Hello, welcome to this lecture of Biomathematics.
In the last few lectures, we have been
discussing about Diffusion like last couple
of lectures, we discussed the diffusion
equation and then, we discussed how do we
calculate the rms distance from this equation
without really solving the equation, even
though diffusion equation is a second order
differential equation. At this moment, we
will not discuss how to solve it, it is a
little
more you need to learn some more mathematics
too.
Figure out, how to learn this or to understand,
how you solve this diffusion equation, but
without really solving it something very useful,
some relation that is very useful we got,
which is the rms distance and time the relation
between time and the x square average.
So, this we will continue to discuss something
about the relation diffusion, initially the
first part we will discuss a bit about another
quantity, which is average x average.
And then, we will go on to discuss with the
one another very important relation as far
as
the diffusion is concerned. So far, we took
diffusion coefficient as a constant, today
we
will discuss, what is diffusion coefficient
related to, that is how is that how diffusion
coefficient related to temperature, viscosity,
of the etcetera of the medium. So, this
relation is a famous Einstein relation, so
we will discuss the Einstein relation in today’s
lecture, towards the end of this lecture.
..
So, the topic of we are continuing to discuss
on section on Applications of calculus and
vector algebra in biology and under this,
we are discussing this diffusion.
.
So, we said that, if you start with, if you
take a tube as you see here.
..
If you take a tube and if you put a few number
of particles in this or a protein molecules
on the middle of this tube, the concentration
is only t is equal to 0, the concentration
is
only at the middle of the tube and as we go
along it will spread, the molecules will
diffuse.
.
Even at this time, maximum number of molecules
larger bigger number of molecules, the
concentration is still more here at the center,
but there are still there are some still there
are some molecules as we go along the tube
to the left or to the right; so, this is the
.concentration of profile and the diffusion
equation is the equation that deals with this
spread of this concentration is the diffusion
equation.
.
So, the diffusion equation is d C x by d C
by d t is equal to D del square C by del x
square, where C is a concentration as a function
of position and the time and by solving
this equation we expect to get concentration
as a function of position for any time, that
is
what you will get.
.
.And we defined two quantities, x average
and x square average as x is C tilde the
average integral d x C tilde d x minus infinity
to infinity, and x square C tilde minus
infinity to infinity as x square average.
And we also saw that x square average is 2Dt,
this is a very interesting relation and important
relation, the square of the position goes
as
time or in other words.
.
If you take square root both sides, the square
root the rms distance goes as the root of
time, so that is what the important relation
that we saw, that is X rms which is, this
is
proportional to the square root of time.
..
So, this is x rms is proportional to the square
root of time. So, this is the interesting
relation that we found and this is so the
square root of time or you can write this
t power
half. So, this is this t power half is a kind
of synonymous to diffusion on would say like
it
is like something moving like t power half,
something moving square root of time, that
is
called diffusive motion. So, this is an important
relation as far as diffusion is concerned,
now we will calculate x average something
which we want to learn, so the question is
next question is what was x average? So, that
is the question that we want to address next
what is x average?
So, we found that x in the previous lecture
that x square average is 2Dt, we discuss this
relation as we said this is an important relation,
which says that how does the rms
distance, there is a root of X rms and how
does the X rms is related to time. So, the
X
rms is, if you take square root on both sides
the X rms as we wrote previously look at
here, X rms is root of x square average, this
is proportional to time. The square root of
time, in other words X rms I proportional
to square root of time or proportional to
t
power half.
..
So, now, the next question is what is x average?
We will calculate the x average is same
way as we calculated the x square average,
how did we do that? We took this equation,
which is the diffusion equation, we multiplied
both sides by x, we are multiplying both
side with the x and integrating; so, you multiply
here with x and integrating, so this
becomes integral x d C d x with del by del
t outside.
And here also we have the right hand side
also we are multiplying with the x and
integrating, so integral x del x square C
by del x square d x, so with multiplied both
sides
of this diffusion equation with x and integrated
from minus infinity to infinity and what
do we get? So, the left hand side, as we defined
earlier is integral x C tilde x d x is
nothing but x average, so this is the left
hand side.
..
So, what do you get you get on the left hand
side, so let us look at what we get.
.
So, what we have is, del by del t of integral
minus infinity to infinity x 
C tilde of x d x is
equal to D into integral x del square C by
del x square C tilde D x, now this part as
we
said last time is nothing but, del by del
t of, so del by del t. And this is integral
x C tilde x
d x is nothing but, c average, so this is
c average, so this one this one, this part
which is I
writing it in a box, this part can be written
as C average. So, this is equal to D into
integral minus infinity to infinity x del
square C tilde a by del x square d x. Now,
just
.like, we discussed last time this can be
written as integral and you can take x as
u and
this term you can take as del V by del x,
so d V by d x, so this is integral u d V by
d x, so
this is you can do this rule in calculus called
integration by parts.
.
So, integral as we just discussed some time
ago, integral as we just discussed in the
last
lecture, integral u d V by d x d x can be
written as integral u v sorry it can be written
as u
v in the limits minus integral V del u by
del x d x. So, we can use this formula, which
is
the standard formula in calculus, which we
in the last class we figured out how this
formula is coming. And let say, let us use
this formula now, so what did we said is that,
we just said have a look at here, so what
did we just said that del C del by del t or
C
equal to minus this and we call this as u
and this as del V by del x.
So, now, we can apply this formula there,
so if you apply this formula there, if we
apply
this formula there what do we get, so we have
x as u and del square C by del x square as
del V by del x. So, what you would get essentially
is this, if we apply this formula what
you would get is that, del x average by del
t is equal to u is x, so x and V is del C
by del
x in the limits minus D into integral V d
u by d x.
So, now V is del C by del x d u by d x is
1, so this is what you will get, so let us
this is
what precisely, I have written what you would
get is that D in to x d C tilde a y d x in
the
limits minus D into del C by del x del C tilde
a by del x d x, so this is what you would
get. Now, if you apply these limits, just
by arguing there are plus infinity and minus
.infinity the derivative is, so by just arguing
that this term, at you apply this del C by
del x
at plus infinity and minus infinity and calculate
this term you will get this equal to 0.
So, then what here remaining with is just,
so at this limit this is 0, so what you have
is
just minus D integral del C by del x d x,
so we have just this term. So, now, let us
see
what is this term what term gives? So, let
us think about that term a bit.
.
So, what end up essentially is that del by
del t of x average is minus D del minus infinity
to infinity del C tilde by del x d x, so what
is this? So, this is derivative, so this is
essentially minus D into C at the limits.
If you take minus infinity and infinity plus
infinity at the limits, if you calculate the
C, C
at infinity and minus infinity is 0, so the
answer that this integral is essentially 0,
because
if you do this derivative integral of a derivative
is just the function itself, so you have C,
C at infinity and C at minus infinity they
are 0. So, essentially this derivative this
integral
is 0, so you will end up with this relation
that del by del t of x average is 0, which
means
that x average is 0.
..
So, what we got essentially is that x average
is 0, we had found that x square average is
2Dt. So, this is two interesting relations
that we get that x average is 0, x square
average
is 2Dt. Now, what does this mean to say that
x average is 0, physically what does that
mean, so let us think about this. So, let
us think about this diffusing in a pipe example
that we thought.
.
So, we let us take this pipe and to begin
with you have some particles here, and let
us say
there is one blue particle here and one particle
which I am circling here, which is in blue
.color, so let say there is one blue particle
here and all other black particles. Now, if
you
look at this blue particle and in one experiment
you might have see this blue particle is
going this way, so it will diffuse some distance
x in this way in one experiment. Let us
say, you are doing the same experiment let
us say, you are doing the same experiment
in
a different day or you repeating this experiment,
now you have again you start with the
same condition.
So, now you have the blue particle you put
here, when you put the blue particles
becomes here and this time it might move in
this way, so it might move the distance x
in
this particular way. So, if you just keep
repeating this experiment of. The experiment
that
we discussed, that is adding some amount of
proteins at the middle of the tube and
looking at where is it going, which way this
is going, so at one experiment you might see
that, this the basically what is this is like
imagine that you have just one particle, that
we
can detect let say this it is fluorescing
or it has some different color.
So, you will see that, in one experiment this
particular particle might be going this way,
in another experiment this particle might
be going this way, in some other experiment
it
might be going this way. So, if you do many
experiments and average over all this, so
some time it would go in the plus direction,
some time it will go in the minus direction,
so minus direction plus direction finally,
the average you will get 0.
So, the first experiments it might have moved
minus 3 centimeter, in the next experiment
it might have moved plus 3 centimeter. So,
if you just repeat this experiment many times
on an average, if you find the average of
this, you would have, you will get 0 that
is what
it precisely means, it means that if you look,
so if you in a in a diffusion experiment,
if
you look the position of one particle or many
many experiments, the average over all
experiments will give you the average position
as 0.
On the other hand, if you calculate this square
average minus 3 square is 9, 3 square is
also 9, so there is no way this x square can
average out to 0, so you will get some
quantity which is 2Dt. So, the meaningful
since, the x average is 0 the meaningful
quantity is x square average, this is the
meaningful quantity that we can that, we should
know or that is useful, physically useful
quantity is x square average in other words
the
rms distance.
..
So, the important formula as far as the diffusion
diffusion motion is concerned, its x
square average is 2Dt or X rms is equal to
root of 2Dt, so it is a square this is the
important formula, this the root mean square
distance that a particle would travel in a
time t. If you do average over many many experiments,
this is the root means square
average things would travel in a time t. Now,
what is this d, so that is the question what
is D? We have been discussing we have been
having this thing called D for a long time,
so from this, we found that D as a D as a
unit, if you dimension of D is length square
by
time length square by time, so it will have
a unit meter square per second.
So, now, what is this D? D is the diffusion
coefficient, what is that mean diffusion
coefficient mean? Diffusion coefficient essentially
that contains the property of the
medium that in which you are putting this
protein, if you are doing in water it contains
the property of the water like viscosity of
the water. It also contains the temperature,
so
you can imagine that, in the temperature is
very large things will diffuse out very fast
higher temperature, higher diffusion.
So, the information about the temperature
viscosity all the property of the medium is
put
into this one quantity called D. So, the D
contains the property of the medium, so now,
how do you find out how does the D depends
on the property of the medium, if the
viscosity is more how does the D change, if
the temperature is more how does the D
change, how do we find it out.
.So, just by learning, just by knowing, what
you learnt in mathematics so for and with
some intuition with some with some simple
thinking, one can figure it out. Actually
this
was discovered by none other than Albert Einstein
in 1905 in 1905. So, this is this is
what we are going to discuss, so the relation
of the D.
.
So, the relation between the relation between
the relation between D temperature and
viscosity let me call this eta as the viscosity,
so this relation is called Einstein relation.
.
.So, that is what we are going to discuss
now, we will discuss this relation between
diffusion coefficient and temperature and
viscosity as and this relation is known as
the
Einstein relation, this was discovered by
Albert Einstein in 1905. Albert Einstein,
for his
PhD, he was studying about Brownian motion
of particles and he discovered this
relation. I will tell you in a simple way
how do we calculate, how do we derive roughly
what Einstein did about 100 plus years ago,
it about 100 and 106 years ago.
Einstein derived this relations, this as you
might have also heard this 1905 is a very
famous year for Einstein like, he wrote three
very famous papers: one paper is related to
this Einstein’s relation, it became very
famous and this relation became one of the
most
popular relations like very highly slighter
relations in science because, this is application
on biology, in chemical engineering, in chemistry,
in physics, in all sorts of fields.
Einstein‘s relation related to diffusion
is used in environmental sciences in, you
can think
of any field, which virtually virtually any
field and this relation will be or is this
relation
is being used. Then, he discovered the, or
he explained the photo electric effect and
he
also explained, he also he also had his famous
paper on relativity, so this three paper
made him world famous like all this papers.
So, one of the paper even got him noble
prize, so this is miracle here as for the
Einstein his concerned and the world of science
is
concerned.
So, we will discuss one of his contributions
in that year 1905. It is interesting that,
just
by understanding this simple mathematics we
can derive this relation. It is very similar
to
what we did for Nernst equation. So, we will
go in the same line as we did went for
Nernst to understand Nernst equation. So,
let us go ahead and think about Einstein’s
relation.
..
So, Einstein thought about the following example,
so he thought there are some particles
in water in a beaker, and this particle is
subjected to some external field. So, let
us say
there is gravity downwards, so if there is
gravity on all this particles with some mass,
they will be forced to come down, because
of the external force gravity for example,
it
could be either gravity or it could be if
they are charged particle even electric field.
So,
you could think of this is electrophoresis,
if you wish. So, basically this are you can
even
think of this as charged particles and then
there is some force exerted on this charged
particles due to electric field, this could
be like some protein molecules under electric
field.
Now, let this force be minus g times x mathematically,
this force is minus g times x,
where x is the distance from bottom to top,
so x is the distance starting from the bottom
to the top, so x cap has this particular direction
and the force has this particular direction;
so, force is acting down words and the distance
is going upwards.
So, the f and x are having opposite direction,
so that is why this minus sign, so g is the
amount of force. So, g could be the amount
of electric field, electro force due to electric
field, it could be amount of force due to
gravity whatever you wish, but g is some force
and the magnitude of the force is g; and f
is the force of the vector force, so let f
is equal
to minus g x. So, we can say that the energy,
so for every particle if you if you want to
.this particle to go up a distance x it has
to spend an energy f dot x, so it if you have
a
force f.
.
So, look at here the force f, so which is
let us say, let us say it is g x cap and let
us say
that the energy, so if you have such particles
and each of this particle is experiencing
a
force, so if this particle wants to go up
a distance. If it wants to reach a distance
x from
the bottom, it has to spend an energy f with
f dot x. So, which is nothing but, f is minus
g
x cap dot x, so this is minus g x. So, this
has to spend this much energy, so it is not
favorable, so the magnitude of the energy
is g x essentially sorry the energy is g x.
So, it is not favorable to go up here, because
the force is in this way, so it has to spend
an
energy, so most of the particle you will find
at the bottom, because there is a force acting
and there is an energy cost to go here. So,
if you look at the concentration, if you think
intuitively, the concentration will be more
at the bottom and less at the top, you can
think
of any particle, if you put something in to
water, you could think it of as sedimentation
something will fall down to the bottom of
the beaker right.
If you put some something, which is some objects
onto water they will fall down,
because the gravity is attracting it down,
so the concentration if of anything will be
more
at the bottom and less at the top.
..
So, if you plot, if you wish if you plot,
the concentration as a function of the distance
from the bottom, you will see some exponential
relation, there is some reason why it is
exponential, but let us intuitively assume
that, C is C is proportional to e power minus
g
x by K B T, so g x is energy and it has to
be divided by another energy K B T to make
it
dimensionless. So, the concentration decreases
as you go along x, this is what it means.
So, if you know this relation, we can derive
the Einstein’s relation. So, this is the
one
ingredient that we need to know, that the
concentration will exponentially decrease
as we
go to the bottom, how do we get this relation?
That we will discuss later, but for the
moment just take this relation for granted,
which is intuitively clear to you that the
concentration will decrease as we go along
x and knowing this we will derive Einstein’s
relation.
..
So, let us say the concentration will be more
here and the concentration will be less here
and this has this particular functional form.
Once we know this, we will follow roughly
what we did for deriving Nernst equation,
we
said that in the case of Nernst equation,
there is a current due to electric field or
the
force. Similarly, here there is a current
due to this force, this gravitational force
or
electric force, which is pulling down these
particles downwards, they will want this
particle to flow down.
.
.So, the flowing down happens in the velocity
v and this v is related to the current or
the
flow j, as we discussed previously in the
case of Nernst equation J f is C concentration
times v, which is velocity. Now, any particle
moving in water will have a velocity, which
is given by f by 6 pi eta a, where f is a
force acting on that particle, pi is the constant,
eta
is the viscosity of the water or the medium
and a is the size of the particle. So, there
are
quantities, which you should remember eta
is a viscosity, a is the size of the particle
and
f is the force acting on this particle; if
you know this much the velocity is this and
the
flow is proportional to the velocity the more
the velocity the more the flow is.
.
So, the flow downwards is C times v. So, this
can be written in a different way the C
times v and C v is forces can be written as
minus g x cap, where x cap is this direction,
so minus g x cap, so substituting this f is
minus g x cap, you get J f that flow has minus
C
g x cap by 6 pi eta a. So, this is the current
that is making, so this is the flow due to
this
attraction, this this force that it could
be gravitational attraction downwards, it
could be
the attraction due to electro static forces
or electric field could be the electric field
down
applying downwards, forcing the proteins to
move in this particular direction.
So, this could be any force whatever be the
force you wish, but that force will flow will
lead to a current or a flow given by this
particular formula, as we saw in Nernst equation
we had similar similar flow. Now, this flow
leads to one interesting thing, that the it
.makes the concentration more here and the
concentration less here. If the concentration
is more here and the concentration is less
here.
Diffusion can happen because, diffusion is
a flow from higher concentration to lower
concentration, so in principle things can
diffuse back from here to here, it can diffuse
back from lower concentration to a higher
concentration. So, here it is sorry you can
diffuse back from higher concentration to
a lower concentration, so here it is higher
concentration here it is lower concentration,
so from here to here you can think of you
can imagine that, there can be some flow due
to diffusion or flow due to concentration
change, you could think of some kind of diffusional
or diffusive flow. So, how much is
that flow.
.
So, we said that due to diffusion there can
be a current or a flow and that current J
D is
related proportional to derivative of the
concentration, as we saw previously J D is
proportional to del C by del x. And as we
go along the x C decreases, so the del C by
del
x is negative.
So, with this minus sign the flow is actually
along the x cap direction, which is in this
direction shown by this blue arrow. So, we
have a diffusion, which is basically taking
this in this particular way, we have a flow
which is in this particular way given by J
D
equal to minus D del C by del x.
..
Now, what is C? We just saw that C is proportional
to e power minus g, so we just said
that C is proportional to e power minus g
x by K B T, so that means, C is some constant
A, it would be some constant times e power
minus g x by K B T, now we also said that
J
is minus D del C by del x.
Now what is del C by del x? del C by del x
of this will be, so let us find the derivative
of
this. So, what is the derivative of this?
So, del C by del x will be A is there, derivative
of
e is e power minus g itself this itself, times
the derivative of this, which is minus g by
K
B T. So, we said that derivative of e power
K X is K e power K x, so we had a K, which
is minus g by K B T. So, that is the K, which
is coming here, so by using this relation
that we learnt that derivative of e power
a x is K e power K x, where our K here in
our K
was minus g by K B T. We have del t by del
x is a e power minus g x by K B T into
minus g by K B T.
Now look at here, what is this A e power minus
g x by K B T, what is this part? This part
is C itself, look at here this is C is a e
power minus g t x, so del C A C is C e power
minus g e x by K B T, so this is C itself.
..
So, del C by del X del C by del X is nothing
but, C itself times minus g by KBT minus c
g by KBT, so what does this mean? This implies
that, we had J which is D del C by del
X is minus D times C times g by KBT.
.
J is minus DCg by K B T, so that is what we
have here, so J D is g is del C by del X and
substituting for D, know the C is proportional
to e power minus g x by KBT and
substituting this, In this, we get J D is
D C, there is a plus sign here I might have
.mistaken there is a type of this, so if you
just substitute this minus, there is a minus
sign
here by taking this minus sign into A account
we will get a plus sign here.
So, essentially you get this, what is shown
in this square, in this rectangle here, what
is
marked here, the current upwards is DCg by
KBT along the J f downwards and J D
upwards. So, we had current downwards and
the current upwards.
(Refer Slide Time: 37: 16)
So, when you have currents in opposing direction,
we said that when both this currents
balance, that the current upwards and the
current downwards when they are, when they
balance, we reach equilibrium we call it equilibrium.
For example, if you look at this
particular point, there will be some current
upward, there will be some current
downwards and when this currents balance.
..
We have net current zero and we reach equilibrium.
This is exactly the argument that, we
discussed in Nernst, in the case of Nernst
equation. So, what does that mean?
Equilibrium means net current zero, net current
is nil, what does it mean? The total
current J D plus J E is 0, in other words
J D is equal to minus J E, there is a what
I meant
here is J f, J this is typo here should be
J f here.
.
So, what I meant here is that, J D plus J
f is 0, in other words J D is equal to minus
J f.
So, the current due to the force is equal
and opposite is equal with the opposite sign,
that
.of currents due to the diffusion, so this
is what, this is this is the condition for
equilibrium.
.
So, what do we how do we, let us try to do
with this, so what we have is J D as DCg by
KBT and J f as minus g x by 6 pi eta a and
we want J D is equal to minus J f. And that
is
that means, DCg by K B T is equal to Cg by
6 pi eta a, so which implies I can take
everything D alone keep this side and take
everything else to the other side; and what
you would get is that, D is equal to K B T
by 6 pi eta a.
So, this is says that diffusion coefficient
is equal to KBT by 6 pi eta by a, so this
is the
famous Einstein’s relation that this relates
the diffusion coefficient to Boltzmann
constant, temperature viscosity and the size
of the particle what does it say? The more
the temperature the more the diffusion coefficient,
the more the viscosity this is inversely
proportional, so if viscosity is very large
the diffusion will be less, which is intuitively
clear, something might diffuse better at water
and much less in honey, where honey has
higher viscosity to compare water. So, the
viscosity of something, which is highly
viscous let say take example of tar or honey,
you will it will be very difficult for things
to
diffuse in tar or a very highly viscous medium.
And if this either the particle is very large
look, if you look at the a is the size of
the
particle, if you have proteins that are very
huge, a very large they will not diffuse the
diffusion coefficient of those objects will
also be very small. So, this is the famous
.relation called Einstein’s relation, which
relates the diffusion coefficient to this
quantities
and the relation that we derive this relation
by the way we derive Nernst equation by
arguing.
.
That, if you have an object if you have a
beaker and particles under a force f, there
will
be two currents the concentration will be
more here, because of this force is attracting
in
it downwards and very few particles up. So,
since the concentration is less here in
concentration is more here, there will be
diffusion in this way, diffusive current and
the
current due to the force and when there are
equal and opposite.
We get by arguing that they equal and opposite
and equating them, we get this relation,
which is D is equal to K B T by 6 pi eta a.
..
So, now, we learn two three things, we learn
that x average is 0 in today’s lecture.
In the
previous lecture, that we learn that x x square
average is 2D t and we also learnt that D
is
K B T by 6 pi eta a. This is true for a spherical
particles typically, if this is not a spherical
particles, this 6 pi eta a could be something
else, but however, let us let us strict limit
ourselves to a case of spherical particles
is proteins could be the thought of us spherical
particles, so this is another relation, that
we learn today.
And this are the three important relations
as far as diffusion is concerned and using
simple arguments from calculus and vector
vectors, we could derive this relations and
this has high, very high significance as far
as as far as diffusion is concerned. Now so,
one more interesting thing I just want to
share with you is that, so in the beginning
we
graphically represented the diffusional profile,
the profile of the concentration in this
particular manner.
..
Now, what is this mathematical function that
can represent this concentration? So, it
turns out, that the mathematical function
C of x is in the case of q at diffusion, that
diffusion with no force this is not the case,
so in the case of diffusion with force we
found that, this is e power minus f x by K
B T. So, this is 
diffusion under external field,
but when there is no external field, that
is diffusion under no external field or free
diffusion C of x will be having some form
which is A e power minus some B x square,
so this function is called Gaussian function,
this has a bell shape curve this is the this
is
the bell this is the bell shaped curve, if
you plot this, so this is called a Gaussian
function.
So, this is diffusion, this is free diffusion,
there is no external, this is free diffusion
and
the free diffusion is, now if you plot this
if plot this we saw that this is this particular
form and if you plot this it will have some
form which is symmetric like this, I am not
properly drawing just go and see how the Gaussian
function plot this function if you
wish like putting some values A and B, plot
it yourself and see how does it look like?
This looks like a bell shape curve symmetric
both sides to the x axis, nicely symmetric
around the peak.
..
Now, the free diffusion is governed by this
equation del C by del t is equal to D del
square C by del x square, it turns out that
the diffusion under some external force is
govern by the following equation, so we have
this equation plus, some effect due to the
external force, so let us say the particle
when they have an external force. So, let
us if
you have just no force, this will be the equation,
but let us say each of this particle is
experiencing some force, downwards or an particular
direction.
Then each of this particles will get some
velocity due to this external force, then
the
equation will be this .. So, this is the diffusional
equation under
an external force and this is free diffusion,
where there is no external force. So, this
two
are two equations and this equation this equation
has a solution, which as of the form this
and this equation has a solution of the form
this.
So, we will we will see how do we get to the
solution of this later, if we may see this
how do we get this relation, but for this
at this movement we would not go to solve
this
equation, we will just understand the there
are such differential equations, when it comes
to probability etcetera we might revisit this
equations in a different context.
But at this movement, it is it safest to say
that, this are very interesting equations
in
biology and there are two important relations,
that as per as this equations from this
equations, from this mathematics essentially
you get two important, three important
relations which you should all remember, that
is the average distance that the particles
.moves. If you just mark a particle and ask
the question, if you do many many
experiments by marking a particle and ask
the question on average where it will go,
some time it will go to the plus 1 minus x,
some other time it will go to plus x, so on
an
average it will not go anywhere and x average
will be 0, that is what this means.
But, the x square average is 2Dt and the diffusion
coefficient is K B T by 6 pi eta a, so by
knowing this relations with this relations
we will summarize todays lecture.
.
So, the important relations are x square average
is equal to 2Dt and D is equal to K B T
by 6 pi eta a, you will come across this relations
many many times in your life at with
this will stop today’s lecture, the some
remember this relations and with this we will
kind
of completing the section on diffusion. We
will go ahead on learn new things in the
coming lectures with this we are stopping
today’s lecture. Thank you.
.
