- 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 ♪
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)