationary
states for each n.
Still not good enough!
A guitar string is 1D,
and we need a 3D wave
to describe the electron
in all of the space
around the nucleus.
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Before we can understand how
to describe an electron in 3D,
we need a better
understanding of how to
describe a standing wave in
1D. To do this we solve a
model called particle in a box.
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Particle in a 1D box
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Particle in a 1D box
The standing wave for an
electron in 1D can be treated
exactly like the guitar string.
There must be nodes at each
end of the string, while the
wave must be described by
some sort of cyclic function,
like a sine function!
We give the cyclic function
description a special name –
the wavefunction!
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For a 1D box (along the x
direction) of size L the
wavefunctions ψn that describe
stationary states of a particle in
a 1D box can be described as
2
nπx
ψn = sin
L
L
where n = 1,2,3,3
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Particle in a 1D box
Particle in a 1D box
h2
n2 h2
Ek =
=
2
8mL2
2mλ
The energy of each stationary
state of a wave can be found
since (using de Broglie)
Ek = ½ mv2 = p2/2m = h2/2mλ2
For a particle in a 1D box
The energy of the particle can
never be zero (the zero-point
energy means never at rest!)
Smaller L (better knowledge
of x) implies larger Ek and
therefore a less certain p
h2
n2 h2
Ek =
=
2
8mL2
2mλ
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Particle in a 1D box
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What is the wavefunction?
h2
n2 h2
Ek =
=
2
8mL2
2mλ
The wavefunction is a mathematical
description of how a particle will behave
under certain conditions (like a box of a
certain size). Ultimately, it must contain
ALL the information of particle behavior
under ALL conditions.
However, the wavefunction is not
something physical we can measure.
Most importantly, the
energy of the particle in
the box is quantized,
based on the allowed
values for n!
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Wavefunction analogy
Particle in a 1D box
Since the particle
behaves like a wave,
we can not exactly
say where the
particle is, but we
can evaluate the
probability of finding
the particle at a given
point in the box.
Consider all of the information available to
us through the internet. We certainly can’t
know all of it in our own heads, but under
the appropriate circumstances (a Google
search), we can find the information we
want.
We’ll see this analogy extend further in a
few slides.
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Particle in a 1D box
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Particle in a 1D box
The probability of
finding a particle is
the quantum
equivalent to the
intensity, and
depends on the
“square” of the
wavefunction ψ2
We saw that the
energy of a
classical wave
depends on its
intensity, which
corresponds to
the square of
the amplitude.
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Particle in 3D box
Problem
In 3D space, we can treat the
wavefunction as a combination of three
1D wavefunctions, each with its own
principal quantum number n. When we do
this, we can get the energies of a particle
in a 3D box
What is the wavelength of the photon
emitted when an electron in a 1D box with
a length of 5.0 x 101 pm falls from the n = 5
level to the n = 3 level?
ℎ " % & =
+
+
8! #" #% #& All media copyright of their respective owners
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Problem answer
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Schrödinger equation
We’ve seen that the wavefunction must
contain information about the particle in all
circumstances, and therefore we should be
able to use it to find many different
properties of the particle. To get the
specific property we’re interested in, we
apply an operator to the wavefunction which
should return the wavefunction multiplied by
an observable such as energy.
λ = 0.52 nm
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Schrödinger equation
Wavefunction of the H atom
Schrödinger equation –
When we saw the energy
solutions of a particle in a
3D box, we did them in
terms of Cartesian
coordinates (x, y, z). It
turns out the math is more
conveniently done in what
is called spherical polar
coordinates (r, θ, φ).
φ
' ψ=Eψ
H
The operator H-hat is much like a Google
search of the internet. It “pulls out” the
property information we are interested in
from the large collection of information that
is the wavefunction.
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Wavefunction of the H atom
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Wavefunction of the H atom
This allows us to break the
wavefunction down into two
parts:
radial wavefunction R(r)
angular wavefunction Y(θ, φ)
This results in
In spherical polar
coordinates, we
define the electrons
position in an atom by
its distance r from the
nucleus, as well as
two angles θ and φ to
some arbitrary line.
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ψ(r, θ, φ) = R(r) Y(θ, φ)
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Wavefunction of the H atom
Quantum numbers
We’ve already seen the principal quantum
number n (based on the angular momentum
of the electron).
The name is interesting, though. It implies
there must be other quantum numbers for
the electron to help describe other aspects
of the electron in the atom. It turns out
solutions for the Schrödinger equation for the
hydrogen atom require two more quantum
numbers! However, they are connected3
Solutions of the Schrödinger
equation for the hydrogen
atom define orbitals that
describe the motion of the
electron in the H atom (not
Bohr orbits). These orbitals
will have features that
depend on both the radial
wavefunction and the angular
wavefunction.
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Quantum numbers
107
Problems
Principal (angular momentum) quantum
number n where
n = 1, 2, 3, 3
Orbital angular momentum quantum
number l where
l = 0, 1, 2, 3, 3, n-1
Magnetic quantum number ml where
ml = -l, (-l+1), 3, -2, -1, 0, 1, 2, l-1, l
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a) Can an orbital have the quantum
numbers n = 3, l = 0 and ml = 0?
b) For an orbital with n = 3 and ml = 1,
what is (are) the possible value(s) of l?
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Problem answers
Physical interpretation of the quantum numbers
a) Yes. For n = 3 the values of l can be
either 0, 1 or 2. For ANY of these
values of l, ml can be 0, since ml must
fall in the range of –l to +l in all cases.
b) If n = 3 then the values of l can be 0, 1
or 2. But since ml must fall between –l
and +l, a ml value of 1 eliminates l = 0
as a possibility, but leaves l = 1 or l = 2
as allowed values.
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Shells and subshells
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Subshells
Since we see the quantum numbers
are tied together, it makes sense to
logically group them together in some
way.
Orbitals that have the same value of n
are said to belong to the same shell.
Orbitals that have the same n and l are
said to belong to the same subshell.
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The quantum numbers tell us something
about the orbitals the electrons move
within
n determines the energy and average
(most probable) distance from the nucleus
l determines the angular shape of the
orbital
ml determines the orientation (direction) of
the orbital
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We saw that l can take integer values
ranging from 0 to n-1, and therefore for a
given shell (given value of n) there can be
a variable number of subshells.
When n = 1 l can only be 0 (1 subshell)
When n = 2 l can be 0 or 1 (2 subshells)
When n = 3 l can be 0, 1 or 2 (3 subshell)
When n = 4 l can be 0, 1, 2 or 3 (4 subshell)
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Subshells
Total orbitals in a subshell
We very quickly see the number of allowed
subshells in a shell equals the principal
quantum number for the shell. Based on
this, we give orbitals special names to
describe the shape of the subshell
When l = 0 (s orbital)
For higher values
When l = 1 (p orbital)
of l the names
When l = 2 (d orbital)
continue as g, h,
i, j, k, When l = 3 (f orbital)
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Total orbitals in a subshell
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Orbital energies
There can only be 1 s orbital when l = 0
since ml can only be 0
There can be 3 p orbitals when l = 1
since ml = -1, 0 or 1
There can be 5 d orbitals when l = 2
since ml = -2, -1, 0, 1 or 2
There can be 7 f orbitals when l = 3 since
ml = -3, -2, -1, 0, 1, 2 or 3.
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Since ml values can range from –l
to +l, for each subshell, there will
exist 2l+1 orbitals of a given
name in a subshell. These
should be similar in shape (same
l) but different in their direction
(different ml)
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The energies of an electron in the orbitals in
each subshell for a hydrogen atom are given by
the energy of the shell (E depends on n only)!
En = -RH (1/n2)
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Orbital energies
If there is no “favorite” direction determined by
an external electric or magnetic field, this means
each orbital of a subshell has the same energy.
We call orbitals at the same energy degenerate.
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