WEBVTT

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[SQUEAKING]

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[RUSTLING]

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[CLICKING]

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GIAN PAOLO BERETTA:
So good morning.

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We're still playing with
mixtures, properties

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of multi-component systems.

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We've defined some
ideal behavior,

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like the Gibbs-Dalton mixture
that applies for ideal gases.

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And today, we'll extend that
to the case of liquid mixtures.

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Remember, this was last--

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the last slide of
the last lecture.

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And we were discussing this
Lennard-Jones interactions

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between particles.

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And remember that in
the gas phase, when

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we can apply the
ideal gas model,

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the particles, on average,
live far away from each other.

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They do experience the
collision very often.

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And so very often, they do
experience this acceleration

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towards the other particle
and then the repulsion

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that pushes it back and
then finishes the collision.

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But the time that they
spend in this area

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is much smaller
than the time they

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spend far away from each other.

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And therefore, for
the calculation

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of the average properties
at stable equilibrium,

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each particle behaves
as if it were by itself.

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And in fact, that's the behavior
that we have observed also

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for mixtures.

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And so also for mixtures,
whether the particles

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are close to similar particles
or to different particles, if we

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are in the Gibbs-Dalton
mixture limit,

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the same holds also for
different particles.

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But you can see that
if there are mixtures

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in which, for example,
the individual components,

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when pure, stay far
from each other,

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but when mixed like to
stay close, in that case,

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you would have a
non-ideal behavior

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because if they spend more time
in this close-by area, then

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this will influence.

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And you cannot neglect anymore
this energy of interaction.

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Now, similar to the
Gibbs-Dalton mixture behavior,

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we can define also
an ideal solution

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behavior for the liquid
phase, like sketched here.

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So in the liquid, of
course, the particles

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live very close to one another.

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And so they live very close to
this dip in the Lennard-Jones

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potential.

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But we can consider
an ideal situation--

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hypothetical, of course.

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But it's a good start
for any kind of modeling,

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or it could be also taken--

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it's actually taken,
typically, as a reference.

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And then we will describe
actual real solution behavior

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as departures with respect to
the ideal solution behavior.

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This ideal solution behavior
is one in which the--

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this effect of the
energy of interaction

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is alike for the
particles for the--

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for-- say, for the A particles
and for the B particles.

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And it's also alike for
the A and B particles.

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So essentially, these particles
are very similar to one another.

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And they don't really care if
the neighbor is similar or not.

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In that limit,
therefore, there is

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no contribution to the enthalpy
of mixing because you remember,

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the enthalpy of mixing has to do
with the fact that, for example,

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here, if I take a gray
particle and I put it

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in a container full
of black particles,

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the enthalpy of mixing is
the change in the energy

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if I keep at the same
temperature and pressure--

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due to mixing-- adding 1
mole of that gray substance

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into the black bulk.

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And so if the particle
lives in this area,

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the enthalpy of
mixing will be related

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to this dip in the energy.

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Now, this other case--

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for the liquid case,
we are somewhere here.

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So the energy is somewhere here.

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But it's the same for all.

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And so even if you add
one more gray particle,

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it doesn't really make a
difference for the energy.

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It doesn't really make a
difference for the volume,

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also, because
being at the same--

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leaving it the same
distance, the volume also

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doesn't change for the mixture.

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But it does make sense--

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I mean, it should make a
difference regarding the entropy

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because as you
remember, when we did

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this entropy of mixing
idea, the fact that you

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have added one more
gray particle-- you

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have diluted the mixture.

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So you have changed, say,
what we call the surprise when

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you see a gray molecule.

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Now you've changed
the probability

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of seeing the gray molecule
with respect to the black one.

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And that is accounted for
by the entropy of mixing.

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That's why the entropy
of mixing does--

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is the only effect.

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So in this ideal
solution behavior,

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we only have this
entropy of mixing effect.

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Now, if you remember,
the chemical potential

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for any constituent
was written this way

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as the chemical potential
of the pure component

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at the same temperature
and pressure of the mixture

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plus the enthalpy of
mixing of that component

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minus the temperature times
the entropy of mixing.

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And we've used this relation
already a few times.

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So if I substitute the
three assumptions up here--

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so let me do first the volume.

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For the volume, I
have to remember

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that the volume of the mixture
is given, as we have proven by--

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in general, as a weighted
sum of the partial volumes

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of the constituents
in the mixture.

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And the partial volumes can be
written as the specific volume

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of the pure component
at the same temperature

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and pressure of the mixture
plus the volume of mixing.

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So in this case,
the volume of mixing

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is assumed to be 0 for
ideal solution behavior.

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And so all we are left is
that the volume is given

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by the weighted
sum of the volumes

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that the single pure
components would

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occupy at the same
temperature and pressure.

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And this is Amagat law of
additive volumes, something

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that we proved is satisfied
also by a Gibbs-Dalton mixture

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of ideal gases.

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For the chemical potential--

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so delta x mix--

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here, there should
be an equal 0.

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I thought I corrected this.

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So equals 0-- and equals 0.

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So delta x-- delta
h is equal to 0.

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And delta s is
equal to minus R log

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of y, which is the surprise
multiplied by the Boltzmann

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constant, or the gas constant.

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So the minus and the
minus become plus.

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And so the chemical potential
becomes plus RT log of y.

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At this stage, ideal
solution behavior

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is when this happens
for all constituents.

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And this is a very rare
practical situation.

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But again, it's a
good starting point,

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a good reference
situation on which

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you can make some calculations.

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Now, this ideal mixture
solution behavior

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can be used to derive an
important aspect of equilibria

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across semipermeable membranes,
which is the osmotic pressure

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concept.

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And osmotic pressure is
nothing but the difference

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between the pressure of the
mixture and the pressure

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that the pure component has
when in mutual equilibrium

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on the other side of a
membrane that is permeable only

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to that component--

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so it's p minus pii.

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Now, the expression that I just
reminded us on the other slide

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just behind this
is rewritten here,

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by remembering that, of
course, across that membrane,

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there is equilibrium,
mutual equilibrium.

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And the condition for
mutual equilibrium

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is that the chemical
potential of the constituent

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in the mixture is equal
to the chemical potential

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of the pure constituent on the
other side of the membrane--

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so up here, where the pressure
is, the partial pressure.

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And also here, I
see another error

00:12:50.800 --> 00:12:52.500 align:middle line:90%
to be corrected on the slides.

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This should be double i
here and double i here.

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So this is equal to-- this is
the chemical potential-- this

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is equal to the chemical
potential of the constituent

00:13:03.900 --> 00:13:04.970 align:middle line:90%
in the mixture.

00:13:04.970 --> 00:13:10.340 align:middle line:90%


00:13:10.340 --> 00:13:13.070 align:middle line:84%
But it is equal to the
pure chemical potential

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at the partial pressure.

00:13:14.990 --> 00:13:19.890 align:middle line:84%
So as you remember, when we
calculate this difference,

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since we know that
the derivative

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of the chemical potential
with respect to pressure

00:13:27.020 --> 00:13:32.930 align:middle line:84%
is the specific
volume, this integral

00:13:32.930 --> 00:13:36.750 align:middle line:84%
can be easily done once we
know the equation of state

00:13:36.750 --> 00:13:38.940 align:middle line:84%
of the substance, of
the pure substance.

00:13:38.940 --> 00:13:42.044 align:middle line:90%


00:13:42.044 --> 00:13:45.760 align:middle line:90%
So this is nothing new.

00:13:45.760 --> 00:13:51.435 align:middle line:84%
But it's the starting point of
this observation made by van

00:13:51.435 --> 00:13:53.470 align:middle line:90%
't Hoff--

00:13:53.470 --> 00:14:00.160 align:middle line:84%
namely, that when we
have a solution and we

00:14:00.160 --> 00:14:02.290 align:middle line:90%
have a dilute solution--

00:14:02.290 --> 00:14:06.290 align:middle line:84%
so a dilute solution means that
there is a lot of a substance.

00:14:06.290 --> 00:14:14.200 align:middle line:84%
Think of, say, a lot of gray
particles and a few solutes.

00:14:14.200 --> 00:14:16.990 align:middle line:90%
One is called the solvent.

00:14:16.990 --> 00:14:19.740 align:middle line:84%
The others are
called the solutes.

00:14:19.740 --> 00:14:24.090 align:middle line:84%
And the solvent is the
predominant amount.

00:14:24.090 --> 00:14:29.340 align:middle line:84%
So that means that the mole
fraction of the solvent

00:14:29.340 --> 00:14:31.890 align:middle line:90%
is very close to 1.

00:14:31.890 --> 00:14:37.270 align:middle line:84%
And 1 minus the mole
fraction of the solvent,

00:14:37.270 --> 00:14:43.550 align:middle line:84%
which is equal to the sum of the
mole fractions of the solutes,

00:14:43.550 --> 00:14:45.770 align:middle line:90%
is much less than 1.

00:14:45.770 --> 00:14:49.610 align:middle line:90%
So that is a dilute solution.

00:14:49.610 --> 00:15:01.590 align:middle line:84%
Now, in a dilute solution,
if you put in a gray particle

00:15:01.590 --> 00:15:08.020 align:middle line:84%
in the box in which there are
already many gray particles,

00:15:08.020 --> 00:15:11.200 align:middle line:84%
it's very likely that
that particle will end up

00:15:11.200 --> 00:15:14.890 align:middle line:90%
with neighbors that are gray.

00:15:14.890 --> 00:15:21.320 align:middle line:84%
So in this situation, there
is no enthalpy of mixing.

00:15:21.320 --> 00:15:23.380 align:middle line:84%
There is no change
in volume of mixing.

00:15:23.380 --> 00:15:26.440 align:middle line:84%
So it's exactly the
kind of behavior

00:15:26.440 --> 00:15:30.490 align:middle line:84%
that describes the
ideal solution behavior

00:15:30.490 --> 00:15:34.320 align:middle line:90%
for the solvent only, though.

00:15:34.320 --> 00:15:38.520 align:middle line:84%
So what we are saying is that
of-- van 't Hoff's observations

00:15:38.520 --> 00:15:45.420 align:middle line:84%
is that in any dilute solution,
even if we are not in that

00:15:45.420 --> 00:15:47.560 align:middle line:84%
limiting situation
in which the blue--

00:15:47.560 --> 00:15:52.050 align:middle line:84%
I mean, the dark and
the gray particles

00:15:52.050 --> 00:15:58.440 align:middle line:84%
are identical in terms of
intermolecular interactions,

00:15:58.440 --> 00:16:01.320 align:middle line:84%
even if they're very
different-- but the fact

00:16:01.320 --> 00:16:05.670 align:middle line:84%
that there are few
black particles

00:16:05.670 --> 00:16:10.800 align:middle line:84%
makes that the
addition of a gray one

00:16:10.800 --> 00:16:18.130 align:middle line:84%
doesn't change the energy and
the volume of the mixture.

00:16:18.130 --> 00:16:22.500 align:middle line:84%
So in other words, the
solvent of a dilute solution

00:16:22.500 --> 00:16:26.345 align:middle line:84%
behaves according to the
ideal solution model.

00:16:26.345 --> 00:16:30.980 align:middle line:90%


00:16:30.980 --> 00:16:37.840 align:middle line:84%
And of course, that
doesn't mean that there

00:16:37.840 --> 00:16:41.170 align:middle line:84%
is no effect in adding
a gray particle because,

00:16:41.170 --> 00:16:44.680 align:middle line:84%
again, if we add
a gray particle,

00:16:44.680 --> 00:16:50.380 align:middle line:84%
it is going to dilute
further the solution.

00:16:50.380 --> 00:16:55.370 align:middle line:84%
And that will have an effect
on the entropy of mixing.

00:16:55.370 --> 00:16:58.440 align:middle line:84%
For example, if you take
the entropy of mixing here,

00:16:58.440 --> 00:17:01.020 align:middle line:84%
which is that expression
that we derived,

00:17:01.020 --> 00:17:07.153 align:middle line:84%
and you write it for
this solution here--

00:17:07.153 --> 00:17:10.440 align:middle line:90%


00:17:10.440 --> 00:17:13.670 align:middle line:84%
the reason I want to
write it carefully

00:17:13.670 --> 00:17:19.020 align:middle line:84%
is that I want to compute
the entropy of mixing.

00:17:19.020 --> 00:17:23.060 align:middle line:84%
And that is the
derivative of S not

00:17:23.060 --> 00:17:24.720 align:middle line:84%
with respect to
the mole fractions,

00:17:24.720 --> 00:17:29.060 align:middle line:84%
but with respect to the
amount of constituent i, 1,

00:17:29.060 --> 00:17:31.490 align:middle line:90%
which is the solvent.

00:17:31.490 --> 00:17:35.510 align:middle line:84%
And so I have to
write explicitly

00:17:35.510 --> 00:17:38.760 align:middle line:84%
how the mole fractions
depend on the amount.

00:17:38.760 --> 00:17:42.720 align:middle line:84%
And you see there is
an n1 in the numerator.

00:17:42.720 --> 00:17:44.970 align:middle line:84%
But there is also an
n1 in the denominator.

00:17:44.970 --> 00:17:50.670 align:middle line:84%
So when you take the derivative,
you have to be careful.

00:17:50.670 --> 00:17:57.490 align:middle line:84%
Well, if you do it, it
turns out that the change

00:17:57.490 --> 00:18:00.280 align:middle line:84%
in this entropy
of mixing by-- due

00:18:00.280 --> 00:18:04.990 align:middle line:84%
to the addition of
a small amount of n1

00:18:04.990 --> 00:18:09.190 align:middle line:84%
is, indeed, equal
to minus R log of y.

00:18:09.190 --> 00:18:11.505 align:middle line:90%
And that is the surprise.

00:18:11.505 --> 00:18:17.560 align:middle line:90%


00:18:17.560 --> 00:18:22.180 align:middle line:84%
So this backs up van
't Hoff's observation.

00:18:22.180 --> 00:18:34.070 align:middle line:84%
So at least for the solvent,
we can use the ideal mixture

00:18:34.070 --> 00:18:34.820 align:middle line:90%
solution.

00:18:34.820 --> 00:18:38.460 align:middle line:84%
So we can set delta
h equal to 0 here.

00:18:38.460 --> 00:18:43.020 align:middle line:84%
And delta smix equals
to minus R log of y.

00:18:43.020 --> 00:18:47.210 align:middle line:84%
So this becomes a
plus RT log of y.

00:18:47.210 --> 00:18:48.190 align:middle line:90%
And here it is.

00:18:48.190 --> 00:18:51.810 align:middle line:90%


00:18:51.810 --> 00:18:55.150 align:middle line:84%
And for the other side,
you see this integral.

00:18:55.150 --> 00:18:57.520 align:middle line:84%
Now, if we apply
it to a liquid--

00:18:57.520 --> 00:19:02.180 align:middle line:84%
so if we are talking about
a dilute liquid solution,

00:19:02.180 --> 00:19:10.300 align:middle line:84%
the specific volume of a liquid
under the incompressible ideal

00:19:10.300 --> 00:19:13.080 align:middle line:90%
fluid is constant.

00:19:13.080 --> 00:19:15.520 align:middle line:84%
It doesn't change with
temperature and pressure.

00:19:15.520 --> 00:19:18.860 align:middle line:84%
So this v goes out
of the integral.

00:19:18.860 --> 00:19:24.340 align:middle line:84%
And it becomes the
specific volume

00:19:24.340 --> 00:19:29.200 align:middle line:84%
of the solvent times the
difference in the integration

00:19:29.200 --> 00:19:29.750 align:middle line:90%
limits.

00:19:29.750 --> 00:19:33.050 align:middle line:90%
And that's the osmotic pressure.

00:19:33.050 --> 00:19:37.860 align:middle line:84%
So we got the expression here
for the osmotic pressure.

00:19:37.860 --> 00:19:42.520 align:middle line:84%
Actually, van 't Hoff makes
an additional approximation,

00:19:42.520 --> 00:19:48.360 align:middle line:84%
noticing that since
the assumption is

00:19:48.360 --> 00:19:51.120 align:middle line:84%
that the solution
is dilute, it means

00:19:51.120 --> 00:19:55.450 align:middle line:84%
that the mole fraction of
solute is very close to 1.

00:19:55.450 --> 00:19:57.630 align:middle line:84%
So we can write it as 1
minus the mole fractions

00:19:57.630 --> 00:19:59.620 align:middle line:90%
of the solutes.

00:19:59.620 --> 00:20:05.830 align:middle line:84%
And then we can take the first
term in the Taylor series

00:20:05.830 --> 00:20:12.770 align:middle line:84%
expansion of log of 1
minus x, which is minus x.

00:20:12.770 --> 00:20:19.290 align:middle line:84%
So this becomes simply the
summation of the solutes.

00:20:19.290 --> 00:20:22.003 align:middle line:84%
And finally, we have
this expression,

00:20:22.003 --> 00:20:23.670 align:middle line:84%
which is the van 't
Hoff relation-- says

00:20:23.670 --> 00:20:27.760 align:middle line:84%
the osmotic pressure
of the solvent

00:20:27.760 --> 00:20:31.140 align:middle line:84%
is equal to RT divided
by the specific volume.

00:20:31.140 --> 00:20:38.950 align:middle line:84%
And again, here, this should
be a v11, another error,

00:20:38.950 --> 00:20:45.530 align:middle line:84%
times the sum of the mole
fractions of the solutes.

00:20:45.530 --> 00:20:49.220 align:middle line:84%
Notice that it doesn't really
matter what the solutes are.

00:20:49.220 --> 00:20:50.230 align:middle line:90%
It can be any molecule.

00:20:50.230 --> 00:20:55.110 align:middle line:84%
It could be sugar, could
be chlorine, and so on--

00:20:55.110 --> 00:21:00.800 align:middle line:84%
could be salt, except that,
as we will see in a moment

00:21:00.800 --> 00:21:03.920 align:middle line:84%
because that is the
important application

00:21:03.920 --> 00:21:09.970 align:middle line:84%
that we want to discuss, as
you know, some molecules--

00:21:09.970 --> 00:21:15.230 align:middle line:84%
for example, salt-- when you
put it in water, they separate.

00:21:15.230 --> 00:21:16.460 align:middle line:90%
They dissociate.

00:21:16.460 --> 00:21:19.310 align:middle line:90%
And then they get solvated.

00:21:19.310 --> 00:21:21.120 align:middle line:90%
They hydrolyze.

00:21:21.120 --> 00:21:29.740 align:middle line:84%
And so you have to be careful
when counting the molecules

00:21:29.740 --> 00:21:34.360 align:middle line:84%
because when you have,
for example, the mole--

00:21:34.360 --> 00:21:38.535 align:middle line:84%
you put in a certain fraction
of salt and then you--

00:21:38.535 --> 00:21:45.200 align:middle line:90%


00:21:45.200 --> 00:21:48.180 align:middle line:84%
it gets dissolved, for
each mole that you put in,

00:21:48.180 --> 00:21:52.640 align:middle line:90%
you get 2 moles of solutes.

00:21:52.640 --> 00:21:54.055 align:middle line:90%
So you have to double this.

00:21:54.055 --> 00:21:56.930 align:middle line:90%


00:21:56.930 --> 00:22:06.030 align:middle line:84%
Now, clearly, this issue
of osmotic pressures

00:22:06.030 --> 00:22:07.930 align:middle line:90%
is important in biology.

00:22:07.930 --> 00:22:12.250 align:middle line:84%
And as I told you several
times, I'm no expert in biology.

00:22:12.250 --> 00:22:16.800 align:middle line:84%
So I don't want to talk
about it because it's

00:22:16.800 --> 00:22:18.700 align:middle line:90%
too risky to say wrong things.

00:22:18.700 --> 00:22:23.850 align:middle line:84%
But I can point you to
a nice YouTube lecture

00:22:23.850 --> 00:22:29.170 align:middle line:84%
where you can learn
about osmosis in biology.

00:22:29.170 --> 00:22:35.450 align:middle line:84%
And it tells you
how cells receive--

00:22:35.450 --> 00:22:40.100 align:middle line:84%
exchange across the
membrane molecules,

00:22:40.100 --> 00:22:46.840 align:middle line:84%
depending on whether there
is a chemical potential.

00:22:46.840 --> 00:22:49.730 align:middle line:84%
If there is equality of
chemical potential, of course,

00:22:49.730 --> 00:22:50.730 align:middle line:90%
you're in equilibrium.

00:22:50.730 --> 00:22:54.890 align:middle line:84%
But if the concentrations
are different than those

00:22:54.890 --> 00:22:57.170 align:middle line:84%
that you are at
equilibrium, then

00:22:57.170 --> 00:22:59.930 align:middle line:84%
depending on the sign of the
difference in the chemical

00:22:59.930 --> 00:23:03.930 align:middle line:84%
potential, you can have a flow
in one direction or the other.

00:23:03.930 --> 00:23:07.490 align:middle line:84%
And that is helpful
in regulating

00:23:07.490 --> 00:23:11.240 align:middle line:90%
the metabolism of the cell.

00:23:11.240 --> 00:23:16.100 align:middle line:84%
And in this video,
you also are brought

00:23:16.100 --> 00:23:20.060 align:middle line:84%
to understand what the
difference between plants--

00:23:20.060 --> 00:23:23.510 align:middle line:84%
membranes of plant cells
and animal cells are.

00:23:23.510 --> 00:23:25.590 align:middle line:90%
They are kind of different.

00:23:25.590 --> 00:23:32.730 align:middle line:84%
And they behave
in different ways.

00:23:32.730 --> 00:23:36.240 align:middle line:84%
In the plants, they can
sustain more pressure.

00:23:36.240 --> 00:23:40.330 align:middle line:84%
In the animal cells, they are
more fragile, so to speak.

00:23:40.330 --> 00:23:41.615 align:middle line:90%
And they can break.

00:23:41.615 --> 00:23:44.990 align:middle line:90%


00:23:44.990 --> 00:23:48.410 align:middle line:84%
There are also
important applications

00:23:48.410 --> 00:23:51.810 align:middle line:90%
in chemical engineering.

00:23:51.810 --> 00:23:55.500 align:middle line:84%
For example,
recently, of course,

00:23:55.500 --> 00:24:02.540 align:middle line:84%
there is this big
topic of developing

00:24:02.540 --> 00:24:04.550 align:middle line:90%
lithium-ion batteries.

00:24:04.550 --> 00:24:14.730 align:middle line:84%
And here, you have these
lithium-ion molecules,

00:24:14.730 --> 00:24:24.800 align:middle line:84%
these pinkish molecule that are
in dilute solutions of solvents.

00:24:24.800 --> 00:24:27.640 align:middle line:84%
And these solvents, for
example, in this case

00:24:27.640 --> 00:24:30.506 align:middle line:90%
are some kinds of carbonates.

00:24:30.506 --> 00:24:34.430 align:middle line:90%


00:24:34.430 --> 00:24:37.540 align:middle line:90%
So the osmotic pressure--

00:24:37.540 --> 00:24:40.460 align:middle line:84%
of course, if you are
at the osmotic pressure,

00:24:40.460 --> 00:24:45.380 align:middle line:84%
you are in equilibrium and
nothing moves, on average.

00:24:45.380 --> 00:24:50.890 align:middle line:84%
But if you do not have
the concentrations that

00:24:50.890 --> 00:24:53.890 align:middle line:84%
deliver the osmotic
pressure, then you

00:24:53.890 --> 00:25:00.790 align:middle line:84%
can have osmosis, which is the
movement of the ions in search

00:25:00.790 --> 00:25:04.220 align:middle line:90%
to a better equilibrium.

00:25:04.220 --> 00:25:07.030 align:middle line:90%


00:25:07.030 --> 00:25:11.780 align:middle line:84%
So let's talk about
seawater as the example.

00:25:11.780 --> 00:25:14.960 align:middle line:90%


00:25:14.960 --> 00:25:19.630 align:middle line:84%
Seawater, on average,
per kilogram of seawater

00:25:19.630 --> 00:25:28.010 align:middle line:84%
has 35 grams of
salt. And of course,

00:25:28.010 --> 00:25:34.610 align:middle line:84%
there is this fact that
the salt dissociates.

00:25:34.610 --> 00:25:40.730 align:middle line:84%
And so you get ions of
sodium and ions of chlorine,

00:25:40.730 --> 00:25:43.370 align:middle line:84%
chlorine that gets
hydrated, meaning

00:25:43.370 --> 00:25:48.750 align:middle line:84%
that the molecules of
water which are polar get

00:25:48.750 --> 00:25:50.870 align:middle line:90%
attracted from one side.

00:25:50.870 --> 00:25:52.620 align:middle line:90%
The oxygen side is negative.

00:25:52.620 --> 00:26:01.380 align:middle line:84%
So it's attracted by the sodium,
which is a positive charge.

00:26:01.380 --> 00:26:05.960 align:middle line:84%
And the hydrogen
side is positive

00:26:05.960 --> 00:26:11.150 align:middle line:84%
and is attracted by
the chlorine ion.

00:26:11.150 --> 00:26:15.410 align:middle line:90%
And so you get these molecules.

00:26:15.410 --> 00:26:23.630 align:middle line:84%
Typically, around six molecules,
on average, are around each ion.

00:26:23.630 --> 00:26:31.090 align:middle line:84%
However, not all ions
get so far apart.

00:26:31.090 --> 00:26:34.310 align:middle line:84%
Sometimes, they get also--
they form ion pairs.

00:26:34.310 --> 00:26:38.740 align:middle line:84%
In other words,
they also attract--

00:26:38.740 --> 00:26:40.030 align:middle line:90%
still attract each other.

00:26:40.030 --> 00:26:45.480 align:middle line:84%
And they are still also
surrounded by water molecules.

00:26:45.480 --> 00:26:50.460 align:middle line:84%
So suppose we apply the
van 't Hoff equation

00:26:50.460 --> 00:26:56.650 align:middle line:84%
by neglecting these
ion pairs and assuming

00:26:56.650 --> 00:27:09.060 align:middle line:84%
that 100% of the salt goes
into these hydrated ions forms.

00:27:09.060 --> 00:27:12.720 align:middle line:84%
So that means that now I have
to do the counting in order

00:27:12.720 --> 00:27:15.820 align:middle line:90%
to see how much solutes I have.

00:27:15.820 --> 00:27:20.560 align:middle line:84%
So for each molecule of
salt that I put in solution,

00:27:20.560 --> 00:27:28.970 align:middle line:84%
I get two molecules
that are the ions.

00:27:28.970 --> 00:27:32.090 align:middle line:84%
But in some sense, you should
also account for the fact

00:27:32.090 --> 00:27:37.870 align:middle line:84%
that these hydrated
or solvated groups--

00:27:37.870 --> 00:27:41.550 align:middle line:90%


00:27:41.550 --> 00:27:43.840 align:middle line:84%
since they like to
stay near each other,

00:27:43.840 --> 00:27:49.970 align:middle line:84%
they could be also
thought as a big molecule.

00:27:49.970 --> 00:27:53.720 align:middle line:84%
So part of the
water molecules that

00:27:53.720 --> 00:28:00.690 align:middle line:84%
are in solution and get engaged
in these solvation groups--

00:28:00.690 --> 00:28:02.990 align:middle line:84%
and so they're not
available anymore

00:28:02.990 --> 00:28:06.290 align:middle line:90%
as to play the role of solvent.

00:28:06.290 --> 00:28:09.260 align:middle line:84%
So the mole fraction
of water also

00:28:09.260 --> 00:28:16.470 align:middle line:84%
changes because you lose
six molecules per ion

00:28:16.470 --> 00:28:17.540 align:middle line:90%
that is formed here.

00:28:17.540 --> 00:28:21.990 align:middle line:90%


00:28:21.990 --> 00:28:25.920 align:middle line:90%
So let's see.

00:28:25.920 --> 00:28:28.110 align:middle line:84%
So the number of
molecules of water

00:28:28.110 --> 00:28:37.200 align:middle line:84%
initially are the mass, which is
1 kilogram, minus the 35 grams.

00:28:37.200 --> 00:28:40.580 align:middle line:90%
So it's 965 grams.

00:28:40.580 --> 00:28:44.010 align:middle line:84%
And if you divide it by
the molecular weight,

00:28:44.010 --> 00:28:50.940 align:middle line:84%
you get how many moles of
water you have initially.

00:28:50.940 --> 00:28:56.730 align:middle line:84%
And then for the
ions, their amounts

00:28:56.730 --> 00:29:03.060 align:middle line:84%
are the 35 grams divided by the
molecular weight of the salt,

00:29:03.060 --> 00:29:08.720 align:middle line:84%
of the salt, not
the ions, which is--

00:29:08.720 --> 00:29:10.340 align:middle line:90%
gives this much more.

00:29:10.340 --> 00:29:14.760 align:middle line:84%
So these are how many molecules
of salt you have put in.

00:29:14.760 --> 00:29:18.020 align:middle line:90%


00:29:18.020 --> 00:29:23.010 align:middle line:84%
And then you assume that
the mole fractions--

00:29:23.010 --> 00:29:26.360 align:middle line:90%


00:29:26.360 --> 00:29:31.310 align:middle line:84%
that as they go in,
you double this amount.

00:29:31.310 --> 00:29:41.650 align:middle line:84%
So in the end, you can compute
the mole fraction of water and--

00:29:41.650 --> 00:29:44.470 align:middle line:90%
minus its logarithm.

00:29:44.470 --> 00:29:47.590 align:middle line:84%
You can check, of course,
that the Taylor series

00:29:47.590 --> 00:29:52.150 align:middle line:84%
approximation is good because
this is a small number.

00:29:52.150 --> 00:29:55.430 align:middle line:90%
The mole fraction is 0.978.

00:29:55.430 --> 00:29:56.810 align:middle line:90%
So it's very close to 1.

00:29:56.810 --> 00:29:59.780 align:middle line:84%
So that approximation-- but
if you do the calculation,

00:29:59.780 --> 00:30:01.670 align:middle line:90%
there is a little difference.

00:30:01.670 --> 00:30:09.400 align:middle line:84%
1 minus y is a little
different from minus log of 1.

00:30:09.400 --> 00:30:11.620 align:middle line:84%
So let me just keep
the minus log of y

00:30:11.620 --> 00:30:13.420 align:middle line:84%
without that
approximation, which--

00:30:13.420 --> 00:30:15.940 align:middle line:90%
since we can do logarithms.

00:30:15.940 --> 00:30:19.420 align:middle line:84%
And we can compute, therefore,
the partial pressure--

00:30:19.420 --> 00:30:26.200 align:middle line:84%
I mean, the osmotic
pressure using van 't Hoff.

00:30:26.200 --> 00:30:33.140 align:middle line:84%
RT-- I assume 298, just
to put a number here.

00:30:33.140 --> 00:30:35.690 align:middle line:84%
The specific volume--
here, it is correct.

00:30:35.690 --> 00:30:37.216 align:middle line:90%
This is with a double ii.

00:30:37.216 --> 00:30:41.450 align:middle line:84%
Double 1 is the specific
volume of pure water.

00:30:41.450 --> 00:30:47.280 align:middle line:84%
So it's 10 cubics meter, 10
to the minus 3 cubic meters

00:30:47.280 --> 00:30:49.820 align:middle line:90%
per kilogram.

00:30:49.820 --> 00:30:52.550 align:middle line:84%
But since we need
it in moles, we

00:30:52.550 --> 00:30:59.020 align:middle line:84%
have to multiply times the
molecular weight of water.

00:30:59.020 --> 00:31:01.090 align:middle line:84%
And in the end, you
can compute this

00:31:01.090 --> 00:31:04.840 align:middle line:84%
in terms of kilopascals
or bars, depending

00:31:04.840 --> 00:31:10.200 align:middle line:84%
on what makes you have
a better feeling of how

00:31:10.200 --> 00:31:12.180 align:middle line:90%
much this pressure is.

00:31:12.180 --> 00:31:15.690 align:middle line:84%
Atmospheric pressure
is 1 bar, more or less.

00:31:15.690 --> 00:31:22.380 align:middle line:84%
And so the osmotic pressure--
this osmotic pressure

00:31:22.380 --> 00:31:24.840 align:middle line:90%
is 30 bars.

00:31:24.840 --> 00:31:29.750 align:middle line:84%
You can measure it in
terms of a column of water.

00:31:29.750 --> 00:31:33.870 align:middle line:84%
The weight of the column
of water gives a pressure.

00:31:33.870 --> 00:31:37.490 align:middle line:90%
It's 300 meters.

00:31:37.490 --> 00:31:41.420 align:middle line:84%
Now, it turns out that
this number is not precise

00:31:41.420 --> 00:31:44.610 align:middle line:84%
because there-- we've made
some approximations here.

00:31:44.610 --> 00:31:49.070 align:middle line:84%
For example, we have
neglected these ion pairs.

00:31:49.070 --> 00:31:58.680 align:middle line:84%
The true number-- first of all,
it varies from place to place.

00:31:58.680 --> 00:32:00.300 align:middle line:84%
For example, in
the Mediterranean,

00:32:00.300 --> 00:32:04.430 align:middle line:84%
it's higher than in the
Atlantic or in the Pacific

00:32:04.430 --> 00:32:06.230 align:middle line:90%
or in the Arctic--

00:32:06.230 --> 00:32:07.340 align:middle line:90%
is even lower.

00:32:07.340 --> 00:32:09.950 align:middle line:84%
So there is a range
of values of salinity

00:32:09.950 --> 00:32:13.590 align:middle line:90%
that go between 30 and 38.

00:32:13.590 --> 00:32:19.800 align:middle line:84%
And the measured
osmotic pressures

00:32:19.800 --> 00:32:25.140 align:middle line:90%
are between 220 and 290 meters.

00:32:25.140 --> 00:32:26.980 align:middle line:90%
Still, it's not too bad.

00:32:26.980 --> 00:32:30.000 align:middle line:90%
But we can do better.

00:32:30.000 --> 00:32:34.440 align:middle line:84%
So here is-- sorry
for the fine print.

00:32:34.440 --> 00:32:45.200 align:middle line:84%
I will put on Canvas
this Excel file in which

00:32:45.200 --> 00:32:46.710 align:middle line:90%
I've done the calculation.

00:32:46.710 --> 00:32:52.790 align:middle line:84%
So this first column
gives that calculation

00:32:52.790 --> 00:32:54.660 align:middle line:90%
that we have just seen.

00:32:54.660 --> 00:33:03.890 align:middle line:84%
And that leads to 310
meters of osmotic pressure.

00:33:03.890 --> 00:33:09.420 align:middle line:84%
And we have assumed 100%
hydration and 0% ion pairs.

00:33:09.420 --> 00:33:13.010 align:middle line:84%
And also, we have
assumed no water.

00:33:13.010 --> 00:33:14.130 align:middle line:90%
I forgot to say it.

00:33:14.130 --> 00:33:19.070 align:middle line:84%
But we assume that I'm
not counting for the water

00:33:19.070 --> 00:33:22.280 align:middle line:84%
molecules that are lost
because they go and formate

00:33:22.280 --> 00:33:25.040 align:middle line:90%
these groups.

00:33:25.040 --> 00:33:30.740 align:middle line:84%
Now, in this other column,
I do the same calculation

00:33:30.740 --> 00:33:38.010 align:middle line:84%
by assuming six
molecules per ion,

00:33:38.010 --> 00:33:44.520 align:middle line:84%
and also by assuming that some
of these, 18% of the ions,

00:33:44.520 --> 00:33:47.920 align:middle line:84%
are grouped in ion
pairs, which are also

00:33:47.920 --> 00:33:51.970 align:middle line:90%
surrounded by six waters.

00:33:51.970 --> 00:33:55.370 align:middle line:84%
If you take that--
why did I take 18%?

00:33:55.370 --> 00:34:00.670 align:middle line:84%
That was just my only
fitting parameter

00:34:00.670 --> 00:34:06.180 align:middle line:84%
in order to get the
measured value of 288.

00:34:06.180 --> 00:34:08.670 align:middle line:84%
So if that were the only
effect, then my model

00:34:08.670 --> 00:34:11.909 align:middle line:84%
would get the actual
measured value

00:34:11.909 --> 00:34:15.179 align:middle line:84%
by assuming this
percentage of ion pairs

00:34:15.179 --> 00:34:20.040 align:middle line:84%
with respect to
the hydrated ions.

00:34:20.040 --> 00:34:24.989 align:middle line:84%
And these other two columns
give you the same calculation

00:34:24.989 --> 00:34:27.620 align:middle line:90%
with the same parameters.

00:34:27.620 --> 00:34:30.860 align:middle line:90%
But in the Mediterranean--

00:34:30.860 --> 00:34:32.489 align:middle line:90%
for example, in Naples--

00:34:32.489 --> 00:34:40.320 align:middle line:84%
you have this much, 316,
whereas in the Arctic Ocean,

00:34:40.320 --> 00:34:43.770 align:middle line:84%
it's 242 because
the-- because of the--

00:34:43.770 --> 00:34:49.694 align:middle line:84%
changes the percentage of
salt that is dissolved there.

00:34:49.694 --> 00:34:54.610 align:middle line:90%


00:34:54.610 --> 00:35:04.940 align:middle line:84%
Now, first of all, let's
view it in this way.

00:35:04.940 --> 00:35:08.960 align:middle line:90%


00:35:08.960 --> 00:35:14.795 align:middle line:90%
So imagine you have a long pipe.

00:35:14.795 --> 00:35:19.430 align:middle line:90%


00:35:19.430 --> 00:35:22.850 align:middle line:90%
So this is air.

00:35:22.850 --> 00:35:27.050 align:middle line:90%
This is z equals 0.

00:35:27.050 --> 00:35:30.415 align:middle line:90%
And this is z equals minus h.

00:35:30.415 --> 00:35:33.480 align:middle line:90%


00:35:33.480 --> 00:35:43.700 align:middle line:84%
So if the density of
seawater is approximately

00:35:43.700 --> 00:35:52.160 align:middle line:84%
equal to the density of water
and if I have here the pressure

00:35:52.160 --> 00:35:55.960 align:middle line:84%
Pa, here, what
pressure do I have?

00:35:55.960 --> 00:36:01.600 align:middle line:90%


00:36:01.600 --> 00:36:06.520 align:middle line:84%
That's the static Bernoulli
equation or whatever

00:36:06.520 --> 00:36:07.380 align:middle line:90%
you want to call it.

00:36:07.380 --> 00:36:11.810 align:middle line:90%


00:36:11.810 --> 00:36:22.480 align:middle line:84%
So now, this is a porous plug
semipermeable to just water.

00:36:22.480 --> 00:36:25.940 align:middle line:90%


00:36:25.940 --> 00:36:35.110 align:middle line:84%
So suppose I want
to have this level.

00:36:35.110 --> 00:36:38.515 align:middle line:84%
This is called-- let
me call it minus h1.

00:36:38.515 --> 00:36:43.100 align:middle line:90%


00:36:43.100 --> 00:36:44.540 align:middle line:90%
And this is h0.

00:36:44.540 --> 00:36:49.612 align:middle line:90%


00:36:49.612 --> 00:36:51.900 align:middle line:90%
So on the outside--

00:36:51.900 --> 00:36:53.850 align:middle line:90%
so below here--

00:36:53.850 --> 00:36:56.240 align:middle line:84%
I have the pressure
that I just said.

00:36:56.240 --> 00:36:59.570 align:middle line:84%
In the seawater at that
level, that's the pressure.

00:36:59.570 --> 00:37:02.930 align:middle line:90%
And it's h1.

00:37:02.930 --> 00:37:06.920 align:middle line:84%
On the other side
of the membrane,

00:37:06.920 --> 00:37:09.145 align:middle line:84%
I have a pressure
that is given by--

00:37:09.145 --> 00:37:12.280 align:middle line:90%


00:37:12.280 --> 00:37:16.240 align:middle line:84%
I assume that also
here, the pressure

00:37:16.240 --> 00:37:18.120 align:middle line:90%
is equal to atmospheric.

00:37:18.120 --> 00:37:22.460 align:middle line:84%
It's a bit higher because
of the column of air.

00:37:22.460 --> 00:37:25.450 align:middle line:84%
But we can neglect the
density of air, of course.

00:37:25.450 --> 00:37:31.520 align:middle line:90%
And so here, it's Pa plus rho.

00:37:31.520 --> 00:37:35.426 align:middle line:84%
It's, again, assumed that
it's the same rho, gh0.

00:37:35.426 --> 00:37:41.805 align:middle line:90%


00:37:41.805 --> 00:37:48.480 align:middle line:84%
So across the membrane, I
have a pressure difference.

00:37:48.480 --> 00:37:51.710 align:middle line:90%


00:37:51.710 --> 00:37:58.610 align:middle line:84%
So the pressure difference
is rho wgh1 minus h0.

00:37:58.610 --> 00:38:02.000 align:middle line:90%


00:38:02.000 --> 00:38:07.440 align:middle line:84%
And if I am at equilibrium
at the temperature that

00:38:07.440 --> 00:38:13.010 align:middle line:84%
prevails there, that is equal
to the osmotic pressure.

00:38:13.010 --> 00:38:16.800 align:middle line:90%


00:38:16.800 --> 00:38:25.580 align:middle line:84%
So that would be the equivalent
of rho wg times 288 meters.

00:38:25.580 --> 00:38:34.178 align:middle line:90%


00:38:34.178 --> 00:38:38.650 align:middle line:84%
So there is a huge
difference in pressure there.

00:38:38.650 --> 00:38:46.010 align:middle line:84%
So this porous plug must be
able to sustain that pressure.

00:38:46.010 --> 00:38:47.785 align:middle line:90%
And in fact, this is why.

00:38:47.785 --> 00:38:50.540 align:middle line:90%


00:38:50.540 --> 00:38:54.200 align:middle line:84%
As you know, in many
countries, the production

00:38:54.200 --> 00:39:02.030 align:middle line:84%
of freshwater from seawater
or from other saline waters

00:39:02.030 --> 00:39:07.940 align:middle line:90%
is done precisely this way.

00:39:07.940 --> 00:39:14.810 align:middle line:84%
It's not really done in
this-- you can do it in a more

00:39:14.810 --> 00:39:20.966 align:middle line:90%
practical way, this, by having--

00:39:20.966 --> 00:39:22.590 align:middle line:90%
you take some water.

00:39:22.590 --> 00:39:24.410 align:middle line:90%
You take a piston.

00:39:24.410 --> 00:39:26.570 align:middle line:90%
And then you push.

00:39:26.570 --> 00:39:30.330 align:middle line:84%
You generate a
high pressure here.

00:39:30.330 --> 00:39:31.423 align:middle line:90%
So this is seawater.

00:39:31.423 --> 00:39:35.540 align:middle line:90%


00:39:35.540 --> 00:39:42.000 align:middle line:84%
And out on the other side of the
membrane, you get freshwater.

00:39:42.000 --> 00:39:45.390 align:middle line:84%
In order to-- for
this to happen,

00:39:45.390 --> 00:39:51.120 align:middle line:90%
you have to impress a pressure.

00:39:51.120 --> 00:39:54.760 align:middle line:84%
If you want freshwater at
atmospheric pressure here,

00:39:54.760 --> 00:39:57.370 align:middle line:90%
you need at least 288.

00:39:57.370 --> 00:39:58.390 align:middle line:90%
You need more.

00:39:58.390 --> 00:40:02.070 align:middle line:84%
If you just do 288,
you get a drop.

00:40:02.070 --> 00:40:09.460 align:middle line:84%
If you put higher
pressure, you get a flow.

00:40:09.460 --> 00:40:12.140 align:middle line:84%
Of course, you need a
membrane that is capable,

00:40:12.140 --> 00:40:14.540 align:middle line:90%
like in the new graph--

00:40:14.540 --> 00:40:18.860 align:middle line:84%
capable of sustaining
those pressures.

00:40:18.860 --> 00:40:20.450 align:middle line:90%
And that's why you need--

00:40:20.450 --> 00:40:25.410 align:middle line:84%
part of the job is
to make the membrane.

00:40:25.410 --> 00:40:27.960 align:middle line:84%
And part of the job is
to sustain the pressure

00:40:27.960 --> 00:40:32.790 align:middle line:84%
so that the membrane can
resist that pressure.

00:40:32.790 --> 00:40:37.890 align:middle line:84%
And that is how
you make seawater.

00:40:37.890 --> 00:40:39.700 align:middle line:84%
Now, you could
also ask yourself,

00:40:39.700 --> 00:40:45.590 align:middle line:84%
what is the minimum work
needed to do this job,

00:40:45.590 --> 00:40:49.656 align:middle line:84%
because this is a very
energy-consuming activity?

00:40:49.656 --> 00:40:54.440 align:middle line:90%


00:40:54.440 --> 00:40:56.300 align:middle line:90%
And there are various--

00:40:56.300 --> 00:41:02.270 align:middle line:84%
there are more than
20 different methods

00:41:02.270 --> 00:41:05.960 align:middle line:84%
that people have devised in
order to separate freshwater

00:41:05.960 --> 00:41:07.880 align:middle line:84%
from seawater
because, of course,

00:41:07.880 --> 00:41:09.475 align:middle line:90%
it's a matter of survival.

00:41:09.475 --> 00:41:12.890 align:middle line:90%


00:41:12.890 --> 00:41:17.510 align:middle line:84%
People have developed
many ways, even

00:41:17.510 --> 00:41:21.480 align:middle line:84%
in the old times, when you
didn't have electricity.

00:41:21.480 --> 00:41:26.740 align:middle line:90%


00:41:26.740 --> 00:41:29.650 align:middle line:84%
Each of these methods
has an efficiency.

00:41:29.650 --> 00:41:32.830 align:middle line:84%
And if we are talking about
second-law efficiency,

00:41:32.830 --> 00:41:35.050 align:middle line:84%
like we like to
do in this course,

00:41:35.050 --> 00:41:38.740 align:middle line:84%
you have to compare what you
spend with the minimum you

00:41:38.740 --> 00:41:39.950 align:middle line:90%
could have spent.

00:41:39.950 --> 00:41:42.650 align:middle line:84%
And so what is the
minimum that we could--

00:41:42.650 --> 00:41:47.620 align:middle line:84%
we need to spend in order
to produce 1 liter of--

00:41:47.620 --> 00:41:49.730 align:middle line:90%
or 1 kilogram of freshwater?

00:41:49.730 --> 00:41:52.350 align:middle line:90%


00:41:52.350 --> 00:42:03.090 align:middle line:84%
And, well, you have it in
front here because suppose

00:42:03.090 --> 00:42:04.235 align:middle line:90%
I had this long pipe.

00:42:04.235 --> 00:42:10.120 align:middle line:90%


00:42:10.120 --> 00:42:12.740 align:middle line:90%
And this is my boat.

00:42:12.740 --> 00:42:17.170 align:middle line:90%
And the guy takes a bottle of--

00:42:17.170 --> 00:42:19.480 align:middle line:90%
an empty bottle, pulls it down--

00:42:19.480 --> 00:42:22.150 align:middle line:90%


00:42:22.150 --> 00:42:22.960 align:middle line:90%
or a bucket.

00:42:22.960 --> 00:42:24.620 align:middle line:90%
Maybe it's easier.

00:42:24.620 --> 00:42:25.930 align:middle line:90%
Take the bucket.

00:42:25.930 --> 00:42:28.390 align:middle line:84%
Fill it with this
water, which is fresh

00:42:28.390 --> 00:42:34.870 align:middle line:84%
because it's on the other side
of the membrane, and pull it up.

00:42:34.870 --> 00:42:38.230 align:middle line:84%
And then after a while,
this will be replenished.

00:42:38.230 --> 00:42:40.840 align:middle line:90%
And you can get some more.

00:42:40.840 --> 00:42:44.380 align:middle line:84%
So how much work do
you need to do that?

00:42:44.380 --> 00:42:48.885 align:middle line:84%
It's just the work for
lifting from here to there.

00:42:48.885 --> 00:42:53.150 align:middle line:90%


00:42:53.150 --> 00:42:56.370 align:middle line:90%
So it's rho gh, essentially.

00:42:56.370 --> 00:43:00.300 align:middle line:90%


00:43:00.300 --> 00:43:03.900 align:middle line:84%
And it turns out
that it is about--

00:43:03.900 --> 00:43:06.120 align:middle line:90%
I think I have it in the--

00:43:06.120 --> 00:43:08.220 align:middle line:90%
I think it's about 1--

00:43:08.220 --> 00:43:16.140 align:middle line:90%
or I mean-- well, g is about 10.

00:43:16.140 --> 00:43:19.740 align:middle line:90%
Rho is about 1.

00:43:19.740 --> 00:43:22.143 align:middle line:90%
So it's about 20.

00:43:22.143 --> 00:43:26.830 align:middle line:90%


00:43:26.830 --> 00:43:33.310 align:middle line:84%
And this, I think, is
kilojoules per kilogram.

00:43:33.310 --> 00:43:35.360 align:middle line:84%
Is it kilojoules
or is it joules?

00:43:35.360 --> 00:43:36.420 align:middle line:90%
That's the question.

00:43:36.420 --> 00:43:40.150 align:middle line:90%


00:43:40.150 --> 00:43:51.490 align:middle line:84%
Because rho is, say,
1 kilogram per cubic--

00:43:51.490 --> 00:43:52.480 align:middle line:90%
I'm sorry.

00:43:52.480 --> 00:43:58.750 align:middle line:84%
Rho is 10 to the 3
kilograms per cubic meter.

00:43:58.750 --> 00:44:06.850 align:middle line:84%
g-- for simplicity, let's call
it 10 meters per second squared.

00:44:06.850 --> 00:44:11.570 align:middle line:84%
And then the other
is 288 meters.

00:44:11.570 --> 00:44:24.250 align:middle line:90%


00:44:24.250 --> 00:44:25.670 align:middle line:90%
Anybody wants to help?

00:44:25.670 --> 00:44:27.041 align:middle line:90%
[CHUCKLING]

00:44:27.041 --> 00:44:30.240 align:middle line:90%


00:44:30.240 --> 00:44:33.360 align:middle line:84%
So this is kilogram
per-- that's newton--

00:44:33.360 --> 00:44:34.420 align:middle line:90%
kilopascals.

00:44:34.420 --> 00:44:37.140 align:middle line:84%
No, this is pascals
times 10 to the 3--

00:44:37.140 --> 00:44:39.990 align:middle line:90%
is kilopascals per meter.

00:44:39.990 --> 00:44:43.140 align:middle line:90%
I think it's kilojoules.

00:44:43.140 --> 00:44:44.410 align:middle line:90%
Well, you check it.

00:44:44.410 --> 00:44:48.190 align:middle line:90%


00:44:48.190 --> 00:44:53.870 align:middle line:84%
There are other ways
in which, for example,

00:44:53.870 --> 00:44:56.380 align:middle line:90%
if you didn't want to put--

00:44:56.380 --> 00:44:58.100 align:middle line:84%
because this is a
commercial product.

00:44:58.100 --> 00:45:06.490 align:middle line:84%
But if you go and
look up a solar still,

00:45:06.490 --> 00:45:15.130 align:middle line:84%
you find an inflatable
survival object,

00:45:15.130 --> 00:45:17.930 align:middle line:90%
which has something like this.

00:45:17.930 --> 00:45:22.620 align:middle line:84%
So this is an inflatable
donut-shaped thing

00:45:22.620 --> 00:45:24.940 align:middle line:90%
with a plastic dome.

00:45:24.940 --> 00:45:28.300 align:middle line:90%
And here is the seawater.

00:45:28.300 --> 00:45:33.790 align:middle line:84%
So you get sunlight
coming in, which

00:45:33.790 --> 00:45:39.250 align:middle line:84%
makes some of the water inside
evaporate, like in a greenhouse.

00:45:39.250 --> 00:45:43.300 align:middle line:84%
The water vapor that
evaporates-- the salt

00:45:43.300 --> 00:45:44.090 align:middle line:90%
remains here.

00:45:44.090 --> 00:45:46.340 align:middle line:84%
So what evaporates
is just water.

00:45:46.340 --> 00:45:52.240 align:middle line:84%
So the water vapor goes and
forms and condenses on here

00:45:52.240 --> 00:45:55.360 align:middle line:84%
because you also, typically,
have some breeze or some wind

00:45:55.360 --> 00:45:57.770 align:middle line:90%
that will cool that dome.

00:45:57.770 --> 00:46:03.370 align:middle line:84%
And then the drops
by capillarity

00:46:03.370 --> 00:46:08.720 align:middle line:84%
stay and get collected
on this side here.

00:46:08.720 --> 00:46:11.510 align:middle line:84%
So eventually, you get
some water on this side.

00:46:11.510 --> 00:46:15.170 align:middle line:84%
And you put a little
pipe and a little bottle,

00:46:15.170 --> 00:46:17.730 align:middle line:90%
and you collect your water.

00:46:17.730 --> 00:46:21.970 align:middle line:84%
So that's a survival
kit for the fishermen.

00:46:21.970 --> 00:46:27.620 align:middle line:84%
And what kind of
efficiency would that have?

00:46:27.620 --> 00:46:30.550 align:middle line:90%


00:46:30.550 --> 00:46:33.130 align:middle line:84%
That will also
help me figure out

00:46:33.130 --> 00:46:36.990 align:middle line:84%
if this k should
be there or not.

00:46:36.990 --> 00:46:40.590 align:middle line:84%
How much energy, how
much solar energy,

00:46:40.590 --> 00:46:44.615 align:middle line:84%
do I need in order to
evaporate 1 kilogram of water?

00:46:44.615 --> 00:46:48.610 align:middle line:90%


00:46:48.610 --> 00:46:53.780 align:middle line:84%
It's the enthalpy of
vaporization of water,

00:46:53.780 --> 00:47:03.610 align:middle line:84%
which is of the order of
2,200 kilojoules per kilogram.

00:47:03.610 --> 00:47:06.760 align:middle line:84%
So I take this k
away because I know

00:47:06.760 --> 00:47:16.090 align:middle line:84%
that the answer is that that
thing has an efficiency of 0.1%.

00:47:16.090 --> 00:47:22.520 align:middle line:84%
So it consumes 1,000
times the minimum.

00:47:22.520 --> 00:47:27.880 align:middle line:90%


00:47:27.880 --> 00:47:33.200 align:middle line:90%
These ones, the commercial--

00:47:33.200 --> 00:47:33.700 align:middle line:90%
let's see.

00:47:33.700 --> 00:47:36.040 align:middle line:90%
Do I have it somewhere?

00:47:36.040 --> 00:47:37.420 align:middle line:90%
Yeah.

00:47:37.420 --> 00:47:51.890 align:middle line:84%
The commercial ones-- these are
the most typical, most important

00:47:51.890 --> 00:47:55.220 align:middle line:90%
industrial technologies.

00:47:55.220 --> 00:47:58.260 align:middle line:84%
One is called
pressure-retarded osmosis,

00:47:58.260 --> 00:48:00.620 align:middle line:90%
which is exactly like that.

00:48:00.620 --> 00:48:07.190 align:middle line:84%
And the other is called
reverse electrodialysis,

00:48:07.190 --> 00:48:13.080 align:middle line:84%
which uses the fact that since
the ions are electrically--

00:48:13.080 --> 00:48:16.340 align:middle line:84%
so if you put an electric
field, you can separate the ions

00:48:16.340 --> 00:48:20.310 align:middle line:84%
and, therefore, take
them away from the water.

00:48:20.310 --> 00:48:24.740 align:middle line:84%
So that's one way of separating,
which is less efficient.

00:48:24.740 --> 00:48:26.650 align:middle line:84%
There is a factor
of 2 of inefficiency

00:48:26.650 --> 00:48:28.400 align:middle line:84%
there because you have
to separate the two

00:48:28.400 --> 00:48:31.560 align:middle line:90%
ions for each amount of water.

00:48:31.560 --> 00:48:34.520 align:middle line:84%
And so I think that
is the reason why

00:48:34.520 --> 00:48:43.070 align:middle line:84%
there is this factor of 2 in the
theoretical maximum efficiency

00:48:43.070 --> 00:48:47.540 align:middle line:84%
that-- or minimum power
that you need to spend.

00:48:47.540 --> 00:48:58.260 align:middle line:84%
Here, it's for-- to treat
1,000 cubic meters per second.

00:48:58.260 --> 00:49:02.650 align:middle line:84%
The reason I was
interested in this

00:49:02.650 --> 00:49:17.530 align:middle line:84%
is because of the fact
that you see here,

00:49:17.530 --> 00:49:22.480 align:middle line:84%
you have freshwater
in equilibrium with--

00:49:22.480 --> 00:49:24.530 align:middle line:84%
so this is an
equilibrium situation.

00:49:24.530 --> 00:49:27.030 align:middle line:90%
So that's the freshwater.

00:49:27.030 --> 00:49:36.900 align:middle line:84%
If you add some freshwater
from above, that freshwater--

00:49:36.900 --> 00:49:39.300 align:middle line:84%
this will add to the
pressure on the membrane.

00:49:39.300 --> 00:49:44.050 align:middle line:84%
And therefore, the water
will flow into the ocean.

00:49:44.050 --> 00:49:48.000 align:middle line:90%
So that's one way.

00:49:48.000 --> 00:49:53.550 align:middle line:84%
That is-- would be
an intelligent way

00:49:53.550 --> 00:49:58.756 align:middle line:90%
to put freshwater in the sea.

00:49:58.756 --> 00:50:00.960 align:middle line:84%
So when do we put
freshwater in the sea?

00:50:00.960 --> 00:50:02.420 align:middle line:90%
Why would you want to do it.

00:50:02.420 --> 00:50:04.860 align:middle line:84%
Not the fisherman, because
he wants to drink it.

00:50:04.860 --> 00:50:10.280 align:middle line:84%
But the rivers-- rivers
bring freshwater to the sea.

00:50:10.280 --> 00:50:16.130 align:middle line:84%
And usually, they just
dump it at the same level

00:50:16.130 --> 00:50:20.020 align:middle line:90%
that's dissipative.

00:50:20.020 --> 00:50:24.530 align:middle line:84%
The best way would be to
take that water, for example,

00:50:24.530 --> 00:50:25.970 align:middle line:90%
one bucket at a time.

00:50:25.970 --> 00:50:29.800 align:middle line:84%
I take that water
from the river.

00:50:29.800 --> 00:50:31.580 align:middle line:90%
I put it in my bucket.

00:50:31.580 --> 00:50:35.690 align:middle line:90%
Then I let the bucket go down.

00:50:35.690 --> 00:50:40.900 align:middle line:84%
So while doing that with a
pulley, I have an engine--

00:50:40.900 --> 00:50:43.300 align:middle line:90%
will produce electricity.

00:50:43.300 --> 00:50:46.510 align:middle line:84%
And that is-- we
know how much work.

00:50:46.510 --> 00:50:50.590 align:middle line:84%
We get this much work
per kilogram of water

00:50:50.590 --> 00:50:51.700 align:middle line:90%
that we put down.

00:50:51.700 --> 00:50:55.310 align:middle line:84%
And then at this level,
when it is at this level,

00:50:55.310 --> 00:50:59.220 align:middle line:84%
it will cross the
membrane spontaneously.

00:50:59.220 --> 00:51:03.990 align:middle line:84%
And I don't need a big
difference in pressure

00:51:03.990 --> 00:51:06.690 align:middle line:90%
to make it flow.

00:51:06.690 --> 00:51:11.680 align:middle line:84%
Of course, I do need some
difference in pressure.

00:51:11.680 --> 00:51:15.870 align:middle line:84%
Otherwise, the flow will not
be able to win the pressure

00:51:15.870 --> 00:51:19.860 align:middle line:90%
drops in the porous plug.

00:51:19.860 --> 00:51:25.330 align:middle line:90%
So I cannot obtain 100% of this.

00:51:25.330 --> 00:51:28.320 align:middle line:84%
This would be the
zero-power situation.

00:51:28.320 --> 00:51:29.380 align:middle line:90%
If I need power--

00:51:29.380 --> 00:51:31.530 align:middle line:84%
so I need to do it
at a certain rate--

00:51:31.530 --> 00:51:36.420 align:middle line:84%
I need, instead of being here,
let's say, being halfway.

00:51:36.420 --> 00:51:40.710 align:middle line:84%
So I use this much head in
order to push the freshwater

00:51:40.710 --> 00:51:43.320 align:middle line:90%
into the sea.

00:51:43.320 --> 00:51:49.620 align:middle line:84%
In any case, this is what
is called blue energy.

00:51:49.620 --> 00:51:55.630 align:middle line:90%
And it is an enormous potential.

00:51:55.630 --> 00:52:00.790 align:middle line:84%
And it is, of course, a
renewable kind of energy.

00:52:00.790 --> 00:52:04.690 align:middle line:84%
It has an enormous potential
if you look at what

00:52:04.690 --> 00:52:06.645 align:middle line:90%
the rates are for rivers.

00:52:06.645 --> 00:52:10.780 align:middle line:90%


00:52:10.780 --> 00:52:18.340 align:middle line:84%
So even if I take just this
small contribution for the--

00:52:18.340 --> 00:52:24.340 align:middle line:84%
for Po River in Italy, which
is only 6,000 cubic meters per

00:52:24.340 --> 00:52:34.520 align:middle line:84%
second, you see with a
pressure-retarded osmosis power

00:52:34.520 --> 00:52:41.680 align:middle line:84%
plant, we could make 6
times 3,000 megawatts.

00:52:41.680 --> 00:52:44.160 align:middle line:90%
So that's 18,000 megawatts.

00:52:44.160 --> 00:52:46.175 align:middle line:90%
That's a lot.

00:52:46.175 --> 00:52:55.040 align:middle line:90%


00:52:55.040 --> 00:53:00.840 align:middle line:84%
And people have
tried to do this.

00:53:00.840 --> 00:53:03.240 align:middle line:90%
There are pilot plants.

00:53:03.240 --> 00:53:06.120 align:middle line:84%
But the problem with these
technologies that have to

00:53:06.120 --> 00:53:06.990 align:middle line:90%
do with--

00:53:06.990 --> 00:53:16.930 align:middle line:84%
that need membranes is
that membranes get clogged.

00:53:16.930 --> 00:53:21.970 align:middle line:84%
Freshwater from the river
is not really just water.

00:53:21.970 --> 00:53:23.140 align:middle line:90%
There is mud.

00:53:23.140 --> 00:53:25.120 align:middle line:90%
There is dust.

00:53:25.120 --> 00:53:30.580 align:middle line:84%
So you need to keep
these membranes clean.

00:53:30.580 --> 00:53:34.270 align:middle line:84%
You need huge surfaces
in order to treat

00:53:34.270 --> 00:53:37.010 align:middle line:90%
these kind of flow rates.

00:53:37.010 --> 00:53:42.400 align:middle line:84%
And therefore, up to now,
it hasn't developed yet

00:53:42.400 --> 00:53:48.250 align:middle line:84%
in any reasonable
even small fraction

00:53:48.250 --> 00:53:50.793 align:middle line:90%
of the electricity produced.

00:53:50.793 --> 00:53:53.630 align:middle line:90%


00:53:53.630 --> 00:53:56.510 align:middle line:84%
This map also shows the
salinity differences

00:53:56.510 --> 00:54:00.600 align:middle line:90%
in the various oceans and seas.

00:54:00.600 --> 00:54:02.420 align:middle line:90%
So here is the Mediterranean.

00:54:02.420 --> 00:54:06.830 align:middle line:90%
That's about 38.

00:54:06.830 --> 00:54:08.930 align:middle line:90%
Here is the Arctic--

00:54:08.930 --> 00:54:12.960 align:middle line:90%
is below 33.

00:54:12.960 --> 00:54:16.845 align:middle line:90%
Now, the idea of--

00:54:16.845 --> 00:54:19.890 align:middle line:90%


00:54:19.890 --> 00:54:28.220 align:middle line:84%
this idea-- using the
pressure-retarded osmosis

00:54:28.220 --> 00:54:30.320 align:middle line:90%
is this.

00:54:30.320 --> 00:54:33.890 align:middle line:84%
There was a power
plant in Tofte, Norway

00:54:33.890 --> 00:54:35.850 align:middle line:90%
that tried to do it.

00:54:35.850 --> 00:54:38.840 align:middle line:90%
But I think they closed.

00:54:38.840 --> 00:54:41.180 align:middle line:84%
I think that there
is somebody else

00:54:41.180 --> 00:54:45.550 align:middle line:84%
who is starting another
project in trying these things.

00:54:45.550 --> 00:54:49.330 align:middle line:90%
So you get water from the river.

00:54:49.330 --> 00:54:52.030 align:middle line:84%
And instead of putting
it directly-- so

00:54:52.030 --> 00:54:54.640 align:middle line:90%
I put a nice wall here.

00:54:54.640 --> 00:54:59.140 align:middle line:84%
And the water does have
to go a different path.

00:54:59.140 --> 00:55:02.720 align:middle line:90%
It has to go into these pipes.

00:55:02.720 --> 00:55:07.460 align:middle line:84%
So these are pipes made
with a porous material,

00:55:07.460 --> 00:55:12.250 align:middle line:84%
permeable only to freshwater,
to just the water molecule.

00:55:12.250 --> 00:55:15.910 align:middle line:90%


00:55:15.910 --> 00:55:21.620 align:middle line:90%
So what I do is to put inside--

00:55:21.620 --> 00:55:37.220 align:middle line:84%
on the inside, I put seawater,
pressurized, pressurized,

00:55:37.220 --> 00:55:42.860 align:middle line:84%
not to the osmotic pressure
because otherwise, we

00:55:42.860 --> 00:55:48.350 align:middle line:84%
would have no flow,
but to a pressure which

00:55:48.350 --> 00:55:56.300 align:middle line:90%
is halfway the osmotic value.

00:55:56.300 --> 00:55:58.790 align:middle line:84%
That way, there will
be the driving force

00:55:58.790 --> 00:56:04.810 align:middle line:84%
that would push water
through the porous medium.

00:56:04.810 --> 00:56:10.120 align:middle line:84%
And so this pipe
will need to be in--

00:56:10.120 --> 00:56:13.330 align:middle line:84%
with an enlarging cross-section
so as to accommodate

00:56:13.330 --> 00:56:15.920 align:middle line:90%
this increasing flow of water.

00:56:15.920 --> 00:56:19.580 align:middle line:84%
So here is the
seawater that gets in.

00:56:19.580 --> 00:56:23.020 align:middle line:84%
And what gets out
is the seawater

00:56:23.020 --> 00:56:26.050 align:middle line:84%
plus the water, freshwater,
that has been added.

00:56:26.050 --> 00:56:28.870 align:middle line:90%
And then it goes here.

00:56:28.870 --> 00:56:31.460 align:middle line:84%
Now, of course, in order
to pressurize the seawater,

00:56:31.460 --> 00:56:33.855 align:middle line:84%
I have to consume more
energy with a pump.

00:56:33.855 --> 00:56:36.470 align:middle line:90%


00:56:36.470 --> 00:56:39.770 align:middle line:90%
And that pump is--

00:56:39.770 --> 00:56:48.300 align:middle line:84%
the work-- the power is the
product of the pressurization

00:56:48.300 --> 00:56:52.535 align:middle line:84%
that I have managed to obtain
times the volume flow rate.

00:56:52.535 --> 00:56:55.280 align:middle line:90%


00:56:55.280 --> 00:56:58.775 align:middle line:90%
And there should be a dot there.

00:56:58.775 --> 00:57:02.895 align:middle line:90%


00:57:02.895 --> 00:57:10.240 align:middle line:84%
What flows out is, again,
at that pressure, perhaps

00:57:10.240 --> 00:57:14.470 align:middle line:84%
a little less because
of pressure drops.

00:57:14.470 --> 00:57:19.720 align:middle line:84%
And so I don't just
put it in water.

00:57:19.720 --> 00:57:22.580 align:middle line:84%
I put it in a turbine,
hydraulic turbine,

00:57:22.580 --> 00:57:26.200 align:middle line:84%
which is going to elaborate,
though, a higher volume

00:57:26.200 --> 00:57:32.880 align:middle line:84%
flow rate because it's not only
recuperating the volume flow

00:57:32.880 --> 00:57:35.490 align:middle line:84%
rate that I have
pumped, but it also

00:57:35.490 --> 00:57:40.690 align:middle line:84%
gets the volume of freshwater
that is being added.

00:57:40.690 --> 00:57:43.800 align:middle line:84%
And so it's on the difference
between this volume flow rate

00:57:43.800 --> 00:57:44.820 align:middle line:90%
that I gain.

00:57:44.820 --> 00:57:47.970 align:middle line:84%
And again, the
net power is going

00:57:47.970 --> 00:57:52.710 align:middle line:84%
to be the difference between
what I obtain from the turbine

00:57:52.710 --> 00:57:56.790 align:middle line:90%
and what I spent from the pump.

00:57:56.790 --> 00:58:00.880 align:middle line:84%
So in principle, this
is relatively simple.

00:58:00.880 --> 00:58:08.740 align:middle line:84%
But again, the problem
is these porous medium

00:58:08.740 --> 00:58:10.606 align:middle line:90%
and porous membranes.

00:58:10.606 --> 00:58:26.810 align:middle line:90%


00:58:26.810 --> 00:58:28.790 align:middle line:84%
This number that
we obtained here

00:58:28.790 --> 00:58:37.700 align:middle line:84%
is the work of separation of
some solvent from this dilute

00:58:37.700 --> 00:58:38.200 align:middle line:90%
solution.

00:58:38.200 --> 00:58:42.190 align:middle line:90%


00:58:42.190 --> 00:58:48.000 align:middle line:84%
In general, the problem
of separating stuff

00:58:48.000 --> 00:58:50.390 align:middle line:84%
is important for
many applications.

00:58:50.390 --> 00:58:54.670 align:middle line:84%
For example, in the
nuclear power industry,

00:58:54.670 --> 00:58:59.640 align:middle line:90%
you have to separate--

00:58:59.640 --> 00:59:05.870 align:middle line:84%
in order to concentrate
uranium-235, you do--

00:59:05.870 --> 00:59:07.170 align:middle line:90%
you need to--

00:59:07.170 --> 00:59:08.685 align:middle line:90%
you make it into gas--

00:59:08.685 --> 00:59:15.720 align:middle line:90%


00:59:15.720 --> 00:59:17.370 align:middle line:90%
hexafluorine of uranium.

00:59:17.370 --> 00:59:21.870 align:middle line:84%
And then you separate
the gas using the fact

00:59:21.870 --> 00:59:27.520 align:middle line:84%
that the molecules are different
because they have a different--

00:59:27.520 --> 00:59:34.460 align:middle line:84%
they're different
isotopes, the 238 from 235.

00:59:34.460 --> 00:59:42.280 align:middle line:84%
And so that is consuming energy
in order to do this separation.

00:59:42.280 --> 00:59:47.280 align:middle line:90%


00:59:47.280 --> 00:59:55.340 align:middle line:84%
So we know that we did
something very similar earlier.

00:59:55.340 --> 00:59:58.580 align:middle line:84%
So I'm going to go quickly
on these energy balances

00:59:58.580 --> 00:59:59.910 align:middle line:90%
and entropy balances.

00:59:59.910 --> 01:00:03.360 align:middle line:84%
But the idea here is
that we have a mixture,

01:00:03.360 --> 01:00:07.840 align:middle line:90%
in general, given composition.

01:00:07.840 --> 01:00:10.950 align:middle line:84%
We do it in a given
temperature and pressure.

01:00:10.950 --> 01:00:13.830 align:middle line:84%
So now I want to operate at a
given temperature and pressure,

01:00:13.830 --> 01:00:15.480 align:middle line:84%
which is-- could
be, for example,

01:00:15.480 --> 01:00:19.370 align:middle line:90%
the atmospheric reservoir.

01:00:19.370 --> 01:00:25.996 align:middle line:84%
And I want to obtain fully
separated, pure components.

01:00:25.996 --> 01:00:29.940 align:middle line:90%


01:00:29.940 --> 01:00:31.620 align:middle line:84%
Of course, in
order to do that, I

01:00:31.620 --> 01:00:34.440 align:middle line:84%
need to spend some
work that's represented

01:00:34.440 --> 01:00:38.330 align:middle line:90%
by the weight that goes down.

01:00:38.330 --> 01:00:43.510 align:middle line:84%
So I do volume balance
for the overall system,

01:00:43.510 --> 01:00:47.280 align:middle line:90%
including the reservoir.

01:00:47.280 --> 01:00:50.710 align:middle line:84%
Here I call B and A.
B is the final state.

01:00:50.710 --> 01:00:52.540 align:middle line:90%
And A is the initial state.

01:00:52.540 --> 01:00:56.040 align:middle line:90%


01:00:56.040 --> 01:01:00.220 align:middle line:84%
So the energy balance,
the entropy balance,

01:01:00.220 --> 01:01:06.140 align:middle line:84%
and then the equation
for the stable equilibria

01:01:06.140 --> 01:01:13.530 align:middle line:84%
of this reservoir with variable
volume, which is this--

01:01:13.530 --> 01:01:16.530 align:middle line:84%
then I combine, as
usual, these equations

01:01:16.530 --> 01:01:21.970 align:middle line:84%
by eliminating the
reservoir variables.

01:01:21.970 --> 01:01:24.850 align:middle line:84%
And here, I'm also using
the fact that the final--

01:01:24.850 --> 01:01:29.440 align:middle line:84%
since we are in equilibrium
with the reservoir,

01:01:29.440 --> 01:01:32.440 align:middle line:84%
it fixes also the
final temperatures

01:01:32.440 --> 01:01:35.420 align:middle line:84%
of the pure
components at the end.

01:01:35.420 --> 01:01:39.550 align:middle line:90%
So TB, TA is equal to TR.

01:01:39.550 --> 01:01:44.280 align:middle line:90%


01:01:44.280 --> 01:01:47.460 align:middle line:84%
And the linear combination
of these balances

01:01:47.460 --> 01:01:55.490 align:middle line:84%
yields this expression
for the work,

01:01:55.490 --> 01:01:57.930 align:middle line:84%
which is one of those
characteristic functions.

01:01:57.930 --> 01:02:01.045 align:middle line:84%
I think it's the one that
we call with the letter phi.

01:02:01.045 --> 01:02:05.070 align:middle line:90%


01:02:05.070 --> 01:02:09.970 align:middle line:84%
And then there is a plus T
times the entropy generated.

01:02:09.970 --> 01:02:12.840 align:middle line:84%
So if I do it in an
irreversible process,

01:02:12.840 --> 01:02:18.760 align:middle line:90%
this will make me consume more.

01:02:18.760 --> 01:02:22.640 align:middle line:84%
Well, let's talk about the
limit in which I do the best.

01:02:22.640 --> 01:02:24.020 align:middle line:90%
So this is not there.

01:02:24.020 --> 01:02:26.420 align:middle line:84%
So that would be
the reversible case.

01:02:26.420 --> 01:02:30.400 align:middle line:84%
And then I recognize that this
linear combination of properties

01:02:30.400 --> 01:02:32.210 align:middle line:90%
is just the Gibbs free energy.

01:02:32.210 --> 01:02:35.380 align:middle line:84%
So essentially, the minimum work
for separating at constant T

01:02:35.380 --> 01:02:39.040 align:middle line:84%
and P is the difference
between the Gibbs free energy

01:02:39.040 --> 01:02:42.220 align:middle line:84%
of the final separated state
and the Gibbs free energy

01:02:42.220 --> 01:02:44.715 align:middle line:90%
of the initial mixture.

01:02:44.715 --> 01:02:49.000 align:middle line:90%


01:02:49.000 --> 01:02:54.490 align:middle line:84%
And now, since we know how to
do things, we know how to--

01:02:54.490 --> 01:02:58.340 align:middle line:84%
we know, first of all, that
the Gibbs free energy--

01:02:58.340 --> 01:03:01.820 align:middle line:84%
now, in general, it's
not an additive property.

01:03:01.820 --> 01:03:06.010 align:middle line:84%
But if all the systems
are at the same T and P,

01:03:06.010 --> 01:03:09.880 align:middle line:84%
then the Gibbs free
energy is additive.

01:03:09.880 --> 01:03:14.740 align:middle line:84%
And so in the final state,
the Gibbs free energy

01:03:14.740 --> 01:03:17.620 align:middle line:84%
of the composite is equal to
simply the Gibbs free energy

01:03:17.620 --> 01:03:22.240 align:middle line:84%
of the sum of the Gibbs
free energy of the parts.

01:03:22.240 --> 01:03:24.790 align:middle line:84%
And again, here
I've used the fact

01:03:24.790 --> 01:03:27.370 align:middle line:84%
that the Gibbs free energy
of the pure component

01:03:27.370 --> 01:03:32.620 align:middle line:84%
is also equal to the chemical
potential of the pure component,

01:03:32.620 --> 01:03:35.530 align:middle line:84%
whereas for the
mixture, the Gibbs

01:03:35.530 --> 01:03:38.170 align:middle line:84%
free energy due to
the Euler relation

01:03:38.170 --> 01:03:41.770 align:middle line:84%
is related to the
chemical potentials.

01:03:41.770 --> 01:03:46.130 align:middle line:84%
So it's the weighted sum
of the chemical potentials.

01:03:46.130 --> 01:03:49.640 align:middle line:84%
And up to here, we
made no assumptions.

01:03:49.640 --> 01:03:54.650 align:middle line:84%
If we further make an assumption
of either being in the case

01:03:54.650 --> 01:04:01.390 align:middle line:84%
of a Gibbs-Dalton mixture
of ideal gases or an ideal

01:04:01.390 --> 01:04:05.270 align:middle line:84%
solution-- and this time,
it's ideal for all components,

01:04:05.270 --> 01:04:06.210 align:middle line:90%
not just the solvent--

01:04:06.210 --> 01:04:09.390 align:middle line:90%


01:04:09.390 --> 01:04:14.920 align:middle line:84%
then if we apply the rules that
we have seen so many times now,

01:04:14.920 --> 01:04:17.640 align:middle line:84%
this gives us a
formula, which is

01:04:17.640 --> 01:04:20.500 align:middle line:84%
similar to the
entropy of mixing.

01:04:20.500 --> 01:04:24.108 align:middle line:90%


01:04:24.108 --> 01:04:31.530 align:middle line:84%
So suppose, for example,
that we want to separate air.

01:04:31.530 --> 01:04:33.940 align:middle line:84%
We want to separate air
into its components.

01:04:33.940 --> 01:04:37.160 align:middle line:90%


01:04:37.160 --> 01:04:39.870 align:middle line:84%
Now, air is a mixture
of many gases.

01:04:39.870 --> 01:04:44.130 align:middle line:84%
It's not just
nitrogen and oxygen.

01:04:44.130 --> 01:04:51.522 align:middle line:84%
The typical composition
at sea level is this one--

01:04:51.522 --> 01:04:59.360 align:middle line:84%
78% of nitrogen, almost 21%
of oxygen, 0.9%, almost 1%,

01:04:59.360 --> 01:05:04.140 align:middle line:90%
of argon, and 0.04%--

01:05:04.140 --> 01:05:07.005 align:middle line:90%
these are the 400 PPMs of CO2.

01:05:07.005 --> 01:05:17.920 align:middle line:90%


01:05:17.920 --> 01:05:20.055 align:middle line:90%
I want to compute this--

01:05:20.055 --> 01:05:23.050 align:middle line:90%


01:05:23.050 --> 01:05:25.020 align:middle line:90%
the summation.

01:05:25.020 --> 01:05:28.430 align:middle line:90%
And it's simple.

01:05:28.430 --> 01:05:37.700 align:middle line:84%
And then I want to divide
it also by nM so that I get

01:05:37.700 --> 01:05:42.070 align:middle line:90%
the result per kilogram of air.

01:05:42.070 --> 01:05:44.467 align:middle line:84%
So I have to divide by
the molecular weight.

01:05:44.467 --> 01:05:46.300 align:middle line:84%
And the molecular weight
is the weighted sum

01:05:46.300 --> 01:05:51.370 align:middle line:84%
of the molecular weights of the
various constituents weighted

01:05:51.370 --> 01:05:52.670 align:middle line:90%
on the mole fractions.

01:05:52.670 --> 01:05:57.260 align:middle line:84%
So it's 28.97 with
this composition.

01:05:57.260 --> 01:05:59.420 align:middle line:84%
And then I choose a
temperature-- say,

01:05:59.420 --> 01:06:05.390 align:middle line:84%
298 K. So the minimum
work to separate air

01:06:05.390 --> 01:06:14.710 align:middle line:84%
into its components, 1
kilogram, is 48.5 kilojoules.

01:06:14.710 --> 01:06:17.400 align:middle line:84%
We could have modeled
air in a simpler way.

01:06:17.400 --> 01:06:20.440 align:middle line:90%
Suppose air is just made of--

01:06:20.440 --> 01:06:24.010 align:middle line:84%
suppose we have a mixture in
which I have only nitrogen

01:06:24.010 --> 01:06:27.340 align:middle line:84%
and oxygen with the same
relative composition

01:06:27.340 --> 01:06:30.880 align:middle line:84%
as they have in the
actual air, but there

01:06:30.880 --> 01:06:33.310 align:middle line:90%
are no other components.

01:06:33.310 --> 01:06:40.390 align:middle line:84%
If it were only those two,
then I do the same calculation.

01:06:40.390 --> 01:06:45.340 align:middle line:84%
I see that the work
for separation is 44.

01:06:45.340 --> 01:06:49.520 align:middle line:90%
It's quite a bit of difference.

01:06:49.520 --> 01:06:54.140 align:middle line:84%
And these additional about
4 kilojoules per kilogram

01:06:54.140 --> 01:07:00.230 align:middle line:84%
here are needed because here,
I have insisted in separating

01:07:00.230 --> 01:07:03.260 align:middle line:90%
all, including argon and CO2.

01:07:03.260 --> 01:07:07.970 align:middle line:84%
And those are very
few molecules.

01:07:07.970 --> 01:07:16.050 align:middle line:84%
And they require a lot for
separating per unit of amount

01:07:16.050 --> 01:07:19.060 align:middle line:90%
that you want to separate.

01:07:19.060 --> 01:07:23.470 align:middle line:84%
So this formula applies
not only in the gas phase.

01:07:23.470 --> 01:07:28.430 align:middle line:84%
You can also apply for an
ideal solution liquid phase.

01:07:28.430 --> 01:07:32.660 align:middle line:84%
For example, let's
take a mixture

01:07:32.660 --> 01:07:41.870 align:middle line:84%
of water and ethanol, which, at
the appropriate mole fractions,

01:07:41.870 --> 01:07:43.645 align:middle line:90%
is like wine--

01:07:43.645 --> 01:07:46.160 align:middle line:90%


01:07:46.160 --> 01:07:47.820 align:middle line:90%
a model for wine, of course.

01:07:47.820 --> 01:07:49.255 align:middle line:90%
It's not just these elements.

01:07:49.255 --> 01:07:52.430 align:middle line:90%


01:07:52.430 --> 01:07:58.400 align:middle line:84%
So if I take a wine 14
degrees by volume, that's--

01:07:58.400 --> 01:08:01.820 align:middle line:84%
phi is the mole fraction--
it's-- is the volume fraction

01:08:01.820 --> 01:08:04.435 align:middle line:90%
of alcohol in wine.

01:08:04.435 --> 01:08:08.850 align:middle line:90%


01:08:08.850 --> 01:08:10.920 align:middle line:84%
If you do the
calculations to go--

01:08:10.920 --> 01:08:15.550 align:middle line:84%
to pass from volume
fraction to mole fraction,

01:08:15.550 --> 01:08:19.590 align:middle line:90%
it's 4.7 mole fraction.

01:08:19.590 --> 01:08:22.229 align:middle line:84%
And if you want to
separate entirely

01:08:22.229 --> 01:08:27.420 align:middle line:84%
the water from the alcohol,
you apply this formula.

01:08:27.420 --> 01:08:31.450 align:middle line:84%
And again, if you want to do
it for 1 kilogram of wine,

01:08:31.450 --> 01:08:36.524 align:middle line:84%
it's 24 kilojoules,
what you spend.

01:08:36.524 --> 01:08:41.090 align:middle line:90%


01:08:41.090 --> 01:08:50.359 align:middle line:84%
Now, the question
is, well, if I want

01:08:50.359 --> 01:08:55.680 align:middle line:84%
to separate-- for example, if I
want to produce oxygen because I

01:08:55.680 --> 01:08:58.890 align:middle line:84%
need it for medical
applications or I

01:08:58.890 --> 01:09:01.109 align:middle line:84%
want to produce nitrogen
because I need it

01:09:01.109 --> 01:09:07.590 align:middle line:84%
for agricultural applications
or if I want to produce argon,

01:09:07.590 --> 01:09:12.890 align:middle line:84%
which is needed in the lab,
or if I want to separate CO2

01:09:12.890 --> 01:09:15.300 align:middle line:84%
from the atmosphere because
I want to segregate it,

01:09:15.300 --> 01:09:18.620 align:middle line:84%
which is something that
people are considering

01:09:18.620 --> 01:09:23.200 align:middle line:90%
these days, how much do I do?

01:09:23.200 --> 01:09:27.260 align:middle line:84%
And do I have to separate
entirely my mixture?

01:09:27.260 --> 01:09:30.340 align:middle line:90%
Well, the answer is no.

01:09:30.340 --> 01:09:32.779 align:middle line:90%
What you could do--

01:09:32.779 --> 01:09:37.090 align:middle line:84%
you could do exactly
just what you need, not--

01:09:37.090 --> 01:09:40.120 align:middle line:84%
or if you do something--
if you separate something

01:09:40.120 --> 01:09:42.760 align:middle line:84%
that you don't need,
then you'll spend energy

01:09:42.760 --> 01:09:45.370 align:middle line:90%
to do the separation.

01:09:45.370 --> 01:09:48.470 align:middle line:84%
And you're left with something
that-- since you don't need it,

01:09:48.470 --> 01:09:49.520 align:middle line:90%
you mix it again.

01:09:49.520 --> 01:09:52.510 align:middle line:90%
So it's wasteful.

01:09:52.510 --> 01:09:54.450 align:middle line:90%
So concentrate on what you need.

01:09:54.450 --> 01:09:57.440 align:middle line:84%
So for example, take
your initial mixture--

01:09:57.440 --> 01:09:58.720 align:middle line:90%
say, air.

01:09:58.720 --> 01:10:07.110 align:middle line:84%
And suppose you want to extract
some oxygen. So the best way--

01:10:07.110 --> 01:10:09.840 align:middle line:90%


01:10:09.840 --> 01:10:14.520 align:middle line:84%
the minimum expenditure
would be to extract

01:10:14.520 --> 01:10:21.220 align:middle line:84%
a little bit of oxygen, not
the entire amount of oxygen

01:10:21.220 --> 01:10:21.910 align:middle line:90%
in your air.

01:10:21.910 --> 01:10:24.820 align:middle line:84%
So you treat the volume
of air and extract

01:10:24.820 --> 01:10:28.410 align:middle line:84%
only a fraction of it, a
fraction of the oxygen that

01:10:28.410 --> 01:10:31.380 align:middle line:90%
is present in that air.

01:10:31.380 --> 01:10:38.490 align:middle line:84%
So you put out air, which
is leaner in oxygen.

01:10:38.490 --> 01:10:41.550 align:middle line:90%
And you have your oxygen here.

01:10:41.550 --> 01:10:44.510 align:middle line:84%
So essentially, this
is what we're doing.

01:10:44.510 --> 01:10:50.120 align:middle line:84%
We take lots of oxygen, little
oxygen. And then the little

01:10:50.120 --> 01:10:51.110 align:middle line:90%
you separate.

01:10:51.110 --> 01:10:53.810 align:middle line:84%
And the lot-- you
leave it with the air.

01:10:53.810 --> 01:10:55.735 align:middle line:84%
And the other components--
you leave it there.

01:10:55.735 --> 01:10:59.985 align:middle line:90%


01:10:59.985 --> 01:11:03.420 align:middle line:84%
You do the energy balances
and the entropy balances

01:11:03.420 --> 01:11:05.880 align:middle line:90%
and the volume balances again.

01:11:05.880 --> 01:11:08.390 align:middle line:84%
And again, you find
that the difference--

01:11:08.390 --> 01:11:10.650 align:middle line:84%
I mean, the minimum
work of separation

01:11:10.650 --> 01:11:12.570 align:middle line:84%
is given by the
difference in the Gibbs

01:11:12.570 --> 01:11:18.150 align:middle line:84%
free energy of the final
situation minus that

01:11:18.150 --> 01:11:20.620 align:middle line:90%
of the initial mixture.

01:11:20.620 --> 01:11:24.700 align:middle line:84%
Now, the final situation
or the final composite

01:11:24.700 --> 01:11:31.890 align:middle line:84%
of mixture plus the pure has,
therefore, two contributions.

01:11:31.890 --> 01:11:37.510 align:middle line:84%
This is for the mixture, which
now has a final composition, y

01:11:37.510 --> 01:11:40.720 align:middle line:84%
mole fractions B, which
are different from the mole

01:11:40.720 --> 01:11:43.640 align:middle line:84%
fractions that we
started with, plus

01:11:43.640 --> 01:11:50.560 align:middle line:84%
you have the Gibbs free energy
of the pure component here.

01:11:50.560 --> 01:11:54.430 align:middle line:84%
And again, remember
that chemical potential

01:11:54.430 --> 01:11:58.410 align:middle line:84%
is, for the pure substance,
the Gibbs free energy.

01:11:58.410 --> 01:12:03.120 align:middle line:84%
And again, we are using
additivity for Gibbs free energy

01:12:03.120 --> 01:12:06.090 align:middle line:84%
because the two subsystems
are at the same temperature

01:12:06.090 --> 01:12:07.890 align:middle line:90%
and pressure.

01:12:07.890 --> 01:12:13.630 align:middle line:84%
For the mixture here, I would
have to write that the--

01:12:13.630 --> 01:12:17.430 align:middle line:84%
from the Euler relation that
the Gibbs free energy is simply

01:12:17.430 --> 01:12:22.440 align:middle line:84%
the summation of the amounts
times the chemical potentials.

01:12:22.440 --> 01:12:24.330 align:middle line:84%
I have to be careful,
though, that when

01:12:24.330 --> 01:12:27.150 align:middle line:84%
I get to component
i in the summation,

01:12:27.150 --> 01:12:29.290 align:middle line:84%
there is this
extra little piece.

01:12:29.290 --> 01:12:33.190 align:middle line:84%
So I'm going to write this
summation in two pieces.

01:12:33.190 --> 01:12:36.690 align:middle line:84%
One is with the n's
that remain unchanged.

01:12:36.690 --> 01:12:41.760 align:middle line:84%
And the one that is
separated, its contribution

01:12:41.760 --> 01:12:45.280 align:middle line:84%
is multiplied by the
same chemical potential.

01:12:45.280 --> 01:12:46.980 align:middle line:90%
But I put it outside.

01:12:46.980 --> 01:12:50.940 align:middle line:84%
So component i-- I will find
it outside of the summation

01:12:50.940 --> 01:12:54.000 align:middle line:84%
for the amount that I separate
and inside the summation

01:12:54.000 --> 01:12:55.430 align:middle line:90%
for the amounts that remains.

01:12:55.430 --> 01:13:06.484 align:middle line:90%


01:13:06.484 --> 01:13:12.610 align:middle line:84%
Then I use ideal solution or
ideal Gibbs-Dalton mixture

01:13:12.610 --> 01:13:17.960 align:middle line:84%
behavior so that I can express
the chemical potentials

01:13:17.960 --> 01:13:22.400 align:middle line:84%
of the mixture in terms of
those two component times RT log

01:13:22.400 --> 01:13:23.750 align:middle line:90%
of y's.

01:13:23.750 --> 01:13:28.530 align:middle line:84%
And doing that substitution,
I obtain this expression.

01:13:28.530 --> 01:13:32.540 align:middle line:84%
So this is the
general expression

01:13:32.540 --> 01:13:44.400 align:middle line:84%
for the work in terms
of the initial amounts

01:13:44.400 --> 01:13:49.700 align:middle line:84%
and of the amount that I have
extracted of component i.

01:13:49.700 --> 01:13:51.920 align:middle line:84%
Now, there are
several cases that may

01:13:51.920 --> 01:13:53.690 align:middle line:90%
be interested-- interesting.

01:13:53.690 --> 01:13:57.200 align:middle line:84%
For example, I could
consider the complete removal

01:13:57.200 --> 01:14:01.130 align:middle line:84%
of constituent i-- so
a case in which I do

01:14:01.130 --> 01:14:05.370 align:middle line:90%
want to take, for example, CO2.

01:14:05.370 --> 01:14:07.800 align:middle line:90%
Maybe I want to extract it all.

01:14:07.800 --> 01:14:10.650 align:middle line:90%
So I don't want to be that--

01:14:10.650 --> 01:14:14.050 align:middle line:84%
to be left with any
in my final mixture.

01:14:14.050 --> 01:14:16.450 align:middle line:90%
So ni should be 0.

01:14:16.450 --> 01:14:19.660 align:middle line:90%
That's complete removal.

01:14:19.660 --> 01:14:30.410 align:middle line:84%
And that expression here
gives us this result.

01:14:30.410 --> 01:14:35.090 align:middle line:84%
And there is an
interesting possible limit,

01:14:35.090 --> 01:14:39.020 align:middle line:84%
which is one in which the
mole-- initial mole fraction

01:14:39.020 --> 01:14:44.810 align:middle line:84%
of my component, like, for
example, the CO2 case--

01:14:44.810 --> 01:14:50.060 align:middle line:84%
the initial mole fraction is
very small because CO2, we said,

01:14:50.060 --> 01:14:55.370 align:middle line:90%
is 0.04%.

01:14:55.370 --> 01:14:59.570 align:middle line:90%
So that means very less than 1.

01:14:59.570 --> 01:15:03.060 align:middle line:84%
In this case, the second
contribution goes away.

01:15:03.060 --> 01:15:08.580 align:middle line:84%
And so the minimum work is just
minus log of the mole fraction,

01:15:08.580 --> 01:15:14.990 align:middle line:84%
the minimum work per
unit amount extracted--

01:15:14.990 --> 01:15:23.190 align:middle line:84%
or you could use a
partial removal in which

01:15:23.190 --> 01:15:25.680 align:middle line:90%
I don't extract it all.

01:15:25.680 --> 01:15:27.810 align:middle line:90%
I extract only a fraction.

01:15:27.810 --> 01:15:30.920 align:middle line:90%
I could call this fraction chi.

01:15:30.920 --> 01:15:34.010 align:middle line:84%
The amount that I
have initially--

01:15:34.010 --> 01:15:37.040 align:middle line:84%
I want to-- say I want to
extract only a fraction

01:15:37.040 --> 01:15:43.950 align:middle line:84%
chi of the initially
present amount.

01:15:43.950 --> 01:15:48.210 align:middle line:84%
So that expression
rewrites this way.

01:15:48.210 --> 01:15:53.440 align:middle line:84%
And here also, there
is a limit, which

01:15:53.440 --> 01:15:57.190 align:middle line:84%
is the same result,
but which obtains

01:15:57.190 --> 01:16:00.880 align:middle line:84%
in the opposite situation,
when I extract--

01:16:00.880 --> 01:16:06.130 align:middle line:84%
when I separate only a minute
fraction of the originally

01:16:06.130 --> 01:16:07.630 align:middle line:90%
present amount.

01:16:07.630 --> 01:16:16.380 align:middle line:84%
This is typically used in the
oxygen or nitrogen separation

01:16:16.380 --> 01:16:16.880 align:middle line:90%
case.

01:16:16.880 --> 01:16:21.385 align:middle line:90%


01:16:21.385 --> 01:16:31.890 align:middle line:84%
So if we do, again, assume the
same air that we considered,

01:16:31.890 --> 01:16:35.390 align:middle line:90%
standard conditions--

01:16:35.390 --> 01:16:42.260 align:middle line:84%
and suppose we want to do a
minute or minor removal so

01:16:42.260 --> 01:16:44.060 align:middle line:90%
that this fraction chi is small.

01:16:44.060 --> 01:16:46.060 align:middle line:84%
So I can apply
this formula here.

01:16:46.060 --> 01:16:49.330 align:middle line:90%


01:16:49.330 --> 01:16:53.560 align:middle line:84%
So if my objective is
to extract 1 kilogram

01:16:53.560 --> 01:16:57.070 align:middle line:84%
and these are per
kilogram of the molecule

01:16:57.070 --> 01:16:58.220 align:middle line:90%
that I want to extract--

01:16:58.220 --> 01:17:01.060 align:middle line:84%
so I want to extract 1
kilogram of nitrogen. I

01:17:01.060 --> 01:17:05.140 align:middle line:90%
spend 21.9 kilojoules.

01:17:05.140 --> 01:17:12.180 align:middle line:84%
If I want to extract 1 kilogram
of oxygen, I spend 121.

01:17:12.180 --> 01:17:18.930 align:middle line:84%
It's more because oxygen is
more rare-- is by 4 times less.

01:17:18.930 --> 01:17:23.580 align:middle line:84%
If I want to extract
argon, it's 290.

01:17:23.580 --> 01:17:25.590 align:middle line:90%
If I want to extract CO2--

01:17:25.590 --> 01:17:42.360 align:middle line:90%
is 440.

01:17:42.360 --> 01:17:48.900 align:middle line:84%
Of course-- I want to make sure
that I cannot build a perpetual

01:17:48.900 --> 01:17:51.460 align:middle line:84%
motion machine of
the second kind.

01:17:51.460 --> 01:17:58.776 align:middle line:84%
That would be something in which
I use this method of separation.

01:17:58.776 --> 01:18:01.820 align:middle line:90%


01:18:01.820 --> 01:18:06.600 align:middle line:84%
And then once I have separated,
then I mix them again.

01:18:06.600 --> 01:18:09.355 align:middle line:84%
And we know that by mixing,
you can extract work.

01:18:09.355 --> 01:18:15.490 align:middle line:90%


01:18:15.490 --> 01:18:19.940 align:middle line:84%
So clearly, if I start
from air, I separate it.

01:18:19.940 --> 01:18:23.170 align:middle line:84%
And I spend so much
to separate it.

01:18:23.170 --> 01:18:27.550 align:middle line:84%
Then when I mix it again, the
maximum work I can separate

01:18:27.550 --> 01:18:28.950 align:middle line:90%
must be equal--

01:18:28.950 --> 01:18:33.340 align:middle line:84%
I mean, I can obtain must
be equal to the minimum work

01:18:33.340 --> 01:18:36.590 align:middle line:90%
of separation.

01:18:36.590 --> 01:18:39.660 align:middle line:90%


01:18:39.660 --> 01:18:47.610 align:middle line:84%
So for example, if I take
the right amount of nitrogen

01:18:47.610 --> 01:18:56.730 align:middle line:84%
that is needed to form again the
air in its original composition,

01:18:56.730 --> 01:18:59.270 align:middle line:84%
I have-- this is the
right-- that right amount.

01:18:59.270 --> 01:19:02.190 align:middle line:90%


01:19:02.190 --> 01:19:08.410 align:middle line:84%
So I have to multiply these
ones times those amounts.

01:19:08.410 --> 01:19:14.470 align:middle line:84%
And then, of course, the sum of
these works properly weighted

01:19:14.470 --> 01:19:20.050 align:middle line:84%
by those amounts
gives me the same work

01:19:20.050 --> 01:19:23.530 align:middle line:84%
that I can obtain
by mixing, which

01:19:23.530 --> 01:19:29.170 align:middle line:84%
is-- so this 48.5 is the
same work of separation

01:19:29.170 --> 01:19:32.200 align:middle line:84%
that we have used in
the complete separation

01:19:32.200 --> 01:19:34.210 align:middle line:90%
case in the previous slide.

01:19:34.210 --> 01:19:36.060 align:middle line:84%
So this is just a
consistency check.

01:19:36.060 --> 01:19:38.880 align:middle line:90%


01:19:38.880 --> 01:19:42.360 align:middle line:84%
Also here, I have
another Excel file

01:19:42.360 --> 01:19:49.885 align:middle line:84%
that I will share in which
these calculations are done.

01:19:49.885 --> 01:19:59.310 align:middle line:90%


01:19:59.310 --> 01:20:01.890 align:middle line:84%
This is where I
took those results.

01:20:01.890 --> 01:20:08.690 align:middle line:90%


01:20:08.690 --> 01:20:17.370 align:middle line:84%
And for example-- this is the
partial removal, partial removal

01:20:17.370 --> 01:20:19.530 align:middle line:90%
separation.

01:20:19.530 --> 01:20:27.130 align:middle line:84%
These are the amounts
of the mass percentages

01:20:27.130 --> 01:20:31.620 align:middle line:84%
of the various
constituents in air.

01:20:31.620 --> 01:20:36.100 align:middle line:84%
So if I take the weighted
sum of these times that,

01:20:36.100 --> 01:20:39.920 align:middle line:84%
I obtain the same as if I
did the complete separation.

01:20:39.920 --> 01:20:43.285 align:middle line:90%


01:20:43.285 --> 01:20:49.840 align:middle line:84%
Now, this also is in the
same or another Excel file.

01:20:49.840 --> 01:20:54.040 align:middle line:84%
There is this calculation
of the alcohol content

01:20:54.040 --> 01:20:58.000 align:middle line:90%
and the separation for wine.

01:20:58.000 --> 01:21:01.980 align:middle line:84%
It's interesting
that-- takes a while.

01:21:01.980 --> 01:21:04.670 align:middle line:84%
You may have to be
careful with that.

01:21:04.670 --> 01:21:08.510 align:middle line:84%
Volume fractions, mole
fractions, mass fractions

01:21:08.510 --> 01:21:10.290 align:middle line:90%
can get confusing.

01:21:10.290 --> 01:21:14.880 align:middle line:84%
But eventually,
you figure it out.

01:21:14.880 --> 01:21:19.610 align:middle line:84%
So this is already the result
that I mentioned before.

01:21:19.610 --> 01:21:21.020 align:middle line:90%
What was the number?

01:21:21.020 --> 01:21:24.560 align:middle line:90%


01:21:24.560 --> 01:21:29.810 align:middle line:84%
21 kilojoules per
liter if I take wine.

01:21:29.810 --> 01:21:39.050 align:middle line:84%
This is 12 degrees, 12 volume
fraction of ethanol, 12%.

01:21:39.050 --> 01:21:42.220 align:middle line:90%


01:21:42.220 --> 01:21:44.680 align:middle line:84%
This other thing
is an application

01:21:44.680 --> 01:21:48.120 align:middle line:90%
of the partial removal case.

01:21:48.120 --> 01:21:51.760 align:middle line:90%


01:21:51.760 --> 01:22:00.570 align:middle line:84%
Sometimes, winemakers
in the north of Europe,

01:22:00.570 --> 01:22:07.020 align:middle line:84%
where there isn't so much
sunshine, the alcohol

01:22:07.020 --> 01:22:12.430 align:middle line:84%
content, the natural alcohol
content, that you obtain is low.

01:22:12.430 --> 01:22:15.320 align:middle line:84%
And you have to find
ways to raise it.

01:22:15.320 --> 01:22:18.940 align:middle line:90%
One way is to buy wine.

01:22:18.940 --> 01:22:21.700 align:middle line:84%
It used to be that they--
it used to be cheaper

01:22:21.700 --> 01:22:25.930 align:middle line:84%
to buy wine from
the south of Italy

01:22:25.930 --> 01:22:30.550 align:middle line:84%
or from Greece or from
Spain because there,

01:22:30.550 --> 01:22:32.090 align:middle line:90%
you get a lot of sunshine.

01:22:32.090 --> 01:22:34.360 align:middle line:84%
And so the alcohol
content is high.

01:22:34.360 --> 01:22:36.970 align:middle line:84%
And it used to be called
the cut wine, where

01:22:36.970 --> 01:22:41.440 align:middle line:84%
you need to cut the
poorer wines of the north.

01:22:41.440 --> 01:22:46.300 align:middle line:84%
But now, also the people in
the south of Italy, Greece,

01:22:46.300 --> 01:22:49.030 align:middle line:84%
and so on have learned
to make good wines.

01:22:49.030 --> 01:22:52.230 align:middle line:84%
And so they don't sell
it for low anymore.

01:22:52.230 --> 01:22:58.060 align:middle line:84%
So an alternative
is to separate,

01:22:58.060 --> 01:23:04.770 align:middle line:84%
take a little bit of
water away from the--

01:23:04.770 --> 01:23:09.900 align:middle line:84%
so for example, suppose you have
a 12-degrees wine and you want

01:23:09.900 --> 01:23:13.200 align:middle line:90%
to bring it to 14.

01:23:13.200 --> 01:23:16.510 align:middle line:84%
All you need to do is
to take some water away.

01:23:16.510 --> 01:23:18.470 align:middle line:84%
Of course, you have--
you cannot boil it.

01:23:18.470 --> 01:23:20.790 align:middle line:84%
Otherwise, you're
going to ruin the wine.

01:23:20.790 --> 01:23:23.020 align:middle line:90%
But maybe you can flush it--

01:23:23.020 --> 01:23:28.420 align:middle line:84%
low pressures so that at room
temperature, you get some-- so

01:23:28.420 --> 01:23:30.820 align:middle line:84%
there are ways in
which you can extract

01:23:30.820 --> 01:23:37.510 align:middle line:84%
without ruining the rest of
the good part of the wine.

01:23:37.510 --> 01:23:40.640 align:middle line:84%
And so this
calculation gives you--

01:23:40.640 --> 01:23:44.710 align:middle line:84%
with the formulas
that we've given, it--

01:23:44.710 --> 01:23:46.300 align:middle line:84%
all you need to do
is to figure out,

01:23:46.300 --> 01:23:50.460 align:middle line:84%
how much water do I
have to take away?

01:23:50.460 --> 01:23:57.420 align:middle line:84%
And so this is the
delta n1 that I need.

01:23:57.420 --> 01:24:00.510 align:middle line:84%
If you look inside, there
is a formula that shows you

01:24:00.510 --> 01:24:03.210 align:middle line:90%
how I calculated that.

01:24:03.210 --> 01:24:10.440 align:middle line:84%
And once I determine
that, I can apply--

01:24:10.440 --> 01:24:14.280 align:middle line:84%
I can find the chi--
so the cut fraction

01:24:14.280 --> 01:24:17.580 align:middle line:90%
that we defined in the slide.

01:24:17.580 --> 01:24:19.080 align:middle line:90%
And we apply those formulas.

01:24:19.080 --> 01:24:25.770 align:middle line:84%
And it comes out that we need to
spend 0.86 kilojoules per liter

01:24:25.770 --> 01:24:30.060 align:middle line:84%
in order to make this little
improvement in the alcohol

01:24:30.060 --> 01:24:31.380 align:middle line:90%
content.

01:24:31.380 --> 01:24:35.350 align:middle line:84%
And of course, therefore,
you'll find that the wine--

01:24:35.350 --> 01:24:43.480 align:middle line:84%
if you do this way, you find
that the winemaking becomes also

01:24:43.480 --> 01:24:44.783 align:middle line:90%
an energy--

01:24:44.783 --> 01:24:45.450 align:middle line:90%
how do you say--

01:24:45.450 --> 01:24:48.040 align:middle line:90%


01:24:48.040 --> 01:24:50.830 align:middle line:90%
an energy-consuming industry.

01:24:50.830 --> 01:24:51.330 align:middle line:90%
Yes.

01:24:51.330 --> 01:24:59.430 align:middle line:90%


01:24:59.430 --> 01:25:08.010 align:middle line:84%
So let me move on to another
interesting application.

01:25:08.010 --> 01:25:11.760 align:middle line:90%
It has still to do with air.

01:25:11.760 --> 01:25:19.240 align:middle line:84%
It has to do with
stratification of air.

01:25:19.240 --> 01:25:28.430 align:middle line:90%
As you know air is stratified.

01:25:28.430 --> 01:25:33.180 align:middle line:84%
As you go up to high
mountains-- for example,

01:25:33.180 --> 01:25:36.050 align:middle line:90%
top of Mount Everest--

01:25:36.050 --> 01:25:40.130 align:middle line:84%
you can read books about
what they call thin air.

01:25:40.130 --> 01:25:42.740 align:middle line:90%
So why is it thin?

01:25:42.740 --> 01:25:45.270 align:middle line:90%
It's thin for two reasons.

01:25:45.270 --> 01:25:54.220 align:middle line:84%
It's thin because the overall
pressure is low, maybe

01:25:54.220 --> 01:25:58.060 align:middle line:84%
one-third of the atmospheric
pressure at sea level.

01:25:58.060 --> 01:26:03.310 align:middle line:84%
And it's thinner also
because the oxygen content

01:26:03.310 --> 01:26:07.680 align:middle line:90%
percentage-wise is less.

01:26:07.680 --> 01:26:08.710 align:middle line:90%
Why less?

01:26:08.710 --> 01:26:14.190 align:middle line:84%
Because the oxygen molecules
are molecular weight 32.

01:26:14.190 --> 01:26:16.950 align:middle line:90%
Nitrogen molecules-- 28.

01:26:16.950 --> 01:26:20.100 align:middle line:90%
So the oxygen is heavier.

01:26:20.100 --> 01:26:24.450 align:middle line:84%
It likes to stay
more towards the sea.

01:26:24.450 --> 01:26:26.660 align:middle line:84%
So the stratification is
due to these differences

01:26:26.660 --> 01:26:30.800 align:middle line:84%
in molecular weights
of the molecules.

01:26:30.800 --> 01:26:33.650 align:middle line:84%
And since we are so
interested in CO2,

01:26:33.650 --> 01:26:37.340 align:middle line:90%
CO2 is 44 molecular weight.

01:26:37.340 --> 01:26:41.800 align:middle line:84%
So even more, there
is a stratification.

01:26:41.800 --> 01:26:51.380 align:middle line:84%
For example, when we say,
ah, the CO2 concentration

01:26:51.380 --> 01:26:56.390 align:middle line:84%
at Mauna Loa, which is the
place where they measure CO2,

01:26:56.390 --> 01:26:57.360 align:middle line:90%
it is a mountain.

01:26:57.360 --> 01:27:05.830 align:middle line:84%
That mountain is at
3,400 meters altitude.

01:27:05.830 --> 01:27:09.820 align:middle line:84%
It's also maybe representative
of the content of--

01:27:09.820 --> 01:27:12.790 align:middle line:84%
the average content
in the atmosphere.

01:27:12.790 --> 01:27:21.140 align:middle line:84%
But the point is that if there,
there is 426 PPM, for example,

01:27:21.140 --> 01:27:25.660 align:middle line:84%
today, it means
that at sea level,

01:27:25.660 --> 01:27:28.400 align:middle line:90%
you have 500 and something.

01:27:28.400 --> 01:27:32.130 align:middle line:90%
We'll calculate it in a moment.

01:27:32.130 --> 01:27:34.980 align:middle line:84%
There is a big difference
due to these differences

01:27:34.980 --> 01:27:37.860 align:middle line:84%
in molecular weights
of the molecules.

01:27:37.860 --> 01:27:42.220 align:middle line:84%
So how do we go about
and do this calculation?

01:27:42.220 --> 01:27:44.500 align:middle line:90%
We need a model, as usual.

01:27:44.500 --> 01:27:53.210 align:middle line:84%
And the model that we make
is very similar to the model

01:27:53.210 --> 01:27:59.200 align:middle line:90%
that we have made for bulk flow.

01:27:59.200 --> 01:28:01.490 align:middle line:84%
You remember, for
bulk flow, we said

01:28:01.490 --> 01:28:08.770 align:middle line:84%
if you have a fluid
element which finds itself

01:28:08.770 --> 01:28:18.930 align:middle line:84%
at elevation z in the gravity
field and which has a velocity--

01:28:18.930 --> 01:28:20.015 align:middle line:90%
let me call it w--

01:28:20.015 --> 01:28:25.430 align:middle line:90%


01:28:25.430 --> 01:28:36.640 align:middle line:84%
and it has a mass m, when
we do fluid mechanics,

01:28:36.640 --> 01:28:41.290 align:middle line:84%
this is a fluid element
or a fluid parcel.

01:28:41.290 --> 01:28:47.320 align:middle line:84%
And we assign the properties
in the local equilibrium

01:28:47.320 --> 01:28:48.310 align:middle line:90%
assumption.

01:28:48.310 --> 01:28:51.730 align:middle line:84%
We assign the properties
of this fluid element

01:28:51.730 --> 01:29:00.310 align:middle line:84%
by assuming that its energy is
the energy that it would have

01:29:00.310 --> 01:29:04.610 align:middle line:84%
with the same
composition, volume,

01:29:04.610 --> 01:29:14.540 align:middle line:84%
and entropy in a stable
equilibrium state, plus we

01:29:14.540 --> 01:29:17.330 align:middle line:84%
add these mechanical
effects that

01:29:17.330 --> 01:29:24.895 align:middle line:84%
are the kinetic energy
and the potential energy.

01:29:24.895 --> 01:29:35.300 align:middle line:90%


01:29:35.300 --> 01:29:40.160 align:middle line:84%
The next time, we'll assume
also that this particle has

01:29:40.160 --> 01:29:46.976 align:middle line:84%
a charge, electrical charge,
and that finds itself in,

01:29:46.976 --> 01:29:49.975 align:middle line:90%
for example, an electric field.

01:29:49.975 --> 01:29:56.190 align:middle line:90%


01:29:56.190 --> 01:29:59.580 align:middle line:84%
An electric field is
really minus the gradient

01:29:59.580 --> 01:30:04.040 align:middle line:90%
of an electric potential.

01:30:04.040 --> 01:30:10.480 align:middle line:84%
And so here, we will
add q times phi.

01:30:10.480 --> 01:30:14.440 align:middle line:90%


01:30:14.440 --> 01:30:17.470 align:middle line:90%
It's mechanical kind of effect.

01:30:17.470 --> 01:30:21.040 align:middle line:84%
So the thermal aspect
of the properties

01:30:21.040 --> 01:30:25.570 align:middle line:84%
are still contained in
the fundamental relation

01:30:25.570 --> 01:30:29.490 align:middle line:90%
for stable equilibrium states.

01:30:29.490 --> 01:30:33.460 align:middle line:84%
And these are other
additional contributions.

01:30:33.460 --> 01:30:37.390 align:middle line:90%


01:30:37.390 --> 01:30:40.090 align:middle line:84%
Now, for the problem
of stratification

01:30:40.090 --> 01:30:44.670 align:middle line:84%
in the atmosphere
here, all we need to do

01:30:44.670 --> 01:30:47.140 align:middle line:90%
is to forget about motion.

01:30:47.140 --> 01:30:49.980 align:middle line:84%
So we are talking about the
still atmosphere-- of course,

01:30:49.980 --> 01:30:50.970 align:middle line:90%
it's a model--

01:30:50.970 --> 01:30:52.055 align:middle line:90%
and not charge.

01:30:52.055 --> 01:30:59.330 align:middle line:90%


01:30:59.330 --> 01:31:01.490 align:middle line:84%
So we have this
potential energy.

01:31:01.490 --> 01:31:13.200 align:middle line:84%
And m is the mass of the
fluid element, which may vary.

01:31:13.200 --> 01:31:15.930 align:middle line:84%
We're going to assume
that it may vary here

01:31:15.930 --> 01:31:21.030 align:middle line:84%
because since the composition
changes with elevation, also

01:31:21.030 --> 01:31:22.603 align:middle line:90%
the mass changes.

01:31:22.603 --> 01:31:25.420 align:middle line:90%


01:31:25.420 --> 01:31:29.140 align:middle line:84%
So the mass is
given by the moles

01:31:29.140 --> 01:31:32.440 align:middle line:90%
times the molecular weights.

01:31:32.440 --> 01:31:37.580 align:middle line:90%


01:31:37.580 --> 01:31:39.760 align:middle line:90%
So this is sea level.

01:31:39.760 --> 01:31:45.331 align:middle line:84%
This is the generic
layer at elevation z,

01:31:45.331 --> 01:31:47.890 align:middle line:84%
where we assume that there
is a temperature, a pressure,

01:31:47.890 --> 01:31:48.820 align:middle line:90%
and the composition.

01:31:48.820 --> 01:31:53.890 align:middle line:84%
And there is a adjacent
layer, which-- at z plus dz,

01:31:53.890 --> 01:31:56.320 align:middle line:84%
with a different temperature,
different pressure,

01:31:56.320 --> 01:31:58.280 align:middle line:90%
and different composition.

01:31:58.280 --> 01:32:03.390 align:middle line:90%


01:32:03.390 --> 01:32:14.530 align:middle line:84%
Now, when we have this kind
of energy, you remember--

01:32:14.530 --> 01:32:17.260 align:middle line:84%
and when talk about--
when we derive

01:32:17.260 --> 01:32:21.700 align:middle line:84%
the conditions for
mutual equilibrium,

01:32:21.700 --> 01:32:26.610 align:middle line:84%
the conditions for
mutual equilibrium

01:32:26.610 --> 01:32:30.270 align:middle line:84%
were equalities between
some partial derivatives

01:32:30.270 --> 01:32:36.160 align:middle line:84%
of the energy of the
fundamental relation

01:32:36.160 --> 01:32:37.960 align:middle line:84%
that, precisely
because they appear

01:32:37.960 --> 01:32:42.200 align:middle line:84%
in these conditions for mutual
equilibrium, are important.

01:32:42.200 --> 01:32:44.320 align:middle line:84%
And that's why we
give them names--

01:32:44.320 --> 01:32:50.560 align:middle line:84%
temperature, pressure,
chemical potentials.

01:32:50.560 --> 01:32:52.840 align:middle line:84%
But here, when we
take the derivative

01:32:52.840 --> 01:32:56.140 align:middle line:84%
of energy with
respect to S and V,

01:32:56.140 --> 01:32:58.600 align:middle line:84%
we still get temperature
and pressure.

01:32:58.600 --> 01:33:01.000 align:middle line:84%
But in a model
like this, when we

01:33:01.000 --> 01:33:04.100 align:middle line:84%
take the derivative of E
with respect to the amounts,

01:33:04.100 --> 01:33:06.760 align:middle line:84%
I have the amounts
here, which give me

01:33:06.760 --> 01:33:09.020 align:middle line:84%
the chemical potential
contribution.

01:33:09.020 --> 01:33:13.240 align:middle line:84%
But I have the amounts also
inside the mass, which gives me

01:33:13.240 --> 01:33:14.890 align:middle line:84%
an additional
contribution, which is

01:33:14.890 --> 01:33:19.540 align:middle line:90%
a gravitational contribution.

01:33:19.540 --> 01:33:22.420 align:middle line:84%
If I have the charge,
which we'll do next time,

01:33:22.420 --> 01:33:25.840 align:middle line:84%
we'll have also an
electrical contribution.

01:33:25.840 --> 01:33:29.320 align:middle line:90%


01:33:29.320 --> 01:33:34.740 align:middle line:84%
So this more general
kind of potential

01:33:34.740 --> 01:33:39.010 align:middle line:84%
that so far we've
called chemical--

01:33:39.010 --> 01:33:41.190 align:middle line:84%
we will call it
the total potential

01:33:41.190 --> 01:33:43.590 align:middle line:84%
because it has
several contributions.

01:33:43.590 --> 01:33:46.720 align:middle line:84%
And when it is only
chemical plus electrical,

01:33:46.720 --> 01:33:49.860 align:middle line:84%
it's the electrochemical
potential.

01:33:49.860 --> 01:33:54.005 align:middle line:84%
Today, for the stratification
problem, you could call it--

01:33:54.005 --> 01:33:57.650 align:middle line:90%


01:33:57.650 --> 01:34:00.020 align:middle line:90%
what is it, electrochemical--

01:34:00.020 --> 01:34:06.150 align:middle line:90%
gravitochemical potential.

01:34:06.150 --> 01:34:06.650 align:middle line:90%
Whatever.

01:34:06.650 --> 01:34:08.080 align:middle line:90%
The name doesn't matter.

01:34:08.080 --> 01:34:13.610 align:middle line:84%
What's important is that
this total potential is given

01:34:13.610 --> 01:34:15.085 align:middle line:90%
by the chemical potential.

01:34:15.085 --> 01:34:19.790 align:middle line:84%
So it's the derivative of this
one plus the derivative of m.

01:34:19.790 --> 01:34:25.520 align:middle line:84%
And so that brings up
the molecular weight

01:34:25.520 --> 01:34:28.700 align:middle line:84%
of the molecule with
respect to which

01:34:28.700 --> 01:34:31.850 align:middle line:90%
you are taking the derivative.

01:34:31.850 --> 01:34:34.860 align:middle line:90%
So I think we are done today.

01:34:34.860 --> 01:34:39.380 align:middle line:84%
But the idea is that we
will obtain our results

01:34:39.380 --> 01:34:43.810 align:middle line:84%
and, therefore, the
distribution in concentrations

01:34:43.810 --> 01:34:50.290 align:middle line:84%
by imposing chemical potential
equality between layers

01:34:50.290 --> 01:34:54.350 align:middle line:84%
because the layers are
attached to one another.

01:34:54.350 --> 01:34:57.600 align:middle line:84%
So they are in
mutual equilibrium.

01:34:57.600 --> 01:34:59.850 align:middle line:84%
In practice, they are
not because as you know,

01:34:59.850 --> 01:35:05.760 align:middle line:84%
the temperature in the
atmosphere is not uniform.

01:35:05.760 --> 01:35:12.600 align:middle line:84%
We will assume that this change
in temperature in the atmosphere

01:35:12.600 --> 01:35:16.940 align:middle line:84%
can be neglected locally,
but not globally.

01:35:16.940 --> 01:35:19.980 align:middle line:84%
So that will be a little
bit of an assumption.

01:35:19.980 --> 01:35:24.420 align:middle line:84%
So we assume that we make an
equilibrium between these two,

01:35:24.420 --> 01:35:29.600 align:middle line:84%
assuming that T at z plus
dz is equal to T at z.

01:35:29.600 --> 01:35:33.590 align:middle line:84%
But we will not assume that
the chemical potentials are.

01:35:33.590 --> 01:35:36.693 align:middle line:84%
And that's how we
will get our results.

01:35:36.693 --> 01:35:39.220 align:middle line:90%


01:35:39.220 --> 01:35:40.700 align:middle line:90%
But the time is up.

01:35:40.700 --> 01:35:47.470 align:middle line:84%
And so I think I have to
simply wish you a good spring

01:35:47.470 --> 01:35:51.300 align:middle line:90%
break and see you in 10 days.

01:35:51.300 --> 01:36:01.000 align:middle line:90%