WEBVTT
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so after doing the angular integration which
comes out to be one third the total pressure
00:31.860 --> 00:41.140
due to atoms of all speed ranges from zero
to infinity is given by p times v is one third
00:41.140 --> 00:49.930
which is due to the angular integration 0
to 2 pi d phi 0 to pi by 2 cos square theta
00:49.930 --> 00:57.880
sine theta d theta divided by 2 pi this will
comes about 2 this will be this will come
00:57.880 --> 01:04.510
about to be one third m 0 to infinity square
into da
01:04.510 --> 01:09.830
the average of the square of atomic speed
which is known as the root mean square speed
01:09.830 --> 01:19.350
is defined to be v square average is 1 by
n 0 to infinity v square times dn this is
01:19.350 --> 01:25.490
known as the root mean square velocity which
is defined as the average of the square of
01:25.490 --> 01:33.970
the velocity so that we have pv equal to n
capital n which is the total number of atoms
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or molecules times mass m divided by 3 into
root mean square velocity
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this is the famous equation of the kinetic
theory of perfect or ideal gases now let me
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write down this equation in other form also
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now a small m into capital m equal to small
n m into capital n equal to capital m which
02:04.600 --> 02:12.090
is the mass of the gas so in terms of this
so which is nothing but the m by v which will
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give you the rho where rho is the density
of gas then the above equation can be rewritten
02:18.629 --> 02:26.650
in terms of p equal to one third rho v square
average so where rho is the density of the
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gas or equal to two third e where e the kinetic
energy per unit volume which is half times
02:36.150 --> 02:38.319
root mean square velocity
02:38.319 --> 02:45.959
so this is another way to write down the equation
in the of the kinetic theory of gases first
02:45.959 --> 02:52.870
way is to write down in terms of pv second
way you can write down in terms of p in terms
02:52.870 --> 02:59.849
of the density of the gas thus the pressure
of a perfect gas is numerically two third
02:59.849 --> 03:05.969
of the kinetic energy per unit volume this
equation is called the basic equation of the
03:05.969 --> 03:09.290
kinetic theory of gases
03:09.290 --> 03:16.400
in deriving equation 1 we have made no assumptions
about the nature of molecular impact against
03:16.400 --> 03:23.819
the wall we have not made any assumptions
what is the nature of molecular impact against
03:23.819 --> 03:24.870
the wall
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now let me see what is the mean square velocity
of the molecules of a gas mean square velocity
03:31.829 --> 03:38.870
from that equation 1 we can see it is 3 p
by rho where p is the pressure and rho is
03:38.870 --> 03:45.750
the density of the gas hence the root mean
square velocity is given by is square root
03:45.750 --> 03:53.599
of 3 p by rho that means if the density of
the gas is large in that case root mean square
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velocity will be small if the density of the
gas is very low root mean square velocity
03:59.069 --> 04:02.439
will be very large
04:02.439 --> 04:08.349
knowing the pressure and the density of a
gas experimentally people have calculated
04:08.349 --> 04:13.359
root mean square velocity and hence c can
be calculated
04:13.359 --> 04:22.470
kinetic interpretation of temperature till
now i did not talk about what is the concept
04:22.470 --> 04:29.790
of temperature how the concept of temperature
has come so let me start it let us consider
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a gram molecule of a gas which occupies a
volume b from equation 1 we can get pv equal
04:37.230 --> 04:44.640
to one third m times average values of v square
which is nothing but the root mean square
04:44.640 --> 04:45.640
velocity
04:45.640 --> 04:53.300
in the kinetic consideration of a perfect
gas we use the experimental equation of state
04:53.300 --> 05:00.620
which is pv = rt in that case where r is the
gas constant and t is the absolute temperature
05:00.620 --> 05:06.912
if you will compare the equation 4 and 5 what
you will see average values of v square which
05:06.912 --> 05:14.880
is the root mean square velocity which comes
out to be 3 rt by m where r is the universal
05:14.880 --> 05:18.319
gas constant m is the mass of the gas
05:18.319 --> 05:25.030
that means if r and m are constant and which
is indeed true then average values of v square
05:25.030 --> 05:29.439
is proportional to t where t is nothing but
the temperature
05:29.439 --> 05:36.610
or other way around v square is proportional
to the absolute temperature this is the kinetic
05:36.610 --> 05:44.300
interpretation of temperature that means till
now we did not know what is temperature now
05:44.300 --> 05:50.699
we come to know temperature is nothing but
it is a measure of the average values of the
05:50.699 --> 05:57.310
kinetic energy average values of the kinetic
energy means how you have got v square by
05:57.310 --> 06:03.759
3 this is because of the isotropycity of momentum
in the momentum space
06:03.759 --> 06:09.860
where average values of vx square equal to
average values of vy square equal to average
06:09.860 --> 06:16.599
values of vz square that means in other way
around temperature is nothing but whenever
06:16.599 --> 06:25.020
there will be equilibration in the momentum
space that situation is known as he is coined
06:25.020 --> 06:31.490
as the temperature ok so according to the
kinetic theory all the molecules should have
06:31.490 --> 06:35.320
zero velocity at the absolute zero of the
temperature
06:35.320 --> 06:44.550
however this result is not justified as c
just to behaves as an ideal gas before reaching
06:44.550 --> 06:53.039
the absolute zero of temperature because whenever
you approaches the absolute zero temperature
06:53.039 --> 07:00.030
that times the concept of perfect gas ceases
to exist at various small temperature there
07:00.030 --> 07:06.790
is no gas which can be called as an ideal
gas only ideal gas the realization of ideal
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gas holds good at very large temperature and
very low density
07:12.370 --> 07:20.539
so that is the reason you cannot simply extrapolate
that as temperature tends to zero the velocity
07:20.539 --> 07:29.550
tends to zero so simply you cannot extrapolate
it again if you write capital m equal to small
07:29.550 --> 07:36.860
m into na where na is the avogadro number
then the above equation equation 6 can be
07:36.860 --> 07:46.590
written as half na times m into have average
values of v square equal to 3 by 2 rt
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where r by na equal to k where k is the boltzmann
constant so finally we got the equation half
07:54.590 --> 08:02.090
m average values of v is v square equal to
3 by 2 kt thus the mean kinetic energy of
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translation of molecules can be regarded as
a measure of temperature however in that case
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the
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average energy per atom is wholly kinetic
energy of translation now the concept of mono
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mono atomic gas will play the major role
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in case of mono atomic gas the kinetic energy
is purely translation in character whereas
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if you will go to the diatomic gases although
the diatomic atoms not interacting with each
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other but it can rotate it can vibrate with
respect to the axis passing through its equilibrium
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position so that is the reason the average
values of kinetic energy whether it will be
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the 3 by 2 kt or other values it depends on
whether it is a mono atomic gas or whether
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it is a diatomic gas whether it is it triotomic
gas
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in our cases since we have started the kinetic
theory for the mono atomic gas mono atomic
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gas have only translational kinetic energy
it cannot have rotational or vibrational kinetic
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energy because it does not have those degrees
of freedom so this is the only kinetic energy
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that a hard spherical atom on influenced by
its neighbor or field can have therefore we
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have limited ourselves to mono atomic gas
only
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diatomic and polyatomic molecules can only
can also rotate and vibrate and may therefore
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be expected to have energies of rotation and
vibration even though there are no forces
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between the interacting molecules but till
now we will not talk about the kinetic
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theory of gases for the diatomic or polyatomic
gases
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in the next part of my lectures we will confine
ourselves only to the mono atomic ideal gases
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whenever we used to claim that we are giving
a new theory new microscopic theory so we
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have to define we have to calculate some quantities
and let and tally with the experimental result
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so let us calculate the internal energy of
a gas of atoms using the kinetic theory of
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gases
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in kinetic theory it is assumed that atoms
behave as non interacting particles as we
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have told you that there is no interactions
among atoms only interaction are occurring
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between the atom between the collision of
atoms with the walls of the gas so the potential
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energy of their interest and may be neglected
the only form of energy these particles may
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possess is the translational kinetic energy
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they may not possess for example rotational
or vibrational degrees of freedom however
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in case of mono atomic gas there are no rotational
and vibrational degrees of freedom so there
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is no kinetic energy which are associated
with the rotational vibration okay so therefore
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the internal energy of the ideal monoatomic
gas is the sum of kinetic energies of all
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its atoms that means u is summed over half
mvj square who are j runs from all atoms in
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the gas okay
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so i can take out so which is nothing but
the n times half m average values of v square
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so using the above equation to replace the
kinetic energy we obtained a calculated expression
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for the internal energy of a monatomic ideal
gas which is u = 3 by 2 nr t where n is the
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number of moles r is the universal gas constant
t is the temperature so the internal energy
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of ideal monoatomic gas calculated from the
kinetic theory of gases is proportional to
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the thermodynamic temperature only
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and this agrees with the experimental result
however we have already seen from the thermodynamics
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that internal energy depends only on temperature
but we do not know how to get it so this is
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the this is the beautiful aspect of the kinetic
theory through which we have started from
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the microscopic point of view which deals
with the constituent of the system and finally
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we have lamp landed up with a beautiful expression
which is which dependent which shows that
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internal energy does depend only on temperature
only
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in the kinetic theory of gases the concept
of temperature is primarily a foreign element
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since in fact the individual atoms are characterized
by the speed only but it is suggestive that
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we should define the ideal gas temperature
in terms of the mean kinetic energy so last
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but the least i want to mention what is the
beauty of the kinetic theory of gases what
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are the most important point which should
be mentioned in in this lecture
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those above results for whatever we have derived
has been derived from the laws of physics
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and statistics rather than being formulated
from experimental data which usually which
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has been done in the theory of thermodynamics
but in case of kinetic theory of gases we
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have derived it from the laws of physics and
statistics and from the first principle of
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calculation and it has come automatically
that the expression of the canid internal
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energy
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so next we want to convince ourselves
that whatever we have derived it whether it
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can reproduce the known laws which which is
known to us since 4 to 500 years ago so let
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us derive those laws from the kinetic theory
of gases okay
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some fundamental law of ideal of phosphate
gas can be deducted from the basic equation
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which we have already derived first boyles’
and charles law we consider a gram mole of
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gas at absolute temperature t occupying volume
v so that we have pv equal to one third mn
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v square average according to the kinetic
theory the mean square velocity of the gas
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is proportional to the absolute temperature
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so v square of arras proportional to t where
equal to alpha times t where alpha is a constant
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so the above equation we see that pv is one
third mn into alpha into t
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so at a given temperature pv equal to constant
which is nothing but the boyles law as we
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know what does it mean by boyles law for a
fixed amount of ideal gas kept at fixed temperature
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p pressure and volume are inversely proportional
to each other while one increases other decreases
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this is boyles law the boyles law is based
on the empirical observation since 100 and
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100 years
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but from the kinetic theory of gases we have
got this law automatically we did not take
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any assumptions we did not get we did not
analyze any experimental result it has come
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automatically from the kinetic theory of gases
that is the beauty of any microscopic theory
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again from the same equation if you rewrite
in other way v is v is proportional to t when
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p is held constant which is nothing but the
charles law
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so what is charles law at constant pressure
the volume of a given mass of an ideal gas
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increases or decreases by the same factor
as its temperature on the absolute temperature
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scale so that means the gas expands at temperature
increases as gas shrinks as the temperature
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decreases so this is charts law and again
the emphasis is given to the kinetic theory
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of gases the charles law which is based on
the empirical observation
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since 100 and 100 years but from the kinetic
theory of gases these charles law has come
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automatically that is the beauty of the kinetic
theory of gases we have already told how charles
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law boyles law could come out automatically
from the kinetic theory of gases let me tell
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how avogadro laws could come automatically
from the kinetic theory of gases let me see
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how it can it can be derived automatically
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let us see let the two ideal gases gas 1 and
gas 2 have the equal volume at the same temperature
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and pressure when the two gases are at the
same pressure then we have from the kinetic
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theory that this would have the same kinetic
energy because we know that since p pressure
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volume and other pressure and volume remains
the same so that means the average kinetic
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energy of the gas1 should match with the average
kinetic energy of the gas2
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so that means pv equal to one third m1 n1
v1 square average = one third m2 into v square
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average when the two gases are at the same
temperature let us say they are temperature
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is also same so that means their average kinetic
energy of the gas1 and gas2 should be the
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same because temperature is a measure of the
average kinetic energy since we are saying
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temperature remains the same if the temperature
held constant
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then the average kinetic energy of the gases
1 equal to should be the same as the average
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kinetic energy of the gas2 so that means the
average kinetic energy of molecule of both
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the gases are the same so that means half
m 1 v1 square average = half m2 v2 square
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average if you will combine these two equations
we will get n1 = n2 that means number of atoms
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or molecule in gas1 should be equal to the
number of atoms or molecule in the gas2 which
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is nothing but the famous avogadro’s laws
okay
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the beauty again it should be emphasized beauty
of this derivation is that it is not based
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on the experimental data since years and years
it has come automatically from a microscopic
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theory in our case it is the kinetic theory
of gases thus equal volume of all gases under
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the same conditions of temperature and pressure
have the same number of molecules which is
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the celebrated avogadros law
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now next go to the laws of partial pressure
dalton’s law of partial pressure so as we
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have already told you in the first transparency
as we have already told you in the very first
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transparency that gases are infinitely miscible
that means gases mix in any proportion such
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has in air a mixture of gases so when you
will mix the two gases what will be the total
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pressure
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so so let me start let there be a mixture
of number of gases of densities rho1 rho2
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and of mean square velocity is v1 square v2
square in the same volume p that means in
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a given volume you have taken two gases and
you mix them then the total pressure exerted
20:30.230 --> 20:37.180
where the mixture is given by p pressure since
it is a scalar nam quantity so the pressure
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could be added together so pressure due to
the gas
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one will be half one third rho1 v1 square
average where rho 1 is the density of the
20:48.220 --> 20:56.480
gas1 v1 square average is the mean square
velocity of the gas1 +one third rho2 v2 square
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this is due to the gas2 where rho 2 is the
density of gas 2 and v2 square is their average
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velocity of the gas2 which is nothing but
the p1 + p2 which is nothing but the pressure
21:11.600 --> 21:17.420
exerted by the gas1 plus pressure exerted
by the gas2 and the pressure exerted by the
21:17.420 --> 21:18.910
other gases
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so where p1 p2 are the partial pressure exerted
by the different components of the mixture
21:24.690 --> 21:32.611
occupying the same volume v so thus the pressure
exerted by the mixture as a whole is equal
21:32.611 --> 21:41.059
to the sum of the pressures exerted separately
by the individual components so that means
21:41.059 --> 21:49.230
if there are n component gases are there so
that means the pressure of the in component
21:49.230 --> 21:56.390
gas should equal to the sum of the pressure
by the individual components that means p
21:56.390 --> 21:58.480
should be summed over pa
21:58.480 --> 22:05.800
where i ranges from 1 to n where n is the
number of components so this is nothing but
22:05.800 --> 22:12.150
the famous dalton’s law of partial pressure
again the emphasis goes back to the kinetic
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theory it has come these dalton’s law of
pressure has come automatically from the kinetic
22:18.970 --> 22:25.270
theory of gases without without any experimental
data
22:25.270 --> 22:36.610
grahams law of diffusion this is a very famous
law of diffusion what is it so now let
22:36.610 --> 22:44.650
us take two gases gas1 and gas2 and obviously
since there are two different gases they are
22:44.650 --> 22:49.391
root mean square velocities will be different
let us take the root mean square velocity
22:49.391 --> 22:57.690
of the gas1 and gas2 are v1 and v2 respectively
and their densities are also rho 1 and rho
22:57.690 --> 23:04.230
2 respective then we know from this relation
of this relation of group mean square velocity
23:04.230 --> 23:10.690
which is nothing but the square root of 3
p by rho we can write down the root mean square
23:10.690 --> 23:13.490
velocity for the two gases separately
23:13.490 --> 23:23.610
first one is v1 = root over 3p by rho 1 where
rho 1 is the density of the gas1 and velocity
23:23.610 --> 23:30.450
would mean square velocity v2 for the gas2
is 3 p by rho 2 where rho 2 is the density
23:30.450 --> 23:38.580
of the gas 2 now if i will take this ratio
what i will get v1 by v2 is equal to square
23:38.580 --> 23:46.090
root of rho 2 by rho 1 so that means root
mean square velocity is proportional to proportional
23:46.090 --> 23:53.820
to 1 by square root or it is inversely proportional
to the square root over their density
23:53.820 --> 23:59.440
that means if the density of the gases are
large their root mean square velocity will
23:59.440 --> 24:05.970
be very small if the densities are very small
then the root mean square velocity will be
24:05.970 --> 24:12.740
very large so using the density dependent
or root mean square velocity the gas diffuse
24:12.740 --> 24:18.840
accordingly thus the rate of diffusion of
a gas universally proportional to the root
24:18.840 --> 24:24.470
square root of the density this is famous
grahams law of diffusion of gases
24:24.470 --> 24:34.050
using this grahams law of diffusion of gases
we can see in our nature why one gas are diffuses
24:34.050 --> 24:41.120
more rapidly compared to the other gas because
of their origin of the root mean square velocity
24:41.120 --> 24:45.030
depends on their density inversely
24:45.030 --> 24:51.760
we can calculate some various constant using
the kinetic theory of gases however we know
24:51.760 --> 24:59.250
the values of constant earlier because of
some experimental data but the values of some
24:59.250 --> 25:05.780
constant will come beautifully from the kinetic
theory of gases those are very famous constants
25:05.780 --> 25:12.830
first is avogadro number the number of atoms
or molecules in one gram mole of gas so if
25:12.830 --> 25:19.190
m is the mass of a molecule then the molecular
weight is capital m equal to smaller into
25:19.190 --> 25:23.309
number of particles which could be they avogadro
number
25:23.309 --> 25:30.800
so avogadro number na equal to molecular weight
by the mass of a molecule for a hydrogen gas
25:30.800 --> 25:40.850
m = 2 molecular weight and small m mass of
a hydrogen molecule is 332 into 10 to the
25:40.850 --> 25:50.220
-24 gram if we will put there then num avogadro
number will comes out to be 2 by 332 into
25:50.220 --> 26:01.330
10 to the -24 = 6024 into 10 to the -23 number
of particles per mole so this is the avogadro
26:01.330 --> 26:06.960
number which comes out to automatically from
this
26:06.960 --> 26:13.850
second universal gas constant which is a very
famous constant which is known as the universal
26:13.850 --> 26:22.300
gas constant r for one gram mol gas we have
the relation pv = rt because n has become
26:22.300 --> 26:30.700
equal to 1 so r = pv by t let us calculate
at normal pressure and temperature ntp we
26:30.700 --> 26:40.170
know at normal pressure p is 76 centimeter
of mercury so which is nothing but in terms
26:40.170 --> 26:51.220
of calculating the uni cgs units 76 into 136
into 78 1 dynes per centimeter square
26:51.220 --> 27:00.410
which is equal to 102 into 10 to the power
15 newton per meter square volume as we know
27:00.410 --> 27:10.020
in the normal pressure and temperature one
mole gas occupy 224 liter volume so v is 224
27:10.020 --> 27:22.250
into 10 to the 3 cc temperature is 220 mm
273 degree kelvin so if you substitute all
27:22.250 --> 27:31.530
these values in the equation r = pv by t we
will r will comes out to be 831 joule per
27:31.530 --> 27:39.500
mole kelvin in terms of other in terms of
calorie it will comes out about 19 and calorie
27:39.500 --> 27:43.059
per mole degree kelvin
27:43.059 --> 27:51.760
first last but the least boltzmanns constant
which is a very fundamental in the sense these
27:51.760 --> 28:00.160
boltzmann constant is the first constant which
connects to the microscopic world through
28:00.160 --> 28:05.590
the famous relation of entropy equal to entropy
is proportional to the logarithmic of the
28:05.590 --> 28:10.830
number of accessible microstates and proportionality
constant is boltzmanns constant
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this relation is fundamental in the sense
this is the first relation which connects
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the macroscopic world means thermodynamics
to the microscopic world the number of possible
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microstates through the boltzmanns constant
you can calculate boltzmanns constant through
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this beautiful relation which we have already
come across in the kinetic theory of gases
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earlier this here let me show you the earlier
transparency
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this r by a equal to k where k is the boltzmann
constant so k = r by na na is the number avogadros
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number r if you will substitute the values
of r and na then k will comes out to be 1
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38 into 10 to the power 16 r per degree kelvin
in terms of joule it will comes out to be
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138 into 10 to the -23 joule per kelvin