- [Instructor] Let's talk
about Boltzmann's constant.
It's named after, first of all,
this guy, Ludwig Boltzmann,
who was a genius.
He lived in the late
1800s and early 1900s,
and he was the father
of modern atomic theory,
one of the big proponents,
early proponents
that the world is made out
of atoms and molecules.
This sounds obvious to us now,
but 120 years ago, it was
definitely not obvious
and some of the smartest people of his day
vehemently disagreed with Boltzmann
and Boltzmann had to defend
these ideas over and over.
And what I mean by atomic theory is this.
If you had a container of, say, anything,
could be a cube of metal,
let's just say it's a gas,
let's say it's a container
and it's full of air.
Well, it feels like the
air is continuous in here
or like the gold, if
this was a cube of gold,
the gold is continuous.
But we know now, and Boltzmann knew,
that it's really made out
of atoms and molecules.
That wasn't obvious 120 years ago
'cause you can't see
the atoms and molecules.
If this was a container
of steam, let's say,
and you stuck your hand in here,
so I took my hand,
I put my hand in this container of steam,
I'd notice it, I'd know
something was going on.
My hand would start to feel hot.
There's energy being transferred here,
but it wasn't obvious what
exactly is the mechanism.
Is this a new kind of energy?
Is this one of our old kind
of energies just in disguise?
Boltzmann's big claim and
groundbreaking idea was that
this gas, if it's steam, let's say,
is really made out of atoms and molecules.
These gas molecules are
running around in here.
There's just little particles in here.
And what you're actually feeling
are these particles striking your hand,
so your hand's just getting
bombarded by these particles.
But they're so small and
they're so many of them,
you can't really tell
that there's particles.
It just looks completely continuous.
So for Boltzmann, this heat energy
isn't really a new kind of energy at all.
All this is, this heat
energy that you're feeling
is just kinetic energy,
and if it's steam, it's
just the kinetic energy
in the H2O molecules flying around in here
at some rapid speed.
And the faster they go,
the greater the impact
with your hand, which is
gonna transfer more energy.
So the faster they go, the
hotter it feels in here.
So, for Boltzmann, to say that
something has a high temperature,
if you said that the temperature is large,
if it's hot outside,
that's kind of redundant.
We already had a word for that.
We could just say if
it's a high temperature,
what we really mean is that
the average kinetic energy
of the gas molecules outside is large.
So, if a gas has a high temperature,
the average kinetic energy
of those molecules is large.
That's why it hurts when
they impact on your skin
'cause they're transferring kinetic energy
to the molecules in your hand,
and when your hand
absorbs too much energy,
these molecules move around,
your skin starts to get damaged,
you can get burned.
So this is often referred
to as the kinetic-molecular
explanation of temperature.
And the details of this theory
were one of Ludwig Boltzmann's
biggest contributions to science.
But what does any of this
have to do with Boltzmann's constant?
Well, let's get rid of all of this.
You've probably heard
of the ideal gas law,
PV equals nRT.
So, remember, T is the
temperature measured in Kelvin.
P is the pressure, and I'm
gonna measure this pressure in,
I'm gonna choose to measure it in Pascals.
V is the volume,
I'm gonna choose to
measure it in meters cubed.
And n, little n, remember, little n is
the number of moles of the gas.
And if you've forgotten what moles are,
n, the number of moles, is defined to be
capital N, the number
of molecules in the gas,
the total number of molecules in the gas,
divided by a constant and
that constant's called
Avogadro's number.
And if you've forgotten Avogadro's number,
Avogadro's number is 6.02
times 10 to the 23rd,
and there's that many molecules per mole.
So in every mole of a gas,
what we mean by one mole of a gas
is 6.02 times 10 to the 23rd molecules.
And if you choose these units,
this R, this gas constant, R
is called the gas constant,
and it has a value, R has a value of
8.31 Joules per mole Kelvin.
That's the gas constant
R with these units.
But these are pretty
macroscopic quantities,
pressure and volume and
temperature and moles.
Even moles, talking about one or two moles
is talking about a huge
number of molecules.
You're kinda glossing over some
of the microscopic details,
so an alternate way to
write the ideal gas law is
P times V equals capital
N, so forget moles.
Let's say we want to talk about
how many molecules there are.
Instead of writing little n,
let's write big N, number of molecules.
We'd need a different constant
'cause we're gonna multiply by the same T.
So again, this T is still
temperature in Kelvin.
P is still the pressure in Pascals.
V is the volume, again, in meters cubed.
N, instead of being the number of moles,
is now the number of molecules,
and that means we need
a new constant here.
We need a different constant,
and that constant's gotta
be really, really small.
The rest of this stuff's the same.
P times V and T are all the same.
And all I did was I swapped
out little n, number of moles,
for big N, number of molecules,
so this is gonna be a huge
number we're plugging in
into this spot now.
Instead of plugging in like, say, two,
if I were to plug in two moles
right here, the number two,
down here, I'd plug in two times this.
So I'd plug in 12.04 times 10 to the 23rd.
Since this is a huge number,
I need a constant that's really small
because it's gotta balance out.
We know that n times R
has gotta be the same
as capital N times this constant
because the rest of this is the same.
This left hand side's the
same and the T's the same.
So if this is all consistent
and n times R has gotta be equal to
N times this new constant,
and that new constant
is Boltzmann's constant.
It's a lowercase k with a B on it
to denote Boltzmann's constant.
So what's the value of
Boltzmann's constant?
We can find it pretty easily.
We know that little n times R
has gotta equal big N
times Boltzmann's constant,
so if we just solve this
for Boltzmann's constant,
we're gonna get little
n over big N times R.
But what's little n over big N?
Just look up here, we can figure it out.
Little n over big N,
if I solve this for little n over big N,
what I'm gonna get is, if I
divide both sides by big N,
I get one over Avogadro's number.
Little n over big N right
here is Avogadro's number,
or one over Avogadro's number.
So I get that one over Avogadro's number
times the gas constant, this 8.31,
is Boltzmann's constant.
If you multiply that out, the
gas constant, which is 8.31
Joules per mole Kelvin,
and divide by Avogadro's
number, which is 6.02
times 10 to the 23rd molecules per mole,
you'll get Boltzmann's constant,
which equals 1.38 times
10 to the negative 23rd
Joules per Kelvin.
This is Boltzmann's constant,
this number right here
is Boltzmann's constant.
Why do we care about Boltzmann's constant?
Well, it allows us to write
a more microscopically oriented version
of the ideal gas law that focuses on
number of molecules
instead of number of moles,
and this number pops up
all over statistical
and thermal mechanics.
It's one of the most important constants
in all of thermal physics, in fact.
It was so important that on
Boltzmann's own gravestone,
if you go to Boltzmann's grave,
there's a bust and a gravestone,
it doesn't actually look like a cross
but there's a grave there
with a big inscription and
there's an inscription.
The big inscription is an equation,
S equals Boltzmann's constant times log W.
This was possibly his most
important contribution,
and it says that the entropy of a system
is equal to, this k's
Boltzmann's constant,
we just talked about that.
Log, it says l-o-g, but nowadays we use ln
because really, they meant
the natural logarithm here
and nowadays it's conventional
that l-o-g is log base 10
but this equation is really
referring to the natural log.
And W, W's kinda
mysterious like entropy is.
W is the number of microstates.
So if you had a macroscopic system
and you wanted to know microscopically
what are all the ways I
can arrange my particles
with given speeds and
distributions and positions
such that it looks identical,
the macroscopic state for
someone standing out here.
They would look at this
thing and they'd be like,
that's the exact same state.
But the particles are doing
something different in here.
It's just on a macroscopic
level identical.
How many ways are there to do that
and still make the macroscopic view
identical for this person out here?
That's what this is measuring,
the number of microstates.
And if you take Boltzmann's constant
times the natural log of that number,
it gives you an idea of the entropy.
Entropy's very mysterious and interesting,
has to do with the disorder or
available energy in a system.
I don't have enough time
to describe it right now,
but if you have time, you
should look into this.
This is mysterious and confusing
and wonderful at the same time.
