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