Hi, I'm Toby and welcome back to another
video. Today I'd like to show you through
another document from the life of Albert
Einstein and today that is his PhD
thesis. Now the title of the thesis is in
German, but it translates to: "A New
Determination of Molecular Dimensions." So
Einstein was trying to find out a new
way to find the size of molecules. And he
also comes up with a value for Avagadro's
number. This thesis was published at
the University of Zurich in 1905 and
1905 is a very interesting year because
it's actually known as Einstein's miracle
year and it's the year that while he was
working at the Patent Office he
published four now famous papers that
changed the scope of physics. These
papers were on the photoelectric effect,
Brownian motion, special relativity and
one that introduces the equation E=mc²
but this thesis is really a fifth paper
published that year that not so many
people know about. So I thought we could
have a little look through it today and
see what it's all about. I also have some
other documents that are relevant
including a letter from Einstein to one
of his best friends talking about how
this is his second attempt at getting
his PhD and his first attempt at
submitting his thesis was unsuccessful.
And I have the grading remarks from the
two professors who assessed the thesis
and they agree that the thesis is very
impressive although having said that
there turned out to be quite a few
mistakes in Einstein's original thesis
and some of these mistakes took years to
correct.
So even Einstein's thesis is not perfect
but it is very impressive. So let's have
a look at it so here's the cover page.
Einstein's main supervisor was Professor
Alfred Kleiner from the University of
Zurich. Now that's not the same
University where Einstein did his
undergrad - that was ETH, and that's
because at the time ETH did not have the
ability to grant doctoral degrees so
they had an arrangement where students
could submit their theses to the
University of Zurich. Now I think you're
really going to smile when you
see what's on the next page of the
thesis and that is who Einstein has
chosen to dedicate this work to. This
thesis is dedicated to Einstein's good
friend Marcel Grossman. Now Grossman was
the classmate of Einsteins who would let
Einstein use his notes to study for
exams and that allowed Einstein to skip
many classes. There was also Grossman's
dad that enabled Einstein to get a job
after being desperately unemployed for a
while. So Einstein is really showing his
gratitude here with this dedication. If I
keep going, here's the introduction to
the thesis and you'll notice it's in
German so for the sake of my
english-speaking viewers and for the
sake of myself I have an English
translation over here that I'll also be
reading through. I want to remind you of
a few definitions before we get started
looking at the thesis. If this is a
beaker filled with water and I have
dissolved in it a bunch of sugar
molecules, then this water is the solvent,
what's dissolved in it is the solute and
the combination of these two things all
of this is the solution. Now with that in
mind let's read what Einstein was doing.
He says, "It will be shown in this paper
that the size of molecules of substances
dissolved in an undissociated dilute
solution can be determined from the
internal viscosity of the solution and
of the pure solvent, and from the
diffusion rate of the solute within the
solvent..." Viscosity is kind of like the
thickness of a fluid. Honey would have a
higher viscosity than water and it's
kind of the resistance of deformation of
that fluid. A dilute solution is just the
opposite of a very concentrated one,
it's kind of a weak solution there's not
much solute in there, and diffusion
refers to
the movement of particles from an area
of high concentration to an area of low
concentration like the way that spray of
perfume would spread throughout a room.
So when this work Einstein finds the
size of molecules by finding two
important equations. He spends most of
this thesis deriving a relation between
coefficients of viscosity for a solution
that does have dissolved particles in it
and for one that doesn't. And then he
also derives this diffusion rate
equation to explain how the particles
would diffuse within the solvent, and
through that he can work out how big
atoms should be based on how they move.
It says here that he will be modelling
the molecules of the solute using solid
spheres. Now there are a few main
sections of the thesis and I'll just go
through and give the main idea and main
result from each section. In this first
section we are seeing how a very small
sphere suspended in a liquid influences
its motion. So you can think about these
models as being like sugar dissolved in
water. So in this instance we just have
one molecule of sugar dissolved in our
water. It says that there are three
different ways that a liquid can move.
One is a parallel displacement of all
the particles of the liquid without a
change in their relative position so
they're all moving across. Two is a
rotation of the liquid without a change
in the relative positions.
Number three is a dilation of motion in
three mutually perpendicular directions
which would be called the principal axes
of dilation. Now we go on to see that a
single sphere suspended in a liquid
doesn't impact these first two motions
of the liquid but does impact this third
motion. So, much of the math that happens
next
involves looking at those axes of
dilation and applying hydrodynamics.
I'll show you the math
the original German version, it'll be a
little harder to understand but at least
we'll see, I guess, Einstein's original
copy. The fluid that we're working with
is considered to be incompressible. So
things like this might indicate that the
divergence of the flow velocity is zero.
I won't go through all of the maths that
we can look at some of the interesting
things. On his way to deriving 'P', which is
the hydrostatic pressure, there is an
equation in here which the note says
comes from Kirchhoff. And it comes from a
book by Kirchhoff that Einstein read as
an undergrad, so obviously he's kept that
with him and found it quite useful in
his thesis. Towards the end of this
section, Einstein starts to look at the
energy per unit time that is converted
to heat, also known as the work done on
the liquid. It's a dissipation of energy
and for our single sphere suspended in a
liquid we do work out a value for this
work, and it is given here. Section two
applies this work to many spheres. I'll
give you the English translation and it
is: "Calculation of the Coefficient of
Viscosity of a Liquid in Which Very Many
Irregularly Distributed Small Spheres
Are Suspended." So some of the math will
be the same for this section, just a
little bit more general. Given that we've
already worked out the diffusion of
energy from one sphere, if we have many
spheres together then this here will be
the amount of energy converted to heat
in a certain region. The symbol at the
end there denotes the fraction of the
total volume that is occupied by our
spheres. Einstein then goes on to derive
a second expression, another way to get
this work done on the fluid, and with two
expressions for the work he can equate
them and get a very important formula in
here which is regarding viscosity. This
is a relation between the coefficients
of viscosity for liquid with and
without suspended molecules
that's K* and K respectively.
Section three is:
"On the Volume of a Dissolved Substance
Whose Molecular Volume is Large Compared
to That of the Solvent." What he does is
try to use his new formula here and
apply it to some experimental results
regarding dissolving sugar in water,
although, he doesn't end up with a very
good calculation - his numbers aren't
really that precise. And he blames this
discrepancy on the fact that you can say,
"a sugar molecule in the solution
impedes the mobility of water in its
immediate vicinity." He's thinking that
water molecules are attaching to the
sugar molecules. It turns out that
there's actually a mistake in this
equation but we'll come back to it later
in the video. Section four is quite an
important one, it is: "On the Diffusion of
an Undissociated Substance in a Liquid
Solution." This is the section where he
derives his diffusion formula and it's a
really powerful result. To get there, he's
thinking about the velocity of a
molecule and to get this he's actually
used Stoke's law which gives the force of
viscosity on a small sphere moving
through a viscous fluid. In thinking
about diffusion, he also has to think a
lot about osmotic pressure. Osmotic
pressure is the pressure which would
need to be applied to a solution to
prevent the inward flow of pure solvent
across a semipermeable membrane.
I'll admit that it's a bit complicated
and there's parts of it that I don't
understand including a discussion of
fictitious forces regarding osmotic
pressure. But we can gloss over that and
go back to what we're really interested
in, which is this *juicy* equation here
for the diffusion of molecules. In here, 'R' is
the gas constant, 'T' the temperature, 'N' is
known as Avagadro's number, it's the number
of elementary particles that make up one
mole of a
substance, and 'P' is the radius of a
molecule, so that's out indicator of size.
These are our two unknowns that we were
wanting to find and now we've derived
two equations. This one for diffusion and
the previous one for the viscosity. So
with that we're able to put everything
together in section number five, which is
the determination of molecular
dimensions with the help of the obtained
relations. And we can go through here and
see that we're actually able to give a
value of the molecular size
and of Avagadro's number. Avogadro's number is
known today to be 6.02 × 10²³ per mole.
This value here is 2.1 × 10²³ so it is the correct order
of magnitude, which is mentioned here in
the note, but it's actually not that good
for a value. At the time they didn't know
Avogadro's number much more precisely
but in 1909 a physicist called John
Perrin did some experiments and got quite
a different value from what Einstein
quotes. Him and Einstein talked about
this and after Einstein suggested that
maybe he's made some errors in his
experiments Einstein asked one of his
students
Ludwig Hopf to check his calculations. He
found that back here in the relation
between the coefficients of viscosity
that this term here is actually missing
a coefficient of 2.5. Einstein submitted
a correction in 1911 and recalculated
his value for Avagadro's number to
be 6.56 × 10²³.
So that matched up a lot better. Perrin,
the guy who did the experiments, actually
went on to get the Nobel Prize in 1926
for his work on Brownian motion. His
experiments verified Einstein's
explanation of Brownian motion and
that's relevant to us because Einsteins
paper on Brownian motion was published
two weeks after
he published this thesis and a major
component of that paper was this here
the diffusion equation.
I'd like to show you that paper now
just so you can see it.
This was the second of those four more famous papers published in Einstein's miracle year in 1905.
Brownian motion is all about the movement of
small particles suspended in a
stationary liquid demanded by the
molecular kinetic theory of heat.
Or put another way, it's the fact that a
particle suspended in a fluid, say a
pollen grain on top of water would be
moving around in random motion because
of their collision with fast-moving
molecules in the water.
You see that a chapter of this paper is the "Theory of Diffusion of Small Spheres in Suspension"
very familiar to us now and
we have here after another derivation
the diffusion equation. Perrin's
experiments that verified this work on
Brownian motion were important enough to
get a Nobel Prize because what he was
really doing was verifying the atomic
nature of matter. At that time it wasn't
widely accepted that things were made of
atoms but this theory of Brownian motion
and the experiments that verified it did
give us a lot of insight into the nature
of matter. I'd now like to show some of
the comments from the people who
evaluated the thesis. This is the
supervisor Alfred kleiner and also a
mathematician. We can see down in the
comments here that they thought it was a
pretty good thesis. Kleiner says that, "The
arguments and calculations to be carried
out are among the more difficult ones in
hydrodynamics..." He says that, "Mr. Einstein
has proved that he is capable of working
successfully on scientific problems; and
I would therefore recommend that the
dissertation be accepted." Burkhart agrees
and says that, "What I checked I found to
be correct
without exception," although that's a
little awkward now knowing that there
were mistakes that made it through the
cracks. Now this wasn't the first
dissertation submitted by Einstein. He
apparently sent one in back in 1901
and it was on the kinetic theory
of gases, but I can find no trace of that
thesis, nor a clear reason as to why it
wasn't accepted. I read one source that
said that they were trying to avoid a
confrontation with Boltzmann and that's
why they decided to withdraw it. Maybe it
even had something to do with the
professors that Einstein no longer got
along with back at ETH. But anyway, this
is a letter to one of Einstein's best
friends, it's Michele Besso, and down
the bottom of it after talking about
random things in his life. Einstein says
that, "I will not go for a doctorate
because it would be of little help to me
and the whole comedy has become boring.
In the near future I'll concern myself
with the molecular forces and gases and
then I'll do comprehensive, extensive
studies in electron theory. So at this
point Einstein had quit his PhD which
is #relatable, but it would seem
that he eventually changed his mind. I
have another letter to the same person
here from later in the year and Einstein
is already talking about some of the
ideas that would eventually go on to
make up his thesis. He asked his friend,
"Have you already calculated the absolute
size of the ions under the assumption
that they are spheres and that they are
large enough to permit the application
of the equations of hydrodynamics?" That
there is essentially what he did in his
thesis. "Given our knowledge of the
absolute size of electrons this would
certainly be a simple matter." He then
says, "you could also make use of
diffusion to learn something about
neutral salt molecules in solution." And
that second proposal was the really
important one for Einstein. I'm not sure
what the rest of this letter is about
maybe it's relevant, maybe it's not, but
you can see that Einstein's been cooking
up plenty of scientific ideas and
talking to his friends about them. And
this is just two years before his
miracle year where he would make some
huge contributions to physics. So thank
you for watching till the end of the video.
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Thanks for watching and I'll see you next time.
