Intermolecular Forces (Lec - 12) - Introduction to Solid State Chemistry Video Lecture - Chemical Engineering

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Let's go to the lesson.
So the last day we studied
VSEPR, which allowed us to
infer the shapes of molecules,
covalent molecules.
Today I want to talk about
the state of aggregation.
What do I mean by state
of aggregation?
Is something a solid, a liquid
or a gas at a particular
temperature?
So 3.091 is introduction to
solid state chemistry.
One of the things we need to
know is, under what conditions
is the solid state stable?
And the state of aggregation,
when it applies to covalent
molecules, is going to lead
us to a discussion
of secondary bonding.
So today's lecture is
state of aggregation
or secondary bonding.
Now let's just reflect
upon what we've
learned up until now.
We studied ironic bonding and
we knew that ionic bonding
necessarily leads to crystal
formation because we have
unsaturated bonds.
And that leads to an ion array
of unlimited size until you
run out of reagent.
And something honking big made
of ions is going to be a solid
at room temperature.
Last day we appreciated
with covalent
bonding we have two options.
We can either make discrete
molecules such as HCl.
And HCl as a discrete molecule,
depending on a
number of factors that I'm going
to show you today, could
be a solid, liquid or gas.
And we're going to understand
more deeply how to sort out
between the three of them.
Or, I showed you at the end of
the last lecture, you can make
a three-dimensional network and
diamond was one example.
If it's a three-dimensional
network that is a large array
of solid then that means that
you're going to end up with a
formation of a crystal, which is
going to give you a solid.
Diamond is solid at room
temperature, graphite is solid
at room temperature.
So let's now go to the one case
that we haven't dealt
with, and that's the formation
of discrete molecules.
So let's look at discrete
molecules.
And what we want to understand
is whether they're going to be
solid, liquid or gas at room
temperature and what's the
relevant physics here in order
to make that determination?
The relevant physics
is the following.
We're going to compare
two forces.
We're going to compare the
cohesive force between
molecules versus the
disruptive force.
And the disruptive
force in 3.091 is
always going to be thermal.
It's thermal energy.
We know this.
As we heat things they go from
solid to liquid to vapor as
temperature increases.
So thermal energy plays the role
of the destructive force
whereas the bonding is
something that is
the cohesive force.
So let's go back to HCL.
Last day we looked at HCl.
So here's one HCl molecule.
We have a covalent bond.
It's a covalent bond within
the molecule.
We know this is a polar covalent
molecule, with the
chlorine having greater
electronegativity and pulling
the electrons towards its end.
And furthermore, this bond I
want to categorize as within
the molecule so I'm going to
call it intramolecular.
Intramolecular.
Now if I want to answer the
question, is hydrogen chloride
going to be a solid, liquid or
gas at room temperature, I
don't have the information
on the board.
The only way I can answer that
question is to put another
hydrogen chloride.
This simply forms the bond
between hydrogen and chlorine.
To determine whether hydrogen
chloride is a solid, liquid or
gas, I need to look at how one
hydrogen chloride molecule
bonds to another hydrogen
chloride molecule.
So now this end is the delta
plus this end is delta minus.
And we have again an
intramolecular bond.
So the question is, how does one
hydrogen chloride bond to
another hydrogen chloride?
This is the intermolecular
bond.
Intermolecular bond.
So intermolecular bonds govern
the state of aggregation.
And I'm going to show you
the different types of
intermolecular bonds.
This intermolecular bond
isn't the primary bond.
The primary bond is here
inside the molecule.
So the intermolecular bond is
known as a secondary bond.
And there three types
of secondary bonds.
We're going to look
at them in turn.
So the first one is depicted
here on the board.
And it's called dipole-dipole
interactions.
And these obviously occur only
between polar molecules.
Operative and polar molecules.
Not in.
Between.
Take out in.
Operative between
polar molecules.
So we've made something covalent
and now how does one
stick to the other?
Alright and I think I've got a
cartoon here that shows this.
There we go.
That's taken from an old
book that I had.
So you see the H and the Cl and
we have a dipole moment
and the negative end of one
dipole attracted to the
positive end of another
dipole.
So we have a net dipole moment
here and we can measure the
distance between the center
of the dipole--
center to center spacing
and call it r.
Center to center
dipole spacing.
And say that the energy in
dipole-dipole interaction--
I don't expect you to know this
by heart and do anything
with it, but I just want
you to realize
that it's a weak force.
This is very weak.
It's proportional
to the magnitude
of the dipole moment.
You'd expect that.
Weak dipole moment,
weak energy.
Strong dipole moment,
strong energy.
Turns out it goes as
the fourth order
of the dipole moment.
And it's inversely proportional
to the
separation.
Only it's not coulombic, it
goes as 1 over r cubed.
And there's a temperature
factor.
1 over t.
As temperature goes up,
this energy goes down.
And these are very weak forces
on the order of about 5
kilojoules per mole.
You remember what the
crystallization energy of
sodium chloride was?
787 kilojoules per mole.
So this is very, very weak.
It's operative at, between,
at low temperatures.
At very low temperatures.
So for example, hydrogen
chloride, in the case of
hydrogen chloride, the melting
point of hydrogen chloride is
minus 115 degrees C and
the boiling point is
minus 85 degrees C.
You can see at very low
temperatures we already have
enough thermal energy
to disrupt.
So if we have solid hydrogen
chloride, the forces between
the hydrogen chloride molecules
in the solid are
these very weak dipole-dipole
interactions.
And I think there's a couple
more cartoons here.
This is taken from your text.
So there we go.
There it is.
That's the soup that might
be HCl liquid.
OK so that's the first type
of secondary bond.
Dipole-dipole interaction.
Let's look at the second one.
The second one is called induced
dipole-induced dipole.
And it's operative in
non-polar species.
Dominant in non-polar species.
Why am I using the
word, species?
I'm trying to be a pedant
and use fancy words?
No.
Because I'm going to
make it generic.
I'm going to show it works in
atoms and in compounds.
So as examples, what are
some non-polar species?
Well how about something
like argon?
Argon if you look on the
periodic table it'll show you
that it has a melting point of
84 kelvin and a boiling point
of 87 kelvin.
So if I cool argon to below 84
kelvin, I get argon ice.
So what are the bonds between
one argon atom and another
argon atom?
It can't be this.
There's no net charge.
It can't be ionic.
It doesn't form a covalent
network the way diamond does.
How do you justify the existence
of solid argon?
It's just so cold that it just
sits there and freezes?
I mean, how does it bond?
There needs to be some
kind of bonding.
And other non-polar species.
So for example, the
molecule iodine.
Iodine melting point is above
room temperature.
It's a solid crystal at
room temperature.
Well there's a strong covalent
bond inside iodine.
But how does one iodine bond to
another and to another and
to another?
And we can even go
to polyatomic
species, such as methane.
You know there was data from
the Cassini Probe.
Look at this.
This is an image from
the Cassini Probe.
This is an island group.
The yellow is an island group.
The blue is a methane sea.
A sea of liquid methane on the
moon of Saturn, Titan.
So there's liquid methane.
How does liquid methane form?
What causes one methane to bond
with another methane?
That's the question that
we're wrestling with.
So let's look at it first
in a simpler context.
Let's look at it with argon.
So I put argon here.
It's a spherical atom.
And the question we had before
says, how does one argon bond
to another argon, as has to be
the case in the solid or
liquid argon.
So we know that this has a lot
of electrons and the atom is
net neutral.
But the electrons
are in motion.
If I had an attosecond camera
and I went in here and I went,
snap, I could catch a freeze
frame where the electrons
aren't symmetrically
distributed around the nucleus.
This nucleus, the atom is
in constant fluctuation.
But net neutral.
Time average it's symmetrical.
So at some moment I might have
a preponderance of electrons
over here at around
three o'clock.
So this end of the argon is
a little bit delta minus.
Which means the other
end is delta plus.
Now what happens if this
is delta minus,
this is delta plus?
Can you see that the positive
end of this argon atom will
then pull on the electron
distribution of the adjacent
argon atom rendering it delta
minus at three o'clock and
delta plus?
So this is the dipole in one,
induces a dipole in another.
And why does it occur
in the first place?
It occurs because the electrons
are in motion.
Electrons in motion lead to
time fluctuating dipoles.
They're time fluctuating.
Time average, there's no
net dipole moment.
I'm not going back on
what I said before.
There's no net dipole moment
here, but this is time
fluctuating dipole.
Time fluctuating dipole.
So who was the first
to explain this?
The first to explain this was
Fritz London in 1930.
In 1930 Fritz London gave us the
explanation because this
troubled people for
a long time.
And he did so in a quantitative
manner.
And the mathematics of the
treatment are identical to the
mathematics for the dispersion
of light under certain
conditions.
This has nothing to do with
the dispersion of light.
The mathematical formulation
imitates the formulation of
the dispersion of light and
hence the force here is known
as the London Dispersion
Force.
So people will say that solid
argon is held together by
London Dispersion Forces.
Or the bond is known by the
name van der Waals.
We either call them van der
Waals bonds or London
Dispersion Forces.
Van der Waals was Dutch.
I know how to spell
the word wall.
This is the dutch spelling.
W A A L S.
Van der Waals.
So in this system here, the
London Dispersion Force or the
van der Waals bond is in fact
not the secondary bond.
It's the primary
bond, isn't it?
It's the only bond.
So by definition it must
be the primary bond.
But can you see that in every
compound, including diamond,
we have time fluctuating dipoles
in all of the atoms?
It's not just argon that has
time fluctuating dipoles.
Every atom in you and me has
time fluctuating dipoles.
But we're held together by
much greater forces.
So that's why I say that
in this case this is
the dominant force.
But it's operative
in everything.
Because wherever we have
electrons, they're in motion.
So this is the dominant one.
Alright so we can look at it
in a few other instances.
So for example we can look at
it in the case of iodine.
We can look at it in the
case of methane.
All the same.
Time fluctuating dipoles.
And this is a very,
very weak force.
So the energy in London
Dispersion Forces, or van der
Waals bonds, is proportional
to a quantity called the
polarizability.
The polarizability is--
and this was defined
by London--
polarizability is the measure
of how easy it is to induce
this dipole.
A measure of the ease
of electron
displacement within an atom.
And it depends on, it's
influenced by the size and
it's influenced by the number
of electrons in the atom.
So what do I mean by that?
Well let's take two systems.
Suppose I've got argon and
I'll go up two members
in the same series
and I've got helium.
So what are the forces that
hold helium together?
Same thing.
But the boiling point
here is 87 kelvin.
The boiling point here
is 4.2 kelvin.
Why does this have such
a low boiling point?
Well, polarizability.
Size--
helium is smaller.
So the degree, the corral in
which the electrons can roam
is smaller.
So the extent of electron
displacement is smaller.
And secondly there are
only two electrons.
And they're in 1s, so they're
tightly held.
And so the delta minus delta
plus capable of being created
in helium is tiny compared to
the delta minus delta plus
that can be created in argon.
So the energy goes as
polarizability squared and
divided by r to the sixth, where
this is the separation.
Not the radius but this
is separation.
Dipole separation.
And this is a font issue.
This is proportional to alpha.
So you can tell the
difference.
This is a proportional
that's to alpha.
Of course you can't tell
the difference.
This is contextual.
If I just wrote this by
itself, you don't
know what it is.
My goodness you're nervous.
So nervous.
All right let's take
a look at--
here's the cartoons.
Induced dipole.
And here's London's paper.
Here's London's paper as it
first appeared in 1930.
Theory and System of
Molecular Forces.
And here's some cartoons
from your book.
This is helium, showing
helium.
It's kind of funny how
the artist shows.
Like there's these two helium
atoms and they're absolutely
immobile and all of a sudden one
of the helium atom starts
jiggling and then it induces
a dipole moment
in the other one.
That's not how it happens,
but anyway.
Alright this is hydrogen.
Same thing as iodine.
All right so there's delta
plus, delta minus.
Because the electrons are moving
even though there's a
strong covalent bond inside.
So let me just make the point, I
want to make sure people are
very clear about primary versus
secondary bonding.
So if I look at iodine.
I2.
This is a covalent bond.
It's homonuclear.
This covalent bond, there's
no net dipole moment.
Here's another iodine.
But then we have time
fluctuating dipoles.
So this delta minus isn't
because this is the bond here.
This is the primary bond.
This is the primary bond
and it's covalent.
And this is the secondary
bond.
And the secondary bond is London
Dispersion Force or van
der Waals bond because this
is induced dipole.
So now let's look at-- oh
here's another one.
This one gets on my nerves.
Oh actually this is good.
This is this the bad one.
All right so I'm going to
show you polarizability.
So here's a series of sp3
hybridized chains.
Propane, octane, icosane.
They're all the same.
They look like this.
Ch is sp3 hybridized.
And so you just have--
so what is that, C 3 H H.
So it's just--
so you have 1, 2, 3 4. so
there's hydrogen, hydrogen,
hydrogen, and on the fourth one
I'm going to make it flat
instead of trying to
make a tetrahedron.
So 1, 2, 3 hydrogens.
The fourth one is carbon.
1, 2 hydrogens, carbon carbon.
1, 2 3 hydrogens.
So this is C3H8.
And compare that to
the longer one.
Which is octane,
which is C8H18.
So it's 1, 2, 3,
4, 5, 6, 7, 8.
And all you have got to do is,
1, 2, 3, 4, 5, 6, 7, 8.
You put four sticks off
of every carbon.
You can't go wrong.
You count them up.
You got C8H18.
Look at the structures.
They are the same.
So how come propane is a gas
at room temperature?
Whereas octane, which is the
principal constituent of
gasoline, is a liquid
at room temperature?
It all comes down to
polarizability.
This one's got a
longer corral.
So if you like the delta minus
versus the delta plus, it's
basically the same here except
that separation is bigger.
All right, so that's
pretty good.
This is the one that
gets on my nerves.
You see this?
It's exactly what I
just showed you.
So here's methane.
And there's the propane.
Here's butane and somewhere
between butane and pentane we
cross the the line at
room temperature.
So pentane is liquid,
butane is gas.
And you keep going up, up, up,
and finally if you get to C20
you go from liquid to solid
because now the van der Waals
forces are strong enough that
even at room temperature you
make a solid.
It looks like paraffin.
That's why you can melt
paraffin, because it's got
weak van der Waals forces.
So temperature disrupts and it
reforms. If you try to break a
covalent bond, you pyrolyze
the thing and you
don't get it back.
Secondary bonds allow for
ready processing.
Here's what bothers
me about this.
Instead of talking about
polarizability, which is a
physical quantity that
means something.
They say molecular weight.
Well it's true that these
scales, these are heavier and
it's monotonic.
But to me what's the relevant
physics between the ability to
form van der Waals bonds
and the mass.
It's just a dumb thing.
It's not a gravitational
effect.
So that's why this
thing is stupid.
And you see it all over
in chemistry books.
And it's just dumb.
And if you put that on my
exam you're not going to
get points for it.
Because that's dumb.
OK let's go to the next one.
There's a third type
of bonding.
There are certain things I feel
strongly about and that's
one of them.
OK so let's look at the third
type of secondary bonding.
The third type of secondary
bonding is
called hydrogen bonding.
Hydrogen bonding.
It's a type of secondary
bonding.
And it occurs between hydrogen
and the most electronegative
elements, fluorine,
oxygen, nitrogen.
Why these?
Because they have very,
very high average
valance electron energy.
So the average valance electron
energy I'm quoting in
that second column, not
megajoules per mole, but in
sensible units like
electron volts.
And so you can see, when you get
up around 18, 19 electron
volts you cross a threshold.
And that's the electronegativity
as
represented by average valance
electron energy.
So you can see that as the
electronegativity gets beyond
some threshold value, roughly
3, then you can
form hydrogen bonds.
So this is owing to high average
valance electron
energy, or if you like
electronegativity.
Which means very strong polarity
in the covalent bond.
So you say, well there's polar
and there's even more polar.
So let's see what happens.
So I'm going to use a
prototypical value here.
I'm going to look at HF.
So if I look at HF let's go
through the Lewis structure.
There's H with it's
1 electron.
And F with the 7.
1, 2, 3, 4, 5, 6, 7.
And so we have a covalent
bond here.
We know fluorine is the most
electronegative so we have a
dipole moment here.
Now the dipole moment
is very strong here.
This is an accounting procedure
to put the two
electrons, but it no way
represents the physical
position of these electrons.
They're brought in very
close to the fluorine.
So it's not some symmetrically
disposed between H and the F.
So I can't tell anything
about whether HF is a
solid, liquid or gas.
Why?
Because there's only
one sitting here.
I have got to put at
least one more.
Because this is a
primary bond.
And it tells me how
H bonds to F.
It doesn't tell me how one
HF bonds to another HF.
So I'm going to put another
HF over here.
So here's another HF.
And it's also dipole.
But there's something
special about that.
Already there's a dipole-dipole
interaction.
But the hydrogen bond
is much stronger.
It's much stronger.
And why is it stronger?
The electron in this hydrogen on
the right is pulled towards
the fluorine to such an extent
that this hydrogen is so
denuded of its electrons that
it's acting as proton-like.
It's proton-like.
Now don't tell people that
Professor Sadoway said that
hydrogen inside an
HF is a proton.
It's not a proton but it's
starting to get more nearly
like a proton.
Now what do we know
about a proton?
Positive charge.
Tiny high-charge density.
So this hydrogen's looking
forlornly over at its electron
that's being hogged by the
fluorine to a right.
And what do we know
about these?
Oh it's time for
colored chalk.
It's time for colored chalk!
What color are those?
They're red because they're
non-bonding.
And the bonding are the
blue in-between.
And what do we know about
the volume occupied by a
non-bonding pair versus
a bonding pair?
It's larger.
Because they're not
constrained.
So not only do we have
a non-bonding pair,
they're not just here.
They're sort of flopping
around.
Hanging way out and there's this
thing here that's almost
denuded of its electrons, so
it starts looking over here
saying, if I can't get
any action over
here, what about here?
So that proton starts
establishing contact with the
non-bonding pair of electrons
on the adjacent fluorine And
this is the hydrogen bond.
The hydrogen bond is here.
The hydrogen bond is not here.
If you write this I
will give you a 0.
I'll give you a 0 with
a circle around it.
It's called the doughnut.
That's what you get when you
write something so stupid.
This is not the hydrogen bond.
This is the hydrogen bond.
OK?
So it works.
It works.
Now let's see the
effect of it.
Let's see the effect.
Alright so here's a cartoon
showing that in water the
hydrogen-oxygen spacing
is about 1 angstrom.
And the hydrogen-oxygen spacing
in adjacent water
molecules can be less than
2 angstroms. So there's
certainly a difference.
I mean it can't be the same.
If it were the same it would
be a covalent primary bond.
You'd have a network.
Now this is interesting here
because this shows the values
that are given on your periodic
table, which were
obtained without the
use the average
valance electron energies.
These trends are correct.
But look at this one.
They've got chlorine
up at 3.16.
And all of these have
been revised.
Now on a test, just use what
you've been given on the
periodic table.
But I want you to understand how
this is rationalized with
better data coming from
photoelectron spectroscopy.
So now I want to show you the
implications of hydrogen bonding.
The implications.
So I've got 4 homologous
series.
So they're all element
plus hydrogen.
So let's start with
the group 14.
That's shown here in purple.
And what do we have?
All of these, we'll start
with methane.
CH4, so that's a central atom.
1, 2, 3 4.
They're all tetrahedral.
Hydrogens at the corners.
Non-polar.
And no hydrogen bonding
capability.
So one methane bonds to another
methane by weak van
der Waals forces.
That's how it does it.
It doesn't matter if I
substitute the carbon with
silicon or germanium or tin.
I can put SnH4.
And how does SnH4 bond
to another SnH4?
It's just by London
Dispersion Forces.
That's all that's
operative here.
And so we have, what makes
sense here, is that the
heavier, more massive--
no, the ones that have greater
polarizability have a higher
boiling point.
Because their van der Waals
forces are stronger.
That means you have to go to a
higher temperature to achieve
disruption.
Same temperature, weak van
der Waals force: gas.
Same temperature, strong
van der Waals
force: liquid or solid.
And you see the boiling
point here.
Monotonic from the lightest to
the heaviest. Now let's go to
the next one.
Let's go to the green
line, group 15.
Well group 15, what's
that look like?
Let's look at the structure
of group 15.
Group 15 is--
ammonia is one of them.
So we can look at the structure
of ammonia.
And if we use VSEPR we'll
end up with something
that looks like this.
I'll go through the
whole thing.
You're going to end up with
three bonds like this.
It's a tetrahedral skeleton
with a lone pair.
Ah, colored chalk.
So now, what happens?
I can't say anything
about this.
Why can't I say anything about
whether this is a solid,
liquid or a gas?
It's the only one there.
I have got to put another one.
So put another one up here.
N.
H.
H.
H.
Same gambit with the
hydrogen fluoride.
This hydrogen sees this lone
pair and establishes a
hydrogen bond.
And that adds to what otherwise
would have been a
dipole-dipole.
This has a net dipole along
moment, agreed?
There's a dipole-dipole
moment here.
But this bond is
even stronger.
The hydrogen bond is even
stronger than dipole-dipole
interactions.
So now let's look at
the graph up here.
So what about in
phosphene, PH3?
No phosphorous isn't
electronegative enough.
So in phosphene, in arsine and
in stibine we don't have
hydrogen bonding.
So you see this series
in the group 4 here?
It goes monotonic from the
heaviest element down to the
lightest element.
But here, heaviest, less heavy,
less heavy, and the
lightest element that should
be down here is up here.
Why is the lightest compound
not down here?
Because of the addition
of hydrogen bond.
So ammonia is off the line.
This line shows the trend
based on dipole-dipole
interactions only.
And you can see the
difference.
See if you have van der Waals
forces alone, versus
dipole-dipole, dipole-dipole
are stronger.
So SnH4 has a lower boiling
point than SbH3.
I can't predict this but I could
ask you, if I gave you
this data, I'd say, can you
explain this to me?
Let's keep going.
Let's go to group 16.
So, telluride, selenide,
suphide,
the oxide, H2O, water.
It should have a boiling point
of minus 100 centigrade were
it not for hydrogen bonding.
How does water work?
Again, SP3 hybridization.
Oxygen, 1, 2, 3, 4.
Two lone pairs.
1, 2, H, H.
And now what happens?
I bring another water
molecule over here
and hydrogen bonding.
And that hydrogen bond raises
the temperature, raises the
requirement for thermal
disruption and moves the
boiling point of water up
to 100 degrees celsius.
It it weren't for this
we wouldn't have this
conversation.
Because we wouldn't have evolved
as a species capable
of conducting business at room
temperature if water boiled at
minus 100 celsius.
Hydrogen bonding is critical.
It's absolutely critical.
So now you know.
And this isn't just
some little bit of
pedantry for a professor.
This is very important.
Because we're going to learn
later that, when it comes to
biochemistry, we will
appreciate that most
biochemicals are made
of carbon, oxygen,
nitrogen and hydrogen.
And that means you can have
hydrogen bonds here.
And you can have hydrogen
bonds here.
Hydrogen bonding is
critical to life.
So this is a very important
thing to know about.
OK we have a minute or two.
So you can see polarizability
increases here but hydrogen
bonding operative here.
That's explains the mystery.
All right we're going to
jump over this because
we are out of time.
And so I will simply show
you a few pictures.
Pictures!
So this is a conference in
Copenhagen in June of 1936.
And there's Fritz London.
But look at who else is
at the conference.
Niels Bohr, Wolfgang Pauli,
Werner Heisenberg, Max Born.
Remember the Born Exponent?
You better remember
it for Wednesday.
This is Lise Meitner, you
haven't met her yet.
We'll get to her.
This is Walter Stern.
Stern-Gerlach.
And this is James Franck.
That's just the first
two rows!
That's quite a conference.
Here's a picture of Fritz London
sitting on a bench in
Berlin with Erwin Schrodinger.
Schrodinger is brooding;
he's thinking.
Fritz London is smiling.
You know why?
Because he's figured it out.
He's figured out.
It's time-fluctuating dipole,
but he's not going to tell
Schrodinger.
He's going to say, you have
to read about it.
But you have to go
to the library.
Because if you don't read the
primary sources, if you go to
Wikipedia, you won't
find this.
Because this is not going to
be in Wikipedia in 1913.
OK, last thing-- hold on, hold
on, hey wait a minute!
Where are you going?
We're not done yet.
Did the professor say
class is dismissed?
No.
So here's a biography of Fritz
London, which I would
recommend if you have
a few minutes and
you'd like to unwind.
Go out, sit in the sun and
read something like this.
It tells the story of how he
came up with these ideas.
The rise of fascism
in the '30s.
He comes to the United States,
takes a teaching position at
Duke University.
All of the people that
he met along the way.
All these people that we study,
all this stuff, it's
all there from his
perspective.
And the perspective
of his biographer.
Plus the pictures.
So anyway, primary sources.
Go read the primary sources.
All right, class dismissed.