Note: the following is NOT part of any A-level chemistry specification, but it is included because it interesting and wonderful stuff that should
help you master more chemistry content!!!
Matter Waves
The dual nature of electromagnetic energy as proposed by Planck and Einstein leads to a rather obvious link....if radiant energy (electromagnetic waves can behave as a wave or a stream of particles) then can matter under appropriate conditions show the properties of a wave? This idea was developed by the French scientist Louis De Broglie.
Consider the famous equation used by Einstein:
E=mc2
Here E= energy, m = mass and c= speed of light (3 x 108 m/s).
We can simply rearrange this formula to make the mass, m, the subject of the formula:
m=E/c2 equation 1
The equation provides a way of converting mass directly into energy.
The Planck equation links energy to the frequency of a photon.
E=hf
(I have used f to denote frequency instead of the usual symbol 饾湀 (mu from the Greek alphabet) for reason that
will become obvious shortly.
Now simply substitute hf for E in equation 1. This gives:
m=hf/c2 equation2
Also remember that:
c=位 x f
This rearranges to give
f=c/位
Now substitute for f into equation 2, this gives:
m=(hc/位)/c2 = h/位c
In this equation c represents the speed of light. However we can replace it by the speed of an electron, let's call this v.
So simply replace c for v in the above equation.
m=(hc/位)/c2 = h/位v
We now have:
m=h/位v or 位= h/mv This is often called the De Broglie equation
From De Broglie's equation we now have a way to calculate the wavelength of a so called matter wave that is
the wave characteristics of any object. The quantity mv is simply the momentum of the object in question.
So any object no matter its size or mass will exhibit wave characteristics which are dependent on its
mass and velocity.
However these effects are tiny when we are dealing with everyday objects e.g.
What is the wavelength of a golf ball, mass 100g travelling at 200 m/s?
位= h/mv = (6.626 x 10-34)/(0.1 x 200)=3.31 x 10-34
This answer is very very very small, smaller than the diameter of atoms!
However if we carry out a similar calculation on say an electron, mass 9.11 x 10-34 Kg travelling at 2 x 106 m/s
inside a hydrogen atom:
位= h/mv = (6.626 x 10-34)/(9.11 x 10-31) ) (2 x 106)
=3.63 x 10-10
This answer in terms of scale is much more reasonable when we are considering the size of atoms, it is similar to the
wavelength of x-rays, however it is larger than the size of a hydrogen atom?
The Uncertainty principle
On the atomic scale the world is a very different place to what we are used to seeing and dealing with. It
seems odd that light can be a wave one minute and a suddenly behave as a particle the next. However on
the atomic scale light and matter behave differently from how we perceive them or expect them to behave.
At best the wave/particle model is simply that, a model to explain the properties of light and matter on
the atomic scale.
There is another problem in making measurements of very small objects such as atoms or electrons. Werner
Heisenberg stated in his theorem in 1927 that it is impossible to know both the location and the velocity of an
electron with complete certainty.
The problem arises when trying to make an observations of the electron, e.g. if you were trying to measure
the position of an electron at any exact time then photons of energy would have to somehow interact with the
electron in order for you to make some kind of measurement on it. This would transfer energy to the
electron and increase its velocity, so the very act of trying to determine its position would cause this to change.
The uncertainty principle states that we cannot know the electrons position and velocity (or momentum) beyond
a certain level of precision. If we know one property accurately then the other will be known less accurately.
The principle is often stated as:
(螖x)(螖mv) = h/4蟺
螖 is the Greek symbol delta, which is often used to mean change in. So 螖x is the change in x, the electron's position.
Mass (m) x velocity (v) is simple the momentum of the electron. So 螖mv is the change in momentum, h is Planck's
constant, 6.626 x10-34Js-1. So the uncertainty principle says we cannot know with a precision greater
than h/4蟺 the product of the momentum and the position of the electron. If we know the electrons position
with a great deal of accuracy then we cannot know velocity with a great deal of certainty and vice versa.
This means that the electron will always appear "fuzzy" when we try to make any measurements of it, e.g.
if we know the velocity of the electron to be 1.3 x 106 m/s, then what is the uncertainty in
its position?
(螖x)(螖mv) = h/4蟺
Rearranging we get:
螤 = 3.14 mass of electron = 9.11 x 10-34 kg velocity = 1.3 x 106 m/s
螖x= h /4蟺(螖mv) = 6.626 x 10-34/4(3.14)(9.11 x 10-31)(1.3 x 106)
=4.45 x 10-11 m
Since the diameter of a hydrogen atom is 5 x 10-11m then the uncertainty in the position of the
electron is of the order the size of the hydrogen atom itself.
As an everyday example consider a speed camera at the roadside. If it records the velocity of the
car as 300 m/s and the car has a mass of 1500kg. If the uncertainty in the velocity of
the car is 2%, that is 6m/s what is the uncertainty in the position of the car?
螖x= (h )/4蟺(螖mv) = (6.626 x 10-34)/(4(3.14)(1500)(6))
=5.86 x 10-39 m
In our everyday world this is a totally inconceivable and irrelevant number, it is so small!
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