WEBVTT
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right now i will start how to derive
vanderwaals equation of state from more fundamental
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point view means how to how to start from
the basic interaction of the among the constitution
00:45.140 --> 00:51.980
and how to derive it from a more fundamental
point of view or from the first principle
00:51.980 --> 01:02.480
of stages so let me try the how to deduce
the equation of state for the real gases using
01:02.480 --> 01:08.780
the virial theory this is not the rigorous
virial theorem very often encountered in the
01:08.780 --> 01:10.220
statistical mechanics
01:10.220 --> 01:17.300
so here we have started just from the simple
classical mechanics approach now let me start
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it clausius first did this equation of state
for an imperfect or real gas more rigorously
01:28.490 --> 01:36.750
using this virial theorem so how he had done
it let me start it step by step so now let
01:36.750 --> 01:45.410
us consider a gas of n identical molecules
each having mass m in a cell volume v the
01:45.410 --> 01:54.410
total the total energy e of the system consists
of the total kinetic energy + total potential
01:54.410 --> 01:55.410
energy
01:55.410 --> 02:03.550
total kinetic energy means summed over kinetic
energy of individual molecule plus the interaction
02:03.550 --> 02:09.190
energy obviously here we will take simple
pearwise interaction means which depends only
02:09.190 --> 02:17.470
on the relative separation between the molecules
ok so total energy of the system e = summed
02:17.470 --> 02:26.690
i = 1 to over n half mci square + 5 however
ci is the velocity of the ith molecule and
02:26.690 --> 02:29.520
phi is the total potential energy of the system
02:29.520 --> 02:39.250
any way we are not going to deal the any real
gas still we have taken from assumptions closer
02:39.250 --> 02:47.430
to ideal gas slightly deviated from the ideal
gas that in the case of verified the potential
02:47.430 --> 02:55.450
energy is assumed to be pear wise additive
that means phi summed i less than j u of ri
02:55.450 --> 03:03.709
j where u of r ij pear potential between the
molecules i amd j separated by distance which
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is nothing mode of ri – rj
03:07.340 --> 03:18.000
let xi yi zi be the cartesian coordinates
of the molecular xyz component of the position
03:18.000 --> 03:26.879
of the molecule ith molecule and similarly
f of xi f of yi f of zi be the components
03:26.879 --> 03:34.870
of the force acting on the molecule i then
we have three separate newton’s law m d2xi
03:34.870 --> 03:49.739
by dt square = f of xi m d2yi by dt square
= f of yi m d2zi by dt square = f of zi newtons
03:49.739 --> 03:56.060
law in three co-ordinate x coordinate y coordinate
and y coordinate
03:56.060 --> 04:03.260
similarly the kinetic energy of the ith molecule
is am half mc square which is nothing but
04:03.260 --> 04:11.260
the summed over m because it is the identical
molecule same will come out m x i dot square
04:11.260 --> 04:21.329
+ yi dot square + zi dot square where xi dot
yi dot zi dot t represent the velocity of
04:21.329 --> 04:30.770
the molecule in the x y z direction of the
ith molecule ok using the relation simple
04:30.770 --> 04:43.330
identity differential calculus half d2 by
dt2 xi square = xi dot square + d2 dxi by
04:43.330 --> 04:49.240
dt ok dtxi dt square ok
04:49.240 --> 04:58.400
if you will use this relation in the kinetic
energy equation so for individually for individual
04:58.400 --> 05:06.150
components means x component kinetic energy
is half m xi dot square equal to half times
05:06.150 --> 05:19.860
half equal to 4 so one fourth m by d2 by dt
xi square – half m d2 xi by d2 square xi
05:19.860 --> 05:35.910
ok so 1 by 4 m d2 by dt xi dot xi square – half
md2 xi by dt2 is nothing but f of xi in so
05:35.910 --> 05:45.471
half f of xi to xi so this is the kinetic
energy of the x component kinetic energy can
05:45.471 --> 05:47.370
be written in this way
05:47.370 --> 05:54.319
similarly we can follow the same identity
for the y component and z component also and
05:54.319 --> 05:55.349
get them together
05:55.349 --> 06:07.849
what will get it half mci square = 1 by 4
m d2 by dt2 xi square + yi square + zi square
06:07.849 --> 06:21.600
– half of f of xi x component of force into
displacement xi in the x direction + y component
06:21.600 --> 06:28.509
of force multiplied by the displacement in
the y direction + z component of the force
06:28.509 --> 06:35.290
multiplied by the z component of the displacement
obviously of this is for the repeated indices
06:35.290 --> 06:39.979
tells that summation which is for the ith
molecule
06:39.979 --> 06:50.470
so if you rewrite this in terms of the r co-ordinate
spherically what we can get this 1 by 4 m
06:50.470 --> 07:01.700
d2 dt2 ri square - half ri dot fi ok we now
sum the expression of all molecule of the
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gas and take a average over a sufficiently
long time so we get half summed over average
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over mci square that means we are taking the
time average equal to 1 by 4 m summation average
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of dt by dt2 ri square - half time average
of summed over ri fi fi dot
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time average has come from very fundamental
fact that the motion of the atoms or molecule
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in the gas completely random they are always
under the experience of random force so we
07:49.349 --> 07:57.639
are interested over the time value of those
average quantities over a large time scale
07:57.639 --> 08:06.930
obviously that time is always much much greater
than the collision all time over and asymptotic
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time scale so this so that is that gives the
concept of time average
08:13.449 --> 08:19.990
because the force each and every molecule
or atom always experiencing by random force
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so we have to take some time average of it
okay so finally where fi is the force acting
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on the molecular which is coordinating with
ri so that is a ri dot fi dot the summation
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extent over all molecules and the bars represent
the time average which starts from 0 to tau
08:43.120 --> 08:48.740
obviously that tau is larger than the collision
time scale ok
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so if you all though this is not i do not
want to tell this thing in this lecture but
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this thing if you look for the lange dynamics
or faulkner plain equation there you will
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see there are two type of time scale one time
scale which are less than the collision time
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scale and random force still exist another
time scale where time is much much greater
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than the collision time scale that actually
we are looking for the time average over a
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very long infinite time which is known as
the steady state system
09:27.230 --> 09:35.390
so the first time if you take the time average
you will get it what i will get it the first
09:35.390 --> 09:41.460
time in the right hand side of equation can
be written down that 1 by 4 just use see it
09:41.460 --> 09:53.180
1 by 4 m average of dt2 by dt2 ri dot square
bar means time average if you write down the
09:53.180 --> 10:00.100
summation in terms of the integral form of
from the definition of obtaining any average
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quantities 1 by 4 tau basically 1 by t t is
the time 0 to t f of t dt that is the general
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definition of time average
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so same thing applies here also 1 by 4 tau
1 by 4 is already there so 1 by 4 tau m 0
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to tau summed over i dt by dt2 ri square dt
so if you do it then i can always write down
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this thing as 1 by 4 m summed over ri 1 by
tau d2 ddt of ri square 0 to tau so this thing
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has come after doing it by part by part if
you do this integration by parts then this
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thing you can get it so then it will put the
upper and lower limits of time so you will
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get it 1 by 4 m summed over i 1 by tau 2ri
dri by dt because ddt of ri square is 2ri
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by dt 0 to tau
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since dri by dt is nothing but the velocity
of the ith molecule since i told you that
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each molecule are always experiencing very
random force so the velocity which is nothing
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but the dri by dt the velocity of the molecules
are always fluctuating rapidly because the
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velocity arises because of the random forces
so obviously velocity will fluctuate very
11:40.630 --> 11:48.100
irregularly so if you average it over a long
time velocity you will get 0
11:48.100 --> 11:57.070
so that is the reason dri by dt for a molecule
fluctuating irregularly with time hence 1
11:57.070 --> 12:04.730
by tau dri dt 0 to tau obviously reduces to
0 when tau is very large so that finally first
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term is 1 by 4 m d2 dt2 riy square average
will be the 0 this is because of the force
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motion is completely random this is the basic
physics behind this average so the equation
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finally reduced to half average of the total
kinetic energy average summed over i mci square
12:34.900 --> 12:46.150
bar means average is nothing but - half summed
over i ri dot fi which is let us all say x
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so this is varial theorem of clausius and
where x is called the virial of this system
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so what does it mean by virial theorem the
total kinetic energy of all the molecules
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is nothing but is equal to the x which is
nothing but the virial of the system once
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if i will calculate x then i will get what
is the total value of the kinetic energy average
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kinetic energy ok
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according to this theorem the total kinetic
energy of translation of molecules of a gas
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in the steady state is equal to its virial
x as i have already told where x is nothing
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but the - half summed over i ri dot fi and
then take its time average this is nothing
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but virial of the system so in terms of means
square velocity c square bar equation 6 can
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be rewritten has half n is the total number
of molecules nmc square
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this is the definition of mean square velocity
which is nothing but the half summed i mci
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square which is nothing but the x each molecule
in the gas experiences the restoring straining
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force of the container and also the force
exerted by all other molecules in the container
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so each molecule in the gas are experiencing
two kinds of force one is the restraining
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forces of the container and other force which
is exerted by all the remaining molecules
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in the container so virial obviously since
forces to component so virial x virial of
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the system can be decomposed into two parts
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one part is due to the external force which
is another xe another part is the due to the
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intermolecular forces so x can be written
as x = xe + xi e stands for the external forces
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i stand for the intermolecular forces so if
the pressure now calculate xe and xi re separately
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then i will add together and then i will get
this equation of state so what is xe if the
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pressure exerted by the gas is p then the
virial of the external forces like that of
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an ideal gas is nothing but xe is nothing
but the 3 by 2 pv that you that is well understood
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second part on the other hand virial of the
internal forces is given by xi = half summed
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over i ri dot del phi by del ri average ok
what does how this has come let me explain
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it as you know xi is half summed over i ri
fi dot intermolecular force and intermolecular
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forces derived by the potential energy so
fi is nothing but derived by the potential
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energy which is the as you have told you phi
is the total potential energy
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phi is the total potential energy of the system
and the internal intermolecular force arises
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because of the inter because of the potential
energy so f is derived from the potential
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energy which is nothing but dphi by dri ri
bar is nothing but the average value of r
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ok substituting these values of i and using
ri = mod of ri – rj so the virial of the
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internal forces can be rewritten as half summed
over i ri ri dot del u rij by del ri bar take
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its total time average
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so what if you skip this notation and notation
is well understood because of this repeated
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indices that half summed over r du del u r
by dr then take its average so i skip the
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indicess which is well understood which is
the repeated indices so substituting these
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values of xi and xe so the total x means total
viral of the system is which is 3 by 2 pv
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which is nothing but due to external force
which is just like the ideal gas 3 by 2 pv
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+ d2 the intermolecular force intermolecular
potential which is half summed over r del
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u r by dr and take its time average
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according to kinetic theory of gases the mean
kinetic energy of translation of a molecule
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is half mc bar square = 3 by 2 kt that we
have already talked about in the in my earlier
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lectures that kinetic energy is a measure
of the temperature is a measure of the reference
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of kinetic energy half mc bar square = 3 by
2 kt obviously this kinetic energy only to
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the translational motion ok so then pv =nkt-
then if you rewrite this equation that means
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you will bring pv in the left hand side then
you will get it
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pv = nkt- one third summed over r dur by dr
then take its time average this is the general
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form of the equation of state for an imperfect
gas derived from the virial theorem and holds
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goog accurately so if you feel it where is
the first glance if you see this if you will
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put what is the u is the measure of intermolecular
interaction if in the ideal gas equation of
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state we know there are no intermolecular
interaction
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so if you put u of r = 0 then it reduces to
the ideal equation of state so this is the
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first check up of this equation of state that
we are going in the right direction so if
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you want to evaluate exactly then you have
to know what is the form of intermolecular
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interaction so if you know the form of intermolecular
interaction then you will plug in the expression
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in form of dur by dr then calculate is time
average do all sorts of summation and finally
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you can get the equation of state for the
real gases which is which is true in principle
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it is true
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but as you know that its exact evaluation
of the equation is not possible since du by
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dr is not known precisely which is why it
is not known precisely because we do not know
20:41.750 --> 20:51.620
what is the exact form of intermolecular interaction
because we know atom is formed by one of the
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most fundamental interaction which is nothing
but the electrostatic coulomb interaction
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because suppose let us take hydrogen molecule
hydrogen atom which is bound because of the
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electrostatic coulomb interaction between
electron and proton
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so this is well understood because you know
in quantum mechanics this problem analytically
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solved but how molecule is formed by two hydrogen
atom is x2 exactly cannot calculate how you
21:28.029 --> 21:34.471
cannot calculate how two hydrogen atom from
a hydrogen molecule so you have to look for
21:34.471 --> 21:41.720
some empirical equation of state because here
exactly in principle the formation of molecule
21:41.720 --> 21:47.450
should be obtained through the electrostatic
are electromagnetic interaction between all
21:47.450 --> 21:49.830
point like electric charges
21:49.830 --> 21:56.970
but since here two atoms are not electrically
charged object so in principle they are not
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interacting electromagnetically directly but
they are interacting electromagnetic through
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in the shielded form which people is to call
leonard jones potential vandervaal forces
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etc they are electromagnetic forces they are
which is in the shielded form in the dipole
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approximation quadruple approximation you
can expand atom in terms of dipole quadruple
22:24.659 --> 22:28.179
then two atom can interact with dipole interaction
or quadruple interaction which is nothing
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but the select form of electromagnetic interaction
so in principal this equation of state this
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is the equation of state for an imperfect
gas but provided if you know u of r exactly
22:46.999 --> 22:56.749
u of r tells you the intermolecular interactions
so but in general u of r is not known to us
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reason is that you u of r is the intermolecular
interaction
23:02.169 --> 23:09.370
so if i know u of r exactly there the i will
substituted there and take its time average
23:09.370 --> 23:17.629
then i can get the equation of state for an
imperfect gas for the readymade for the readymade
23:17.629 --> 23:25.840
check you see if you put u of r = 0 it immediately
deduces to the ideal equation of state that
23:25.840 --> 23:31.820
we are going in the right direction because
in the ideal equation of state interactions
23:31.820 --> 23:35.590
among the constituent is 0 there are no interaction
23:35.590 --> 23:44.029
so it gives rise to the ideal gas equation
of state but if you want to evaluate exactly
23:44.029 --> 23:51.460
this expression which is given equation 24
this is not possible because we do not know
23:51.460 --> 23:57.950
you are exactly the reason is that the reason
why you do not know u of r exactly it has
23:57.950 --> 24:05.519
many reasons because you do not know how the
two atoms or two molecules interacting among
24:05.519 --> 24:12.919
themselves because we know at leastlet us
take a simple example to clarify this statement
24:12.919 --> 24:21.640
suppose take single hydrogen atom was taking
how a hydrogen atom forms how the bounce rate
24:21.640 --> 24:29.019
of hydrogen atom forms we know exactly reason
we know exactly because hydrogen atome form
24:29.019 --> 24:35.539
the due to a one of the most fundamental interaction
which is known as the electromagnetic interaction
24:35.539 --> 24:43.740
between electron and proton and it seems there
this interaction is known fully so it is as
24:43.740 --> 24:46.919
you know although i do not want to tell you
the things in details
24:46.919 --> 24:54.899
schrodinger equation which tells the again
the energy of this hydrogen atom which exactly
24:54.899 --> 25:02.490
soluble analytically for this attractive coulomb
interaction between electron and proton but
25:02.490 --> 25:09.820
how the true hydrogen atom forms hydrogen
molecule that you know exactly analytical
25:09.820 --> 25:15.659
we do not know exactly reason is that two
hydrogen atoms cannot interact fundamentally
25:15.659 --> 25:18.509
electrostatic coulomb attraction
25:18.509 --> 25:25.940
reason is that atom is not an electric charged
particle atom is not an electric charge particle
25:25.940 --> 25:33.400
but about you know that two hydrogen atom
interact each other to form a hydrogen molecule
25:33.400 --> 25:39.779
these interaction nowadays it can be explained
in terms of the dipole or quadruple interactions
25:39.779 --> 25:47.700
among the two hydrogen atom and these interaction
these problem is not solve analytically people
25:47.700 --> 25:54.730
is to take that the two atoms are interacting
through the dipole or quadruple interactions
25:54.730 --> 26:05.990
sometimes people use to call this is a shielded
electromagnetic interactions so
26:05.990 --> 26:13.460
these things are not easily even for a simplest
case which is the hydrogen atom which consists
26:13.460 --> 26:20.390
of only one electron if you want to look for
the intermolecular interaction having that
26:20.390 --> 26:26.990
many multi electron system so this is very
complicated thing however nowadays people
26:26.990 --> 26:32.330
are able to improve not using the density
functional approach
26:32.330 --> 26:38.149
anyway i am not going to digress i am not
going to tell it because it will be too much
26:38.149 --> 26:44.669
digression for this stock nowadays people
have developed a new method which is the remarkable
26:44.669 --> 26:49.260
in the theoretical physics which is known
as the density functional approach through
26:49.260 --> 26:56.919
which people are able to extract intermolecular
almost the exact exact form of the intermolecular
26:56.919 --> 27:04.029
interaction if you can extract intermolecular
interaction they in principle you can get
27:04.029 --> 27:08.889
a exact form of the equation of state for
the real gas
27:08.889 --> 27:17.909
but for the sake of simplicity for the real
gas for the analytical point of view it is
27:17.909 --> 27:27.110
very hard to get an analytical equation of
state for any real gas so let me start it
27:27.110 --> 27:35.759
let me get some of this let me extract some
of this form of this equation of state
27:35.759 --> 27:44.450
for the real gas sohere i would like
to tell you something about the maxwell boltzmann
27:44.450 --> 27:50.529
distribution as you remember as you quite
remember in my earlier lecture maxwell boltzmann
27:50.529 --> 27:52.879
distribution of speed
27:52.879 --> 28:02.049
maxwell boltzmann distribution has two kinds
of thing one it tells how many atoms or molecules
28:02.049 --> 28:09.600
having the velocity in between v and v + dv
or he tells the velocity distribution of molecular
28:09.600 --> 28:17.869
or energy distribution of molecules how atoms
or molecules are distributed among the energy
28:17.869 --> 28:25.509
in terms of e and e + de and this is one maxwell
boltzmann distribution law another maxwell
28:25.509 --> 28:32.019
boltzmann distribution that i have also told
you at the end of the lecture of boltzmann
28:32.019 --> 28:33.019
distribution
28:33.019 --> 28:40.779
there i told you how this molecules are atoms
or molecule are in general how the particle
28:40.779 --> 28:46.950
are distributed according to their potential
energy distribution which obviously potential
28:46.950 --> 28:52.559
energy tells you in the coordinates space
means how the particle are distributed in
28:52.559 --> 28:58.860
the coordinate space so you see there are
two kinds of beautiful distribution of the
28:58.860 --> 29:05.139
same maxwell boltzmann one tells the distribution
of molecular in the momentum space
29:05.139 --> 29:12.160
in its fourier space which is nothing but
the coordinates it can also tell how the particles
29:12.160 --> 29:19.499
are distributed in the code in space between
r and r + dr so let me explain that maxwell
29:19.499 --> 29:28.259
boltzmann distribution ok so let me tell few
things more maxwell boltzmann distribution
29:28.259 --> 29:37.889
in the velocity space tells how quickly system
will be equilibrated or how system will achieve
29:37.889 --> 29:45.309
thermal equilibrium quickly this is the these
will be dealt with the maxwell distribution
29:45.309 --> 29:47.399
law of speeds
29:47.399 --> 29:53.529
the maxwell boltzmann distribution in the
coordinate space is tells how the particles
29:53.529 --> 30:00.320
will be how they diffuse the property of the
particle which is in the coordinates space
30:00.320 --> 30:01.320
ok
30:01.320 --> 30:12.119
so now let me start one can evaluate approximately
this del u by del r times del u by del r and
30:12.119 --> 30:18.409
its average this thing somebody can evaluate
approximately using the maxwell boltzmann
30:18.409 --> 30:24.330
law according to its number of molecules per
unit volume in the region of the potential
30:24.330 --> 30:31.450
energy ur is given n of r subscript r means
it is a function of r n of r = n of where
30:31.450 --> 30:41.369
r = 0 e to the power of –ur by kt that we
have already derived by in my last lecture
30:41.369 --> 30:44.179
in the maxwell boltzmann distribution of speeds
30:44.179 --> 30:51.400
where n naught is the number density in the
reason for potential energy is 0 where u of
30:51.400 --> 30:54.669
r is 0 ok
30:54.669 --> 31:01.879
let us first calculate the contribution of
one molecule the average number of other molecules
31:01.879 --> 31:11.049
lying between r and r + dr from this molecule
is nr 4pi r squared dr and hence the contribution
31:11.049 --> 31:21.169
of 1 molecule to r u prime r u prime means
del u by del r arising due to all other molecule
31:21.169 --> 31:33.100
in the spherical shell of radii r and r +
dr is 4 pi nr u prime r rq dr which is if
31:33.100 --> 31:44.519
4 pi n naught just you substitute n of r so
4 pi nr exponential - ur by kt u prime r qdr
31:44.519 --> 31:50.379
so then the contribution of and molecules
of the gas is if you will take summed over
31:50.379 --> 32:01.700
r del u by del r and its average = n by 2
0 to infinity 4 pi n naught exponential - ur
32:01.700 --> 32:11.259
by kt at u prime r rq dr to testing the account
over counting ok so we have divided this expression
32:11.259 --> 32:18.600
by 2 to avoid the duplication of every pair
of molecules
32:18.600 --> 32:26.140
when n is large and putting n naught = n by
v which is nothing but the density than equation
32:26.140 --> 32:37.019
14 gives that means that equation of state
is pv = nkt – 2 pi by 3 n square by v substitute
32:37.019 --> 32:43.350
n naught it which is nothing but n by v there
is another n outside of the situation that
32:43.350 --> 32:54.899
gives you the n square so pv = nkt - 2 pi
by 3 n square by v 0 to infinity e to the
32:54.899 --> 32:59.090
power of -ur by kt prime r rq dr
32:59.090 --> 33:06.440
so if you integrated by parts this equation
can be written in a simplified form pv =nkt
33:06.440 --> 33:20.549
+ 2 pi n square by v kt 0 to infinity 1 – e
to the power – ur by kt r square dr if you
33:20.549 --> 33:27.409
assume ur tends 0 when r tends to infinity
obviously i should assume this because we
33:27.409 --> 33:34.450
know whenever in the ideal gas approximation
people use to take why there are no interaction
33:34.450 --> 33:36.919
potential between the molecule
33:36.919 --> 33:43.200
reason is that idea behind that assumption
is the since the separation between the molecule
33:43.200 --> 33:49.409
is far far large then the range of interaction
so that is the reason there is no interaction
33:49.409 --> 33:59.519
potential so obviously if you take r tends
to infinity ur tends to 0 so putting to vanderwaals
33:59.519 --> 34:05.970
the molecules are assume to be hard sphere
of diameter sigma and attract each other with
34:05.970 --> 34:10.520
weak force who is rapidly decreases as the
distance increases
34:10.520 --> 34:17.929
that means he has taken some potential energy
form so whenever two atoms or molecules close
34:17.929 --> 34:26.740
together they will stick to whenever they
touch each other they experiencing huge potential
34:26.740 --> 34:32.290
whenever the separation distance is very large
they are not feeling any interaction due to
34:32.290 --> 34:39.150
other so he has taken some kind of form of
this potential which is below which is given
34:39.150 --> 34:48.080
as ur = infinite when r tends to sigma he
takes some finite interaction u0 of r when
34:48.080 --> 34:57.490
r greater than sigma if r is much much greater
than sigma then it will be 0 so if you plug
34:57.490 --> 35:03.990
in this kind of potential energy form in the
above expression it so what you will get it
35:03.990 --> 35:12.060
you will get pv = nkt + 2 pi n square by v
kt then you split this integration into two
35:12.060 --> 35:20.050
form 0 to infinity you can make it you can
make it into two form 0 to sigma and sigma
35:20.050 --> 35:32.430
to infinity so 0 to sigma you know u fo r
= infinity e ot the power minus infinity is
35:32.430 --> 35:39.820
zero so only one will survive so r square
dr so that is the reason first term of this
35:39.820 --> 35:45.260
integration reduces to 0 to sigma r square
dr
35:45.260 --> 35:52.510
second part is sigma to infinity where it
will take the form u0 of r this phone just
35:52.510 --> 36:02.660
you see this from u0 r then sigma to infinity
1- e to the power of –u0 r by kt r square
36:02.660 --> 36:13.030
dr so you do this integration so nkt + 2 pi
n square by v kt one third sigma cube + i
36:13.030 --> 36:21.870
by kt sigma to infinity u0 of r r square dr
so let us take let us say b = 4 pi 2pi by
36:21.870 --> 36:32.190
3 n sigma cube and let us take assume let
usa take a = 2 pi n square 0 to infinity u0
36:32.190 --> 36:34.320
r r quare dr
36:34.320 --> 36:44.250
then the above equation will be deduced in
terms of a and b pv = rt + rt by v into b
36:44.250 --> 36:52.030
– a by rt where a and b are vanderwaals
constants so equation 20 can be written in
36:52.030 --> 37:02.250
the form p= rt by v 1 + b by v – a by v
square so which is rt if you we will 1 + b
37:02.250 --> 37:07.080
by v – a by v square which is rt which you
will 1- b by v when you bring it to the denominator
37:07.080 --> 37:10.130
when b is much much smaller than capital v
37:10.130 --> 37:18.240
so rt by v - b - a by v square if you will
bring a by v square on the left hand side
37:18.240 --> 37:25.630
so you will get p + a by v square into b - v
= rt which is nothing but the famous vanderwaals
37:25.630 --> 37:34.270
equation of state so finally we have derived
vanderwaals equation of state from the more
37:34.270 --> 37:44.170
fundamental description of the atoms and molecules
which constitute a system so that is the beauty
37:44.170 --> 37:52.050
through which which is one kind of first principal
calculation in physics which tells which bring
37:52.050 --> 37:53.840
the vanderwaals equation of state
37:53.840 --> 38:01.390
although it was derived it very herusitically
by incorporating the salient feature of the
38:01.390 --> 38:10.670
real gases but anyway we are able to derive
a form of vanderwaal equation of state from
38:10.670 --> 38:12.740
the virial theorem
38:12.740 --> 38:18.570
which is theoretically correct of two terms
one maybe because if you will see it we have
38:18.570 --> 38:27.570
terminate the series 1 by v at etcetera term
so so that is the reason who is this equation
38:27.570 --> 38:36.160
is theoretically correct up to term i by v
more correct to use equation 20 instead
38:36.160 --> 38:47.680
of 22 but but due to its form of simplicity
this form of equation is more frequently use
38:47.680 --> 38:53.350
so just for the sake of completeness if you
want to write down the vanderwaals equation
38:53.350 --> 38:55.170
of state for n moles of gas
38:55.170 --> 39:03.620
then you substitute small v to capital v by
n so we will get p + n square a by v square
39:03.620 --> 39:17.910
into v – b = nrt