From: Ronald Wong (ronwong@inreach.com)
Date: Wed Mar 23 2005 - 17:44:01 PST
Message-Id: <l03102800be66db5fdb7c@[209.209.14.208]> Date: Wed, 23 Mar 2005 17:44:01 -0800 From: Ronald Wong <ronwong@inreach.com> Subject: RE: Speed of an electron
A while ago, Charlie Bissell From Hillsdale High School asked:
>How fast do electrons travel in their orbits? Do they maintain this
>speed forever? When they reach higher energy levels, is this reflected
>in a change in velocity? Are all these questions irrelevant in a quantum
>mechanical model of the electron?
A. To answer Charlie's first question:
The speed varies depending on the element and the orbital level of the
electron in question.
To simplify matters, consider the hydrogen atom where a single electron
orbits a single proton.
************ The Classical Approach ************
Let's apply the classical physics taught in our high schools to the
simplest case - a circular orbit:
The attractive, electrical force between the electron and the proton is
responsible for the electron's circular motion about the proton. As a
result,
The electrical force = the centripetal force
K(q)(Q)/r^2 = -m(v^2)/r [q = -e = charge of electron,
Q = +e = charge of proton, right hand expression
is negative because the force is centripetal]
-K(e)(e)/r = -m(v^2)
v = e X SqRt(K/(mr))
Since e is about 1.6 X 10^-19 C
and K is about 9 X 10^9 Nm^2/C^2
and m is about 9 X 10^-31 kg
and r is about 0.5 X 10^-10 m (for the smallest orbit),
It turns out that v = around 2.3 X 10^6 m/s.
This is good enough at the high school level - if you are only interested
in it's speed for the smallest circular orbit.
But you should keep in mind that Classical physics FAILS to explain why the
electron should continuously orbit the proton for any length of time at
all.
According to classical physics, any acceleration of a charged body will
produce an electromagnetic wave.
That's how radio waves are produced. The electrons in the
radio antenna are oscillating back and forth in the radio
antenna.
The electron of the hydrogen atom is constantly accelerating as it moves
about the proton.
It's continuously falling towards the proton just like
an orbiting satellite is continuously falling towards
the earth.
Classical physics says that the electromagnetic wave produced by this
acceleration should be carrying energy away from the atom. This continuous
loss of energy should cause the electron to spiral into the proton and
collide with it - something that just doesn't happen.
************ The Quantum Mechanical Approach ************
Enter Quantum Physics and the work of Niels Bohr.
Bohr came up with two postulates.
The first postulate has to do with angular momentum - a topic barely
touched upon in most high school physics classes.
Just like objects moving along a straight line have
linear momentum characterized by the expression, mv
(mass times velocity), objects moving around in a
circle have angular momentum that is characterized by
the expression, mvr, where "r" is the radius.
(for the nitpickers, it's (m)(R X V) where V and
R are vector quantities that form a cross product here)
The first postulate says that the angular momentum must be an integral
multiple of a "magic" number.
That is, mvr = nh/(2Pi) where "n" is 1,2,3,... and "h" is Planck's constant.
This means that the electron can only travel around the hydrogen's nucleus
in prescribed orbits defined by the above equation - the smallest when n =
1.
Since the orbits are now "quantized", any transition from one to another
leads to discrete changes in energy. This leads to the second postulate.
Any change from one orbit to another leads to a discrete amount of radiant
energy being given off or absorbed.
That is, the change in energy = hf where "f" is the frequency of the
radiant energy emitted (or absorbed).
Just like Newton who, in the 18th century, was able to derive Kepler's
empirical laws of planetary motion by postulating that the motion of the
planets were a function of an inverse square law and who subsequently went
on to make predictions based on this idea that proved to be true, Bohr used
his two postulates to derive a famous, empirically arrived at formula
regarding the emission spectra of hydrogen by Balmer (almost 30 years
earlier) and he then went on to make a number of predictions based on these
postulates that also proved to be true (a Newton for the 20th century so to
speak).
Using the first postulate, we can discover the speed of the electron in
it's lowest orbit where n = 1.
v = h/(2mrPi) = (6.6 X 10^-34)/(2 X 9.1 X 10^-31 X 0.5 X 10^-10 X Pi)
v = around 2.3 X 10^6 m/s (sure looks familiar - but wait...)
B. To answer Charlie's second question,"Do they maintain this
speed forever?", you need quantum physics:
Due to the quantized nature of things energy is only emitted or absorbed
when the electrons move from one prescribed orbit to another. As long as
the electron remains in a prescribed orbit it will travel at the speed
stipulated by the first postulate and - most importantly - it will not
radiate any energy away from the atom as required by classical physics.
If the electron is in it's ground state (n = 1) and it is never subjected
to any radiant energy that will allow it to move to one of it's higher,
prescribed states then it will move at the speed calculated above forever.
C. To answer his third question, "When they reach higher energy levels,
is this reflected in a change in velocity?":
We can "quantize" the first expression, K(q)(Q)/r^2 = -m(v^2)/r, by solving
for "v" in Bohr's first postulate and substituting into the first
expression to solve for "r". You'll get:
r = n^2 times another set of constants.
Setting "n" to 1 allows you to determine the radius of the lowest orbit by
putting in all the constants and solving for "r" (that's what I did to get
the 0.5 X 10^-10 used in my calculations).
So r = n^2 (R) where R is the radius of the lowest orbit (so the radii are
quantized, increasing with the square of the quantum number).
Postulate one now looks like this: mvn^2(R) = nh/(2Pi)
Solving for "v" again, we get: v = 1/n X h/(2mRPi) = 1/n X V, where V is
the speed at it's lowest orbit (2.3 X 10^6 m/s).
You can see from the very last expression that the speed is proportional to
the inverse of the quantum number. It too is quantized. At n = 2 it's
around 1.1 X 10^6 m/s and so on...
D. "Are all these questions irrelevant in a quantum mechanical model
of the electron?"
Not if you want to know the speed of an electron orbiting about the nucleus
of a hydrogen atom.
Things do get more complicated (even for the hydrogen atom) when you move
beyond the level discussed here.
It IS irrelevant if you decide to think of the orbiting electron in terms
of it's wave function where it no longer has mass - in the traditional
sense - and is nothing more than a probability distribution within
prescribed regions around the nucleus. But that's a different kettle of
fish.
Cheers.
ron
This archive was generated by hypermail 2.1.3 : Mon Aug 01 2005 - 16:06:47 PDT