Fick's law of diffusion - Chemical Engineering Video Lecture

So I have a picture
for you of Adolf Fick.
And this is probably the
second most well-known Adolf
in history, but this Adolf was
well known for his science.
He actually came up
with some fantastic laws
that we use in all sorts
of different branches
in science today.
And we're going to talk about
one of his laws right now.
So I drew for you
a little box, and I
thought one of the most fun
ways to think about some
of these laws that
Mr. Fick came up
with was to do a little game.
So I'm going to give
you a challenge.
And the challenge
is that, let's say
that you're a person
standing right here,
maybe standing behind this box.
And the part of the box that's
facing you, that's nearest you,
is that blue wall that
I kind of shaded in.
This blue wall is the
back wall of the box.
And on the front
wall, I'm actually
going to put in some
little molecules.
Let's say there are some
molecules I'll put in,
I don't know, let's say
three or four molecules here.
And the challenge is
this, if a molecule
gets from the front of the
box-- I'm going to call it one,
this first side is
side one-- if it
gets from side one, which is
here, over to side two, which
is the back wall, if
it gets from side one
to side two of the
box, then you get
$5 for each particle
that makes it over.
So to put that into words,
we're interested in the amount
of particles that move
over some period of time.
And that period of time can
be one hour or 10 hours,
whatever period of time we want.
But I'm going to-- just
for argument's sake,
let's just say that we're
going to do this for one hour.
So let's say I do it for
an hour and these molecules
started moving around,
they're migrating around,
because, of course,
they're bouncing,
these molecules don't
stay stationary.
And I come back, and it turns
out that only one molecule ever
eventually made it
over to this side.
So I say, hey, good
job, you get $5
because I promised that to you.
And so of course
you get your $5,
and you're happy
and smiling, right?
But I'm feeling in a
generous mood, and I say,
you know what, let's start
this experiment over.
Let's just start over,
and this time I'm
actually going to give you a
chance to tweak the experiment.
You're going to
actually get a chance
to modify the experiment.
And you can do whatever
you want to try
to maximize your profits.
So think about this,
you want to try
to maximize this right here.
And how do you do that?
How do you maximize the amount
of particles that make it
to that blue wall over
some period of time?
And I'm going to write your
good ideas down here, so start
brainstorming some good
ideas for how you might want
to tweak the game, or play the
game, to maximize your profits.
Well, if you're
thinking about this,
you might think, well, maybe
the first obvious thing
is why do you make
this so darn far?
Why not make it a
little bit closer?
So let's get rid
of this back wall
and scooch it nearer,
so that the molecules
don't have to go so far.
And that's a pretty good
idea it seems to me, right?
So let's just make this
a little bit smaller.
That's your first
idea, and I would say,
that's a brilliant strategy.
Now let's just make it half
the size, so it's less thick,
and these molecules don't
have to go nearly as far.
And actually, let me
maintain the blue wall
so we can keep seeing properly
what this should look like.
And this is going to
be dashed back here,
and like that, and of
course the blue wall
is going to look like that.
So now you just basically
make it come closer
and so the molecules don't
have to go nearly as far.
So idea one is less thick wall.
What's another idea?
Well you remember
from Graham's law,
we learned that these
molecules, the big ones,
actually don't move.
Their diffusion rate
is not as quick,
and that it's the smaller
molecules that actually
have a faster diffusion rate.
So if I'm waiting
at that back wall
to see how many
molecules can get over,
I want tiny little molecules
to make their way over
because they're going to
have a faster diffusion rate.
When I say tiny, I really
mean smaller molecular weight.
So small molecular
weight molecules,
that's the second idea.
Change up the molecules, make
the molecular weight smaller,
and as Graham's law tells
us, they'll move faster.
So what's a third idea?
Well, maybe you could
just have more of them.
Maybe in this number
one plane, which
is the leftmost
plane of this box,
why don't you just jam it
full of more little molecules?
If you have more molecules
moving around-- that's
another way of saying just
increasing the pressure, that's
increasing the pressure at
that position one-- then
you're going to
have a better chance
of having molecules move across.
So increased pressure at one.
And what's a fourth idea?
I'm just going to make
a little bit of space.
What's a fourth idea that
we can maybe put on here?
Well, if you're thinking
really outside the box,
and this is actually
literally outside the box,
then you might think,
well, why not just
expand this entire thing?
Make it a larger area.
What about that?
Why not just make a larger area.
So that's the last idea.
Maybe you can actually just
make it a bigger surface area.
So maybe something
like this, you
could expand it
in all directions.
And maybe you can do
something like that.
I'm going to have to make sure
I draw it carefully so don't
confuse you, but basically
something like this,
where you now have
the same thickness,
I'm not changing the
thickness, but I'm
basically-- you're going
to make this wall bigger.
Actually, I screwed
that up a little bit.
Let me just fix that.
So this is what my new back
wall is going to look like
and maybe I should
do it in blue just
to make sure we stay
consistent with the colors.
But of course, this is just
going to extend out like that.
And this is my new back wall.
Right, this whole
thing is my back wall.
And so if I expand
the area, now I
have, of course,
much more chance
of getting some of these
molecules back there.
And the partial pressure
is going to stay the same,
so if I expand the area, I
still have more molecules
on this initial leftmost face.
And so the pressure is
going to stay the same.
This is the P1 that we just
got through talking about.
But because I have
more area there's
more chance that somewhere
along this entire area
a molecule will make its
way across the thickness
and hit that back wall.
So something like that.
And let me fill this in.
So this is a fourth idea.
Let me write that down
as the fourth idea,
is increase the area.
So these are four good
ideas, four good ideas
for how you can maximize the
amount of particles over time
and hopefully make as
much money as you can.
Probably much more than $5.
So this is exactly what
Fick's law talks about.
It talks about the
idea of amounts
of particles moving over time.
So let me write out Fick's
law, and this is actually
how you most commonly will
come across it, although there
are some variations on it.
It's going to look
something like this.
And I'm actually going to try
to color code it to go along
with the ideas that
we already presented.
So we said there's some things
you can do with pressure,
some things you might do
with the surface area,
and also, remember we had
that diffusion constant.
And you divide all this stuff
by the thickness of the wall.
So it's very colorful,
but this is Fick's law
as you usually see it.
There are some other
variations I'll talk about.
So to go through
this piece by piece,
this is V with a dot over it.
This is the rate of
particles moving.
And when I say rate,
you know that that
means that there's
some time component.
So this gets to what
the challenge was.
We said, how many
particles can you
get to go to that blue back
wall over some period of time.
And sometimes when we
talk about particles
we can think of them as giving
that in terms of an amount.
You might think of
that as like a moles
or some numerical
value, or the volume
of a gas that's moving across.
So that's why
sometimes you'll see it
as V, to refer to volume.
On the other side, this
bit makes perfect sense.
If you have more
molecules, that's
going to cause more
pressure, we said earlier.
And if there's a bigger pressure
difference between what's
on the first side versus
what's on the second side--
remember the second side is
the back wall-- then of course
that's going to mean that
more of the molecules
are going to move over.
So this is a bigger difference.
So sometimes you'll
see this as delta P,
and delta just means difference.
This A, we said, refers
to just surface area.
Of course if you have
a greater surface area,
that's going to allow for more
of the molecules to get across.
This D, that's an
interesting one.
This is diffusion constant.
And remember when we think
about diffusion constant,
there are two laws that
might jump into your head.
In fact, the first
one was Henry's law,
and you remember we
talked about solubility,
in terms of the amount
of molecules that
go from, for example,
air to liquid.
This is Henry's law
that told us about that.
And of course if
something is very soluble,
then maybe that would
be an increased P1,
going back to the idea of
pressure at the first wall.
And then you have to divide
by the molecular weight.
The square root, in fact,
of the molecular weight.
So that's the idea we
had, and this actually
comes from Graham's law.
So whenever I talk about
the diffusion constant,
remember there are two
laws that are in play here,
Henry's law and Graham's
law, that are coming together
to offer us some information
about the diffusion constant.
And that's actually
why we said, well,
if you have a small molecular
weight molecule, maybe that'll
help you out, because it's
in the denominator it's
going to cause the rate
of particles moving across
to go up.
And finally, this T,
this is thickness.
This is the thickness
of the wall.
And this is totally intuitive.
If you have a thick
wall, it's going
to be harder for molecules to
make it across very quickly.
So without even knowing it,
you kind of derived Fick's law
all by yourself just
kind of using intuition,
and that's kind of the best
way to learn this stuff.
And sometimes, as
I mentioned, you
might see this formula
written differently.
In fact, let me
actually just rearrange
this formula in a different way.
I'm just going to sketch
out how you might also
see it, which is that
sometimes you see area
on this side of the equation.
Of course, that's just
rearranging it, right,
dividing both sides by area.
And then you might see
the P actually up here,
like P1 and then minus
P2, something like this.
And in the denominator over
here, you'll see the T.
So you'll get this, and
then very separately you'll
see times D.
So this might not look so
different, but what happens
is that then people
lump things, and that's
when things get kind of tricky.
They'll say well, let's
lump this together,
and let's lump this together.
And they'll call this flux.
And this second part
they'll call gradient.
So you may see Fick's law
written this way, where
it says flux equals the gradient
times diffusion constant.
Because of course the diffusion
constant hasn't changed,
it's the same thing, it
just carries on down.
And if you see it
that way, let me just
give you an example of
what these things mean.
Let's start with flux,
which is basically
the net rate of particles
moving through an area,
moving through some area.
And you can kind of follow
through the equation
and that makes sense.
And here the important part
is the idea of a net rate.
It's not total rate,
but you're actually
looking for what is the net gain
or net rate that you're seeing.
And this gradient
over here, this
is just change in
pressure over-- divided
by or over-- some distance,
or over a distance.
And occasionally you'll
see pressure written out
as particles in a volume, which
is kind of the same thing.
Conceptually it's
the same thing.
Particles in a
volume, of course,
are going to exert
some pressure,
so sometimes you'll even
see it written this way.
So I guess the
point is that you'll
see these different
terms, and I just
want you to be
familiar with them.
But at least now you know that
the most common way you'll
see Fick's law is
written out this way,
and that it's
completely intuitive.
In fact if you had to come
up with it, given some time
you'd probably come up
with Fick's law yourself.