- So I just finished a
personal history of Niels Bohr,
and how and why he made his model,
where he skimped on the
details of the mathematics.
And after I published it, I was
asked to make another video,
where I showed you the details
of how he made this model.
And specifically, how did
he get to the restrictions
on the radius, the energy of the electron,
and the energy and the frequency
of the light produced by the electron.
So, you asked for it
Here you go.
Let's go!
♪ Electricity, electricity ♪
♪ Electricity, electricity ♪
Before I start, I'd like
to give a tiny backstory.
Now I often use the word quantize,
which is defined by the
dictionary as quote,
"to restrict a variable
quantity to discrete values
rather than to a
continuous set of values."
This whole idea of restricting
things, specifically energy,
to discrete values start in 1900,
when Max Planck restricted
the energy of light
to be in little energy packets,
where the energy equal the constant H,
now called Planck's constant,
times the frequency of light,
given by the Greek letter nu,
which, yes, looks like a V.
Anyway, 11 years later,
Ernest Rutherford had a theory
that the majority of an atom
is smushed into a tiny nucleus,
which is assumed to be positive
to counter the electrons
which are negative.
Rutherford didn't know what
this meant about the electrons.
Maybe they were evenly distributed.
Maybe they were in a big ring like Saturn.
Maybe there are many rings.
He didn't know, and frankly,
he didn't really care.
But his student Niels Bohr did care.
In fact, Niels Bohr was
attempting to mesh the idea
of quantized energy with the idea
of a nucleus and electrons,
when he was reminded by a colleague
that there was an equation
for the frequency of light
produced by burning hydrogen
called the Balmer formula.
And as he recalled years later, quote,
"As soon as I saw the Balmer Formula,
the whole thing was
immediately clear to me."
Even more impressively, Bohr also found
that he could use his new formula
to explain strange shadows from a star
as being from ionized helium.
Let me explain how we did this.
I'm going to split this
video into five parts.
One, tiny backstory, done!
Two, Bohr's assumptions.
Three, why Bohr made his assumptions.
Four, the math.
And five, his conclusions
for hydrogen and helium plus.
Bohr made six assumptions
in his July, 1913 paper.
One, in this first paper he only dealt
with systems with one electron.
Two, the electron spins around the nucleus
in a stable orbit,
which he assumed to be a circle.
Three, the electron can only
circle around the nucleus
at set position, although the
electron can magically jump
between one state to
another in a quantum leap,
and no Bohr did not use the word magic.
Four, the energy of
light produced is equal
to the change in energy of the electron,
as it jumps between states.
Five, all classical laws of physics apply
with three exemptions.
A, he knew that in classical physics,
objects just don't just
jump between position.
B, in classical physics, a
rotating charged particle
will radiate energy and
spiral into the nucleus,
but Bohr just basically said
nope, not for my electrons.
And C, in classical physics,
the energy of the light produced
is due to the energy of the electron.
In this theory, the light is created
due to the change in energy.
And six, the position of
the electron is quantized,
with the rule, the work
needed to remove the electron
equals an integer,
times Planck's constant,
times the frequency of the
electrons orbit, divided by two.
Part three, why did Bohr
make these assumptions?
Bohr started with a single
electron moving in a circle,
because he wanted to be
as simple as possible.
He also, remember, wanted
to derive Balmer's formula
from simple principles,
and Balmer's formula
is the frequency of light
produced by hydrogen,
and hydrogen only has one electron.
Note that Bohr did not
assume that the nucleus
only has one proton.
So his theories would work for hydrogen
and for helium plus, helium
missing one electron,
or lithium plus plus, lithium
missing two electrons,
et cetera, et cetera, et cetera.
Quantum mechanics, at this
time, consisted of using
the regular rules of classical physics,
but then adding a quantum
twist to some of it,
like the energy.
So it wasn't controversial
to make the energy quantized,
and therefore the position quantized.
What Bohr basically said was,
"Hey, if I can break one rule,
why can't I break a few more?"
So he also made a rotating
electron not produce light,
so that he didn't have
to deal with the electron
spiraling into the nucleus,
and he separated the energy of the light
from the energy of the electron,
so that his equations would work.
Stating that the energy of radiation
is unconnected to the
energy of the electron
was what really shocked many
people, including Einstein.
However, this is also what
has endured from Bohr's model,
we still think that
electrons radiate light
with an energy equal to the
change in energy of an electron.
Since Bohr disassociated, or separated,
the energy of the electron
from the energy of the light it produced,
he could put any limitation
he wanted to on the electron.
He then set the limit that the work needed
to remove an electron from an atom w,
to be n h omega over two,
where omega is the frequency
of rotation of the electron.
Which begs the question,
why divide by two?
Well, in the paper, Bohr made an argument
about averaging the work
from the state of no motion,
which leads to the two.
He also might have divided by two,
because then the equations
work out well for hydrogen.
It turns out that for a circular orbit,
this limitation is equivalent
to quantizing, the angular momentum,
so that m v r equals n h over two pi,
but this was his conclusion,
not his initial proposal,
As I will show you right now in the math.
Let's start with a single
electron of charge e,
and mass m moving in a circle
around the nucleus of charge Z e,
where Z is called the
atomic number and represents
how many protons you have,
at a radius r, and constant speed v.
The only significant force on
the electron is electrical,
so using Coulomb's Law, we
get that the electrical force
is Coulomb's constant k, times
the charge of the electron e,
times the charge of the nucleus Z e,
divided by the distance squared,
which can be simplified
to be Z times k e squared,
divided by r squared.
Now according to Newton's law,
force equals mass times acceleration,
and an object moving in
a circle, constant speed,
has an acceleration of v squared over r.
Let's call this equation one.
The work needed to remove
an electron from an atom,
which we will label W,
is the potential electrical energy
holding the electron in the atom,
which is Coulomb's constant,
times the two charges,
divided by the distance, minus
the electrons kinetic energy,
one half m v squared.
If you multiply equation one by r,
you get that Z k e squared
over r, equals m v squared,
or that the potential energy equals
twice the kinetic energy,
since one m v squared
minus a half m v squared,
equals a half m v squared,
W can be reduced to be
one half m v squared,
which is the same as
half the potential energy
or Z times k e squared over two r.
Let's call this equation two.
This is as far as you can get
without quantum restrictions.
As I said earlier, Bohr
restricted his atom
so that the work needed
to remove an electron
was quantized to equal n h omega over two,
where as I said before, n is an integer,
h is Planck's constant omega
is the frequency of rotation.
Plugging that into equation
two, and multiplying by two,
we get that n h omega equals m v squared.
Now we need an expression for omega,
the frequency of rotation.
If the electron is moving in
a circle at a constant rate,
then the velocity is distance over time,
where the distance is the circumference,
or two pi times the radius,
and the frequency is one over time.
Dividing by two pi r,
we get to the frequency
equals v over two pi r.
And plugging that into the equation above,
and dividing by V, and multiplying by r,
you get that the classic angular momentum,
m v r equals n h over two pi.
This is where most derivations start.
Let's call that equation three.
Now we're ready to get to the results.
First, we're going to solve
for the possible
positions of the electron,
i.e., the allowed radii.
Let us start with equation one,
multiply both sides by r squared,
then divide both sides by Z k e squared.
Finally multiply the
right side by m over m.
The numerator is the angular
momentum m v r squared,
which, using equation
three, can be replaced
with n h over two pi squared,
to get, r equals n squared h squared,
over four pi squared k Z e squared m.
Whew, I would like to
rearrange this equation
by taking n and z out, so we
get r equals n squared over z
times a naught, where a
naught equals eight squared
over four pi squared m e squared k,
a naught is a constant,
we can plug and chug,
and it turns out that a
naught is .53 angstroms.
Side note, I use the word naught
for that little zero next to the a,
because that's the way I was taught,
naught standing for the Old
English term for nothing,
as in he was naught but a worthless fool,
not like not like you're
not going to get any candy,
or not like I'm tying a knot.
We're now in a position to
figure out the work needed
to remove an electron from
a certain energy level n.
Plugging our new definition of the radius
into equation two, we get
W equals Z k e squared,
divided by two n squared a naught,
divided by Z, rearranging
so that the Zs and the Ns
are in front, gives us z
squared over n squared,
times k e squared over two a naught,
which equals z squared over
n squared times, ready?
Two pi squared k squared, e to the fourth,
times m divided by eight
squared, plug the numbers in,
Bohr got that the work is
z squared over n squared,
times 13 electron volts.
With the current values of
e, m, h, and k, however,
it is valued at 13.6 electron volts,
which is what I'm going to
use for the rest of the video.
This also means that for
hydrogen, where Z equals one,
the energy to remove an
electron from the lowest state
is 13.6 electron volts,
which means that the
lowest state has an energy
of negative 13.6 electron volts.
The next state is negative
13.6 divided by two squared,
which is negative 3.4 electron volts,
and the next is negative
13.6 divided by nine,
which is 1.51 electron volts, and so on.
More dramatically, Bohr predicted
that the light was produced
when the electron jumped
from one state to another,
which is equivalent to the
difference of work needed
to remove an electron from one shell, n b,
versus another shell n a, which
equals 13.6 electron volts,
times z squared, times
one over n b squared,
minus one over n a squared.
If, as Planck postulated, the
light has an energy of h nu,
where nu is a frequency of light,
then by dividing by h, you get
that the possible frequencies
of light would be 13.6 electron
volts, times Z squared,
divided by h, times one over n b squared,
minus one over n a squared.
Now astronomers tend to deal
with one over wavelength,
not frequency.
But luckily, the speed of a wave
equals frequency times the wavelength,
so we can replace the
frequency with c over lambda,
where c is the speed of light,
and the Greek letter
lambda is the wavelength.
If I divide by c, we
get one over wavelength
equals 13.6 electron volts
over h c, times z squared,
times one over n b squared
minus one over n a squared.
We can rewrite this as R times v squared,
times one over n b squared,
minus one over N A squared,
where R is two pi squared, k
squared, m e to the fourth,
divided by c h cubed.
Now for the hydrogen atom, Z equals one,
and this is exactly, exactly
the empirical equation,
or the equation from experiment,
without theoretical backing,
called the Balmer series,
for the frequency of light
from burning hydrogen,
where R is called the Rydberg constant.
But Bohr had another trick up his sleeve.
He started reading about other spectrum,
and found this very unusual spectra,
called the Pinkerton series,
named after the boss of the
woman who discovered it,
Wilhelmina Fleming, that discovered
that the star had an unusual pattern,
that looked like a Balmer
series with half integers.
In 1912, the year before
Bohr published his paper,
a scientist named Alfred Fowler
had reproduced these lines
in the laboratory ,with a
mixture of hydrogen and helium,
which he attributed to hydrogen,
or something he called
proto-hydrogen with half integers.
Bohr thought maybe his model
could solve the mystery.
He realized that the Z
squared in the equation
for the possible frequencies meant that
the resulting frequencies
from a helium plus atom,
or a helium atom with two
protons but only one electron,
would look exactly like a
hydrogen atom with half integers.
The reason he thought
this was helium plus,
and not regular helium, is remember,
Bohr assumed there was only one electron,
and regular helium has two electrons.
Anyway, this fact, solving the
mystery of the helium lines,
was what convinced many people
that Bohr had hit on something important.
Interestingly, Alfred Fowler,
the scientist who had made the experiments
with hydrogen and helium,
was not convinced,
and wrote that his new lines
were not exactly equivalent
to multiplying the
Rydberg constant by four,
they were equivalent to
multiplying it by 4.0016.
Picky, picky.
Anyway, this objection made Bohr realize
that he'd made a small
mistake in his calculations.
Even though the nucleus is
far heavier than the electron,
the charge of the electron
does cause the nucleus
to wibble wobble a little bit,
which changes the energy of the electron.
Bohr knew from planetary physics,
that you needed to deal with something
called the effective mass,
defined as m one times m two,
divided by m one plus m two.
Bohr then showed that with
his effective mass correction,
the new Rydberg constant
is multiplied by 4.00163.
Currently, we define Rydberg
constants the same way
as Bohr derived it to be.
Because of the order the
Bohr derived everything,
we often derive Rydberg's constant
for an infinitely big
nucleus with no wobble,
then adjust it with the effective mass
for a nucleus we're dealing with.
Whew, so there's the math
of how he derived his model.
Thanks for watching my video.
Big thank you to my
Patreons for supporting me.
If you want to be thanked as well,
there's a link down below,
and if you're interested in the history
of any of this stuff,
check out my other videos.
Okay, you stay safe out there, bye!
(gentle music)
