I am working my way through Linus Pauling’s General
Chemistry book slowly in a rather haphazard fashion. Sometimes I skim,
sometimes I read… I realize this post is incongruous with the rest of this
blog, but I was rather excited about this derivation and I felt like sharing.
Anyway, I am not an expert on this topic (far from it, I read this stuff before
bed if I’m lucky to read it at all, and am completely half asleep, or as much
as you can be completely half ) so if you have questions I suggest you look
elsewhere (eg: Wikipedia). My professor did go over this at one point, but he
went way above the level of a first year course, and it was invigorating as
well as intimidating. So I am going to break this down in a way that hopefully
will make sense to the general chemistry masses. :) And if not, well, oops. :P
Usually textbooks start off by telling you about the
discrete spectrum of the hydrogen atom, which is a distinctive pattern of lines
of certain frequencies emitted as one electron returns to a lower state
releasing a photon. I enjoy the analogy to an atomic fingerprint, because it
sounds cool. Then the textbook gets more complicated and the coloured line
drawings turn into a set of calculations explaining the properties of the
hydrogen atom.
When Bohr was investigating the hydrogen atom, he was the
link between the classical physics perspective and the new-fangled quantum
mechanics explanation. His explanation was not completely right, but according
to one source I came across, should not be belittled, as it was significant for
his time (such as the cassette player is to the mp3 file today or something). So don’t bully Bohr, not that we were
about to with all his brilliance! Anyway, down to business now.
Let’s recall a concept from classical physics. The hydrogen
atom has two charges, a positive nucleus, and a negative electron. These two
charges should follow the charge force of attraction equation (Coulomb’s Law)
given by:
Note: I expressed the qs by an E only because I was
previously more familiar with the q notation (and anticipate many high
school/first year university students will be as well), but I will continue to
use the E notation to signify charge for the rest of this post series.
BUT classical physics was found to have some problems in
regards to the hydrogen atom.
Sadly, all was not well and orderly with the universe and human
understanding. (It still isn’t, but we’re getting better supposedly, but that
is another discussion)
EM theory states that the electron should produce light.
This frequency of light should equal the frequency of rotation in the atom. And
this should produce a continuous spectrum of light, as the electron would
approach the nucleus more and more closely. The typical description of the motion of the electron in the
atom is to imagine it orbiting like the earth around the sun.
However, observation shows us the hydrogen atom does NOT do
this at all. In fact the hydrogen atom has
1) a discrete spectrum (that cool atomic
fingerprint)
2)
a volume with diameter 1 A (this A is supposed
to have a little circle on the top, I can’t figure out the notation)
This means that the hydrogen atom is not able to have a
range of wavelengths, as it only has a few. Also, the electron’s orbit does not
run towards the nucleus, it is actually pretty fixed. If classical physics
held, the electron would run into the nucleus so quickly that the atom would
fail to exist, and that would be pretty disastrous indeed you must admit.
So classical physics fails to explain these issues in a
satisfactory nature, and another explanation is necessary.
Bohr to the rescue! (Well, almost!)
Bohr had a couple of clues to aid him with developing his
model. First came Planck’s quantum theory of the emission of light and then
Einstein’s theory of the photoelectric effect and the light quantum. Hopefully,
the reader is familiar with these ideas, but perhaps I will do a post about
them later (not that there isn’t heaps of material about this out there
anyway). In brief, 
Bohr’s idea in compact terms was that the hydrogen atom can
exist only in certain discrete states, known as stationary states. He defined
the ground/normal state to be the state of minimum energy possible for the
atom. Any other state of higher energy is called the excited states of the
atom.
The frequency emitted (or absorbed) is given by:
where E" is the excited
state and E' is the lower
state of energy.
The big deal about Bohr is pretty much that he calculated
the energy of stationary states of the hydrogen atom with Planck’s constant –
quantizing the hydrogen atom, or more specifically, its angular momentum. Which
gets me in a pretty excited state just thinking about it!
He assumed the orbits of the electron are circular, and
defined the angular momentum of the electron as follows:
Angular momentum: n ħ n
= 1,2,3…
Where n=1 is the ground state
n is defined as the principal quantum number (this little
blighter will come back and bite you in the pants, trust me)
*** A quick spoiler alert as we will derive this equation
next post, in addition to a formula for velocity of the electron! ***
The atomic radius is defined as
for an atom with
atomic number Z, where 
m is the mass of
the electron, and E is its charge
It is also worthwhile to mention that the total energy is given by
but don’t stress too much about the first form of the
equation – it never showed up on a first year test. The important one to remember is probably the second form of the equation. :)
No comments:
Post a Comment