WEBVTT
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today i will be giving lecture on the maxwell
distribution law of velocities so let me start
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it actually before coming to the topic let
me tell you what are the random quantities
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and the probability first of all i should
familiarize with these quantities then i will
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start what is the derivation of maxwell distribution
law of velocities random quantities and probability
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let us consider a small region in the space
occupied by an ideal gas
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molecules of the gas move arbitrarily we cannot
specify the time when an individual particle
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will be located inside this region in the
course of its motion hence the location of
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a particle in a given region of space is an
random event the motion of micro particles
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like electrons protons photons etcetera is
described by quantum mechanics not by classical
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mechanics it is so it is impossible in principle
to predict their locations as well as they
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are dynamic simultaneously
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which is nothing but the uncertainty principle
between position and momentum hence the position
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of a micro particle in a given region of space
is a random event by nature suppose as we
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know according to the uncertainty principle
any two conjugate variable cannot be measured
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simultaneously accurately suppose if i can
localize a particle suppose if i know the
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position of a particle then its momentum could
be completely uncertain so that is the reason
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simultaneously if i know its dynamics suppose
if i know its momentum then its position is
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completely uncertain
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so that is the reason the position of micro
particle in a given region of space is a random
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event by nature thus most of the events in
a many particle system are random the coordinates
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and velocities of the molecules in a gas are
random quantities in order to describe the
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behavior of random quantities we introduce
the concept of probability in fact in our
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lecture we will derive what is the probability
of finding a molecule having velocities between
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v and v + dv in a gas
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so that is the reason that is the quantity
we are going to calculate now let me start
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what is maxwell distribution law of velocity
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we consider an ideal gas in a vessel of volume
v a gas consists of large number of molecules
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moving arbitrarily in all directions with
all possible velocities ranging from 0 to
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infinity which are changing continuously in
magnitude as as well as directions due to
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collision with other molecules this is the
first assumptions of the derivation of maxwell
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distribution law of velocities
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that means the velocities of the molecules
in a gas can will be from 0 to infinity they
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can take any velocity the infinity means maximum
limit of the velocity of light is obviously
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velocity of light in free space but as we
have already told in the lecture series of
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kinetic theory of gases that that will not
introduce any error so that is the reason
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for the sake of the calculations we are saying
the velocity of the molecules in a gas can
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take values from 0 to infinity
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the distribution of these velocities is governed
by a certain law known as the law of distribution
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of velocities it will be known as the maxwell
distribution law of velocities obviously it
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was derived by maxwell in 1859 now let me
start it by assuming n be the number of molecules
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per unit volume of the vessel and let a molecule
move with a velocity c in an arbitrary directions
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whose components of along the x y z direction
are u v w respectively
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and obviously so the length of this velocity
vector c square = u square + v square + w
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square like the radius vector in the coordinate
space is r square = x square + y square +
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z square obviously the radius is measured
from the origin which is defined as the 0
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0 to x y z similarly here the in the velocity
space the origin coincides with the 0 0 and
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the a point in the xyz plane is uvw so c square
= u square + v square + w square
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if nu the number of molecules per unit volume
having velocity u then the number of molecules
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per unit volume having the velocities lying
between u and u + du is given as a new deal
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so this is the first thing if nu be the number
of molecules having velocities u and u + du
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then the number of molecules per unit volume
having the velocities lying between u and
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u + du is a new deal ok when nu is some function
of u then nu du is f of u du ok
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then the number of molecules per unit volume
having velocities lying between u and u +
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du will be nu du = n into fu du and hence
the probability that a molecule selected at
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random will have the velocities lying between
u and u + du is fu du basically you have to
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divide the equation 4 by a so you will get
the probability which is nothing but fu du
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we assume that a fu is independent of v and
w similarly the probability for molecules
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with velocities lying between v and v + dv
is a vdv and the probability of finding the
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molecules having velocities u w and w + dw
is f of w dw here we have assumed that these
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separate these events of uvw these are independent
to each other then we will use a theorem in
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the probability theory that i am going to
tell it now
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if the probabilities of occurrence of two
independent events a and b are pa and pb then
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the probability of simultaneous occurrence
of two independent event is given by p of
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ab = it is a product of two probabilities
pa times pb this is the probability multiplication
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rule for independent events it can be generalized
for any number of independent events
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thus for example the probability of simultaneous
occurrence of independent events let us say
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abc is defined as p of abc which is equal
to the product of multiple product of individual
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events so equal to pa into pb into pc etcetera
so now we will use this probability theorem
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in our case
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so let us say hence the total probability
that a molecule can have the velocity lying
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between u and u + du v and v + dv w and w
+ dw = f of u times f of v times f of w du
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dv dw which is just the product of the probabilities
three independent probabilities f of udu f
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of vdv f of wdw which is the product of the
individual probability then the number of
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such molecules per unit volume will be n f
of u f of v f of w du dv dw
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these molecules are content in the volume
element du dv dw in the velocity space so
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as shown in the figure from this figure you
can see that these number of molecules are
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confined in the volume element du dv dw now
these molecules have the resultant velocity
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c so the number of these molecules must be
n into f of c du dv dw
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as i have already told you c is the resultant
velocity which is nothing but the income in
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analogy to the position radius vector in the
coordinate space c is nothing but the c square
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= u square + v square + w square where f of
c is the probability for molecules with velocity
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c so equating equation 9 and 10 we get f of
u into f of v into f of w equal to some other
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function which is f of c which is nothing
but the phi of c square say where phi of c
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square is some function of c square
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so for fixed values of c phi of c square is
constant so d of phi should be zero so if
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i will take if we will differentiate d of
phi so what i will get d of f of u times f
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of d times f of w = 0 so if i will take if
i will differentiate them keeping console
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keeping other two values constant suppose
if i want to take different set u so keeping
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constant v and w similarly i can differentiate
v keeping constant u and w similarly i can
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differentiate w you keeping u and v constant
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so if i will do these simple mathematics what
we will get it f of f prime u du f of v f
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of w + f of u prime v dv f w + f of u f of
v times f prime w dw = 0 where prime stands
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for the differentiation f prime u stands for
df by du f prime v stands for df by dv keeping
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other two variable constant similarly f of
f prime w means the derivative of probability
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function with respect to w keeping u and v
constant
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so dividing the equation 14 by f of u f of
v f of w we will get a beautiful equation
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f prime u by fu du + f prime v by fv dv +
f prime w by fw dw = 0 now for fixed values
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of c we know that u square c square = u square
+ v square + w square if i differentiate this
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thing for a fixed values of c square d of
c square should be 0 that means udu + vdv
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+ wdw should be 0 now i have two equations
one equation is obtained by differentiating
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d of phi = 0
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and second equation by demanding for a fixed
values of c square d of c square = 0 so i
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got the two differential equation from there
i have to find out what is f of u? what is
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f of v? and what is f of w? so i have two
differential equations two differential
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equation from there how to solve f of u f
of v f of w this is a new method which is
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known as laplace's method of undetermined
multiplier
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so what to do suppose if you have one mother
equation which is a prime udu just just let
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me show you let me show you there's a equation
15 this is the mother equation other constant
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equation is equation number 16 so what is
the rule to do it if you have 2 3 constant
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equations so you multiply one constant to
each constant condition and then you add with
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the mother equation and then you demand that
the coefficient of each variable will be 0
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separately
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let me start it how to do it so if f prime
u by fu + beta u du + f prime v by fb + beta
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vdv + f prime w by fw + beta wdu = 0 so since
i got this equation then i will demand that
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since du dv and dw the the coefficient of
du dv dw since they are arbitrary so the coefficient
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should be 0 independently if i will do it
what i will get it? i will get first equation
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f prime u by fu + beta u du = 0 that means
f prime u by fu + beta u = 0
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similarly f prime v by fv + beta v = 0 and
f prime w by fw + beta w = 0 so i got three
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ordinary partial differential equation
if i will solve these three ordinary difference
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partial different ordinary differential equations
or not this partial differential equation
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this is three ordinary but differential equation
once i will solve these three ordinary differential
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equations i will get the solution f of u f
of v f of w respectively
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once i got it then i will multiply fu fv fw
to get f of c square ok this is the algorithm
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of our mathematics so let us do it so f prime
u by fu = - beta u f prime v by fv = - beta
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v f prime w by fw = -beta w
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so from the first differential equation if
i will solve it i will get log of f of u = - half
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beta u square + log a where a is the integration
constant if i will rewrite it then i will
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get it f of u = a to the power -beta u square
by 2 let us say beta by 2 is some other constant
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b so i will get a to the power - beta u square
so where a and b are the constant similarly
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i can solve other two ordinary differential
equations
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so i will get f of v and f of w which will
look like f of v = ae to the power -beta bv
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square f of w = ae to the power - beta bw
square where b is nothing but the beta by
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2 where a is nothing but the integration constant
so once i got f of u f of v f of w then i
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can define the combined probability having
the velocity u and u + du v and v + dv and
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w and w+dw
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so hence the number of molecules having the
velocity component lying between u and u +
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du v and v + dv w and w + dw is given as the
product of this theta which i will get dn
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=n aq e to the power -bu square + v square
+ w square du dv dw so so this is the famous
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maxwell distribution law of velocities where
du dv dw is the volume element in the velocity
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space in the spherical polar coordinate system
in the velocity space c theta phi the volume
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element can be written as du times dv times
dw = c squared dc sine theta d theta d phi
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as we know in the coordinate space the volume
element is defined as dv equal to r r square
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r square dr sine theta d theta d phi only
difference is that r will be replaced by the
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resultant velocity vector which is c okay
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so if i will substitute the volume element
du dv dw in the equation 23 we will get this
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dn = n aq e to the power -bc square c square
dc sine theta d theta d phi so this is the
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this is nothing but the number of molecules
having velocities lying between c and c +
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dc theta and theta + d theta and phi and phi
+ d phi if you will integrate this equation
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for theta theta 0 to pi phi 0 to 2 pi what
we will get it
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we will get dn c = n aq e to the -beta square
c square dc 0 to pi sine theta d theta 0 to
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2 pi d phi what we will get 4 pi n aq e to
the power - bc square c squared dc this is
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maxwell distribution of law it gives the complete
knowledge of the gas only in this statistical
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sense so now let me discuss this thing so
if we will rewrite this equation in terms
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of some new variable then things will be clear
much and things will be much easier much clearer
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so let me do it from this the probability
of it is velocity lying between cu and c +
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dc it just you divide it by n what you will
get f of c dc = 4 pi aq e to the power - bc
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square c square dc let us substitute x which
is nothing but the square root of bc square
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then the above equation reduces to f of x
dx = 4 pi to the power minus of e to the power
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-x x square dx or f of x equal to 4 pi to
the power minus of e to the power - x square
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so this is the probability having having some
quantity of x where x is nothing but the velocity
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but it is a new variable so f of x is nothing
but 4pi to the power of minus of e to the
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-x x square this equation is universal in
the sense in this form the distribution function
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depends neither on the kind of gas or nor
on the temperature
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so now if you will plot f of x versus x a
curve is shown in the figure from this figure
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we will obtain the following information so
if you plot f of x versus x the curve whatever
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we obtain i will get the following information
by plotting this curve so i will tell one
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by one the total area between the curve and
the x-axis gives the total number of molecules
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in the gas so this is the definition of integration
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integration f of x dx a to b will give you
the area under the curve which is nothing
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but the result okay the shaded area which
is shown in the figure gives the number of
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molecules having delos having the variables
between x and x +dx or you can say in terms
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of velocity shaded area gives the number of
molecules having velocities some c to c +
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dc the values of fx corresponding to any values
of x gives the number of molecules with velocities
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x = square root of bc square
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the probability when x =1 is the greatest
thus the gas has the most probable velocity
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x = 1 from there you will get c = 1 by square
root of d so these are the features of this
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curve so taking this who will compute the
constants a and b by normalizing it once i
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will get a and b then i will substitute it
back values of a and b we will get complete
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maxwell dis maxwell distribution law of velocities
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since the total number of molecules per unit
volume having velocities 0 to infinity is
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n so we will integrate it we will get the
total number of molecules so if you will integrate
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it the 4 ppi naq will come out of this integration
0 to infinity e to the - bc square c square
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dc = n so if i integrate it this is nothing
but a gaussian type integration if you will
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integrate it by substituting bc square = x
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so then i will rewrite this equation in terms
of new variables what i will get it 2 pi naq
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half b to the power -3 by 2 0 to infinity
e to the power x x to the power of minus of
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dx = n from there if i if i if i will put
the value of 0 to infinity e to the power
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x e to the power x x to the or minus of dx
then i will get a = b by pi to the power 1
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so as we have already derived from the kinetic
theory of gases the pressure p exerted by
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the gas is p equal to one third mn by v average
values of c square where c square is the mean
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square velocity which is defined as c square
average is 1 by n 0 to infinity c square dn
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c so if you will substitute the values of
dn c if you substitute the values of dn c
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from this equation from the equation 26 and
substitute it there then what we will get
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it
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average values of c square we will get 4 pi
aq 0 to infinity e to the power -bc square
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c to the power 4 dc because c square is already
here another c square you will come from dn
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c so c square times c square is c to the power
4 then again i will substitute bc square x
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so i will rewrite the equation for the mean
square velocity is 2 pi aq b to the by b to
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the power 5 by 2 0 to infinity e to the power
–x x to the power 3 by 2 dx which comes
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out to be 3 by 2d
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since if you will substitute average
values of square velocity c square in the
25:29.310 --> 25:37.420
pressure equation so we will get p equal to
small m times n by 2 vb however the ideal
25:37.420 --> 25:45.230
equation of state gives p = nkt by v if i
compare these two equations one is from the
25:45.230 --> 25:55.270
experimental result p nkt by v and p = mn
by 2 vb if you this is this equation is obtained
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from the maxwell kinetic theory of gases
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if you compare this two equation you will
get the values of b once you will get the
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values of b then we have already derived it
some equation in terms of a and b you substitute
26:10.920 --> 26:22.600
b so you will get the values of b so in principle
we have obtained the constants a and b so
26:22.600 --> 26:30.710
b = m by 2kt so if you will substitute a and
b then maxwell distribution law of velocities
26:30.710 --> 26:40.070
can be written in the form dn c del 2 nc dc
= 4 pin m by 2pi kt e to the power 3 by 2
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e to the power - mc square by k2 kt c square
dc
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this is the famous maxwell boltzmann distribution
law this tells the number of molecules having
26:53.460 --> 27:01.990
velocities c and c + dc this is a very famous
equation and probability you just divide it
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by nc you will get the probability of finding
a molecule between c and c + dc which is nothing
27:09.540 --> 27:18.430
but 4 pi n will go away m by 2p kt eto the
power 3 by 2 e to the power -mc square by
27:18.430 --> 27:20.720
2 kt into c square dc
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here i should tell some interesting thing
although it is a heuristic argument e to the
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power some any exponential function with a
negative e to the -mc square by 2 kt you can
27:35.710 --> 27:42.670
think in different ways there are two energy
scale in the system one is since it the system
27:42.670 --> 27:48.540
is at thermal equilibrium so obviously there
is a temperature so there is one scale which
27:48.540 --> 27:53.390
is related to the thermal energy scale which
is of the order of kt
27:53.390 --> 27:58.510
other energy scale which is intrinsic to the
energy or to the particle which is nothing
27:58.510 --> 28:04.400
but the simple kinetic energy half mv square
whenever there is a dominant whenever there
28:04.400 --> 28:11.140
is a computation of two scales here it is
the energy scale so there will be always some
28:11.140 --> 28:17.340
exponential function will come which is the
ratio of the two energy scale mc square by
28:17.340 --> 28:18.340
2kt
28:18.340 --> 28:27.630
the quantity f of c which is i have already
told you dnc by ndc which is 4 pi to the power
28:27.630 --> 28:35.590
minus of m by 2kt to the 3 by 2c square e
to the -mc square by 2 kt is the distribution
28:35.590 --> 28:42.110
function of velocities of molecules so if
you plot these maxwell distribution function
28:42.110 --> 28:49.270
for 3 different temperature of the system
suppose some temperature having thermal equilibrium
28:49.270 --> 28:57.320
at temperature t1 other system having temperature
t2 obviously is in thermal equilibrium
28:57.320 --> 29:03.750
other system is in also obviously in thermal
equilibrium having temperature t2 if you plot
29:03.750 --> 29:10.640
these three figures for the three different
systems so what you will see if you will plot
29:10.640 --> 29:17.800
f of c versus c having temperature t1 t2 t3
who are t1 less than t2 less than t3 what
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you will see the peak of this curve we will
slip to the higher temperature so physically
29:24.360 --> 29:32.080
what does it mean if you if this system has
a system or in a system is in a higher temperature
29:32.080 --> 29:39.240
so obviously probability of finding molecules
having large kinetic energy is large which
29:39.240 --> 29:46.950
is obvious because the square of the mean
of the square of the energy below energy mean
29:46.950 --> 29:52.230
energy is proportional to the temperature
of the system if the temperature is large
29:52.230 --> 29:58.570
so mean energy will be large so that is the
reason the peak of the curve will be shifted
29:58.570 --> 30:02.059
towards the towards towards in the right direction
30:02.059 --> 30:12.429
so now in deriving the maxwell distribution
of velocities of molecules we assume that
30:12.429 --> 30:18.100
the velocity coordinates are independent that
is we disregard the collision between the
30:18.100 --> 30:24.670
molecules the collision effects the velocities
of the molecules as a result the state becomes
30:24.670 --> 30:34.820
stable or thermal equilibrium so that means
somebody could ask that why when thermal
30:34.820 --> 30:37.330
equilibrium will be achieved
30:37.330 --> 30:44.380
so in i have seen i have given in two lectures
in the kinetic theory of gases where i have
30:44.380 --> 30:52.160
told somewhere that temperature is a concept
temperature is a concept of the isotropicity
30:52.160 --> 30:59.440
in the momentum space suppose you have taken
n number of molecules in a container of gas
30:59.440 --> 31:05.610
initially they are momentum in different directions
maybe it is different but after some time
31:05.610 --> 31:06.950
there will
31:06.950 --> 31:14.060
be some isotropic in the momentum space which
is average values of p px square should be
31:14.060 --> 31:19.900
equal to the average values of py square equal
to average values of pz square
31:19.900 --> 31:26.350
this isotropicity momentum space is known
as the thermal equilibrium but in my earlier
31:26.350 --> 31:35.350
two lectures with this equilibrium situation
has been obtained by the collision of a molecule
31:35.350 --> 31:41.200
between between the molecules and the wall
of the container not the collision among themselves
31:41.200 --> 31:47.320
but what maxwell's told the if the collisions
will be more if the collisions will be more
31:47.320 --> 31:54.010
frequent the system will achieve thermal equilibrium
more quickly okay
31:54.010 --> 31:59.650
the maxwell distribution law of velocities
corresponds to this state boltzmann were later
31:59.650 --> 32:05.670
sort that going to collisions between its
molecule a gas will spontaneously pass from
32:05.670 --> 32:12.070
a state of non maxwellian distribution to
a state of maxwellian distribution the maxwellian
32:12.070 --> 32:20.380
distribution sometimes also called the maxwell
boltzmann distribution is an equilibrium distribution
32:20.380 --> 32:22.730
so now i will tell
32:22.730 --> 32:32.350
since it is a very beautiful laws of nature
so one must verify the maxwell distribution
32:32.350 --> 32:39.390
law in a couple of minutes in a couple of
minutes i will try to demonstrate some experiment
32:39.390 --> 32:47.190
through who is it has been proved now that
the disc maxwell distribution is perfect law
32:47.190 --> 32:54.840
who is which is obeyed by the molecules in
a gas okay initially whenever people have
32:54.840 --> 33:02.370
performed some experiment they are experiment
in some sense crude so their experimental
33:02.370 --> 33:04.920
error is not perfect
33:04.920 --> 33:10.620
they are the matches which they experience
the agreement between theory and experiment
33:10.620 --> 33:16.270
up to 15 percent of error so more and more
refined experiment have been done through
33:16.270 --> 33:23.800
this refined experiment now it is confirmed
that there is no deviation between theory
33:23.800 --> 33:26.330
and experiment
33:26.330 --> 33:31.520
so in view of the fundamental importance of
the maxwell distribution law in kinetic theory
33:31.520 --> 33:38.920
of gases it was subjected many times to thorough
experimental verification many attempts have
33:38.920 --> 33:45.640
been made some of them are considered here
so first experiment in this endeavor stern
33:45.640 --> 33:53.890
experiment so let me start it maxwell distribution
of velocities has been verified by the experimental
33:53.890 --> 33:55.930
arrangement due to stern
33:55.930 --> 34:03.470
the principle of stern experiment can be expressed
by the given figure where l is a platinum
34:03.470 --> 34:10.409
or let me show you the experiment this is
the schematic diagram of the experiment where
34:10.409 --> 34:19.730
l is a platinum wear coated with silver the
wire l serves as the source of atoms whose
34:19.730 --> 34:26.179
velocity is to be studied when the wire is
heated by an electric current it emits atomic
34:26.179 --> 34:34.369
silver in all direction the wire l is surrounded
by two cylindrical diaphragms with narrow
34:34.369 --> 34:36.399
slits s1 and s2
34:36.399 --> 34:44.860
these slits are parallel to the wire now this
is the schematic diagram of this experimental
34:44.860 --> 34:52.460
arrangement through these slits is stream
of silver escapes and condenses on the plates
34:52.460 --> 34:59.810
p and p prime okay the whole apparatus is
enclosed in a highly evacuated glass vessel
34:59.810 --> 35:05.830
so that the silver atoms may not suffer in
the collisions in the space that means in
35:05.830 --> 35:12.660
the within the experimental vessel it is maintained
at a very high vacuum
35:12.660 --> 35:19.300
so that there are no air molecules inside
so that while silver atoms foil it is going
35:19.300 --> 35:26.020
to the two p and p prime so it will not suffer
any collision so that is the reason it has
35:26.020 --> 35:34.330
kept in an evacuated chamber okay these slits
s1 and s2 and the plates p and p prime rotated
35:34.330 --> 35:41.260
together as a rigid body about the wire l
as the axis of rotation when the entire system
35:41.260 --> 35:48.859
is at rest the silver stream traverses along
l o obviously and ad and deposited at o
35:48.859 --> 35:56.340
because it will straight hit to o so obviously
it will be deposited at the point o when the
35:56.340 --> 36:04.260
system is rotated around the axis of the wire
so at high speed in clockwise direction which
36:04.260 --> 36:10.140
is shown in the figure the silver molecules
will no longer strike the target at o but
36:10.140 --> 36:18.619
will be displaced from the from o and deposited
at some point above o
36:18.619 --> 36:26.020
the faster moving molecules will condense
near to narrow then slower ones thus the velocity
36:26.020 --> 36:32.630
spectrum of silver molecules will be obtained
when the relative intensity of deposit is
36:32.630 --> 36:38.210
measured with the help of a micro photometer
the ratio of the number of molecules with
36:38.210 --> 36:46.060
different velocities can be deduced and the
maxwell distribution law is verified however
36:46.060 --> 36:50.220
the result obtained by the stern experiment
is quite satisfactory
36:50.220 --> 36:57.050
but still there are some problems in the
experiment because although i have kept it
36:57.050 --> 37:04.610
in a highly evacuated chamber but still that
time the vacuum technology is not so in advanced
37:04.610 --> 37:10.390
situation so you cannot make it completely
evacuated so that is the reason the agreement
37:10.390 --> 37:16.010
between the theory and experiment a tail up
to 15 percent
37:16.010 --> 37:22.301
so and this is due to the difficulty in retaining
the perfect vacuum in the vessel as just i
37:22.301 --> 37:29.490
have told you as a result the maxwell distribution
law has been verified within about 15% so
37:29.490 --> 37:36.800
thus this method needs to be improved so exactly
this method has been improved by many scientists
37:36.800 --> 37:43.820
one of the improvements is due to zartman
and ko in 1930 which is described below
37:43.820 --> 37:53.500
first i will explain zartman and ko’s experiment
zartman and ko in 1930 have modified the stern
37:53.500 --> 38:01.180
method to study the distribution of velocities
of molecules the apparatus consists of an
38:01.180 --> 38:08.970
oval v with a narrow opening a which is shown
in this figure s1 and s2 are the two parallel
38:08.970 --> 38:16.900
slit's through which these through which
these molecules will go
38:16.900 --> 38:23.380
above the slits there is a cylindrical drum
d which can be obviously here some section
38:23.380 --> 38:28.930
of the cylindrical drum is shown and which
is nothing but a circle which can be rotated
38:28.930 --> 38:37.310
in the vacuum about an axis passing through
o it is given in the figure a slit s3 is on
38:37.310 --> 38:44.460
is on one side of the drum and g is the glass
plate mounted on the inside surface of the
38:44.460 --> 38:53.850
drum opposite to the s3 okay so you can see
and that s3 is just opposite to the glass
38:53.850 --> 38:54.850
plate g
38:54.850 --> 39:02.450
it is shown in the figure so bismuth is taken
as the experimental substance in the earlier
39:02.450 --> 39:08.869
experiment we have taken in the stern experiment
we have taken silver but here they have taken
39:08.869 --> 39:15.560
these working substance as the bismuth so
bismuth is taken as the experimental substance
39:15.560 --> 39:21.400
which is heated and vaporized in the oven
a molecular beam of bismuth escaping through
39:21.400 --> 39:27.609
a through this narrow opening is collimated
by these slits s1 and s2
39:27.609 --> 39:33.440
so finally when the drum is stationary the
beam of the molecule entering into it through
39:33.440 --> 39:39.640
the slits s3 strike the glass plate at the
same point obviously since it is at stationery
39:39.640 --> 39:46.950
it will go straight and hit the point in the
drum but when the drum is rotated at a high
39:46.950 --> 39:53.530
speed the molecules with very high speed reach
the glass speed g firstt that is on the right
39:53.530 --> 40:01.540
hand of g and molecules with this slower speed
reach the plate g on the other end of g
40:01.540 --> 40:07.770
so after a short time sufficient quantity
of bismuth molecules is deposited on the plate
40:07.770 --> 40:14.180
whose density varies across g according to
the velocity distribution of molecules the
40:14.180 --> 40:21.200
thickness of the deposit that is the density
distribution is measured by a micro photometer
40:21.200 --> 40:29.040
and this result is obtained by zartman and
ko exactly matches with this almost matches
40:29.040 --> 40:37.099
not exactly almost matches with the theoretical
distribution which is given by the solid line
40:37.099 --> 40:45.810
if you plot the intensity versus the this
displacement is the dot of this straight line
40:45.810 --> 40:53.220
the straight these connected line gives the
theoretical distribution and this circle this
40:53.220 --> 40:58.720
open circle is the result obtained by this
experiment
40:58.720 --> 41:06.760
it almost matches with except at some high
velocity and at the low velocity regime so
41:06.760 --> 41:13.369
these circles denote the observed density
of deposit while the line represents the theoretical
41:13.369 --> 41:20.340
distribution on the basis of maxwell distribution
law so the experiment is good which confirms
41:20.340 --> 41:34.460
the maxwell distribution law