- [Lecturer] 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's 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 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 energy,
it's 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 particle 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 of 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,
P V equals n R T, 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 forgot 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 forgot 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 wanna 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 it 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 gonna 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 is 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
in 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 is gotta be the same
as capital N times this constant,
because the rest of this is the same,
this left-hand side is the same,
and the T is the same, so if this is all consistent,
then n times R is 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 got to 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 could 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
and 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.