here we are going to discuss different molecular
speeds for a gas molecules. you know that gas molecules are in random brownian motion.
so for various kind of mathematical analysis. some standard different molecular speeds are
defined. and, there are vast range of speeds so with which gas molecules move. here we
are going to discuss about some specific speeds, the first one is. average velocity. of gas
molecules. it is the velocity with which gas molecules are moving on an average. second
we use which is quite important which is use for various analysis of energy in pressure
this is, root mean square . velocity. of gas molecules. root mean square is abbreviated
as, r m s velocity of gas molecules. will discuss all these one by one in detail. the
third one in sequence is mean speed. of gas molecules. mean speed is just the magnitude
. magnitude of average or average magnitude of speed of gas molecules . with which the
molecules move. between successive collision, that is considered as mean speed of gas molecule.
and the last important , speed which is use for analysis , is most probable. Speed, of
gas molecules. after this section will study about an article where we’ll see. that different
molecules in a gas moves with different speeds. so the speed with which maximum number of
molecules , would be moving or the probability of. a particular speed, carried by a particular
molecule is maximum is termed as most probable speed of gas molecules. lets discuss these
four , analytical molecular speeds, one by one. to analyze the average velocity of a
gas molecules lets consider a container. in which various gas molecules are moving randomly
in different directions like this. i am just drawing random direction in which the gas
molecules can move. and say they are having there independent speeds. v one. with a particular
direction its velocity vector is v one another is with. vector v two , v three and so on.
now in this situation we can simply state the average velocity. of gas molecules can
be written as v average is equal to. v one vector + v two vector + up to v n vector,
if in total there are , n molecules. then this divided by n that will be regarded as
the , average velocity. and we know well that, as the motion is totally random . then corresponding
to each molecules there will be some other molecule which will be moving with the opposite
, direction with the same speed. in opposite direction with same speed so, in all we can
say, the, random summation of all the vectors which are, having the directions distributed,
randomly through out the container. its summation can be taken as zero because corresponding
to each motion , there would be some opposite motion, this can be found. in this random
motion of particles. so we always assume that, average velocity of gas molecules in a container
is equal to zero. next is , root mean square velocity of gas molecules. this root mean
square means , square root of mean of square of all the velocities. to calculate this r-m-s
velocity of gas molecules first we calculate. mean square velocity. that is the mean of
squares. we talk about mean square velocity is for this we first square all the velocities
and we take mean like. square of velocity of first molecule. + square of velocity of
second molecule and so on up to, the velocity of n-eth molecules, and if divided by the
total number of molecules . this is the mean square velocity. and this, r m s velocity
is defined as, v-r m s is equal to, root of this mean of squares. that is root of. v one
square + v two square up to, v n square. divided by n. basically its mathematical expression
is calculated by maxwellian velocity distribution function which will study, in next article,
but this is the analytical, form, by which you can understand how this r m s velocity
is calculated and this velocity is quite important to use mathematically in calculation of energy
of a gas or pressure exerted by a gas. and. using. maxwells. velocity distribution function.
its expression is. the expression which we, get by using this expression maxwellian-velocity
distribution function is, it is given as root of three r t by m, where m is the molar mass
of the gas. and t is the absolute temperature , this quite important to keep in mind that
r m s velocity is directly proportional to square root of. absolute temperature. and
you can also write this , v r-m-s. is used in , energy. and pressure calculation. of
a gas. we can, state that , for calculation of total energy of a gas which is generally
considered as. total ki-netic energy of all the gas molecules. if we just write down total
energy of a gas is half m . v r m s square, you can see this will give us. m is the total
mass of gas and v r-m-s is velocity of one gas molecule. so here we can write if we substitute
v r m s as v one square + v two square plus v n square , you can see this will transform
into half m, when we substitute this v r-m-s here , total mass upon number of molecules
will give us the mass of one molecule . multiplied by v one square + v two square plus up to
, v-n square. you can see that just by using this v r m s . square, in calculation of total
energy of gas it automatically transforms into, the total ki-netic energy of all the
gas molecules. the next speed of gas molecules is mean speed of gas molecules. mean speed
or we can also call it average speed, can be directly calculated. by calculating the
average of magnitude of all velocities. that will be given as v one magnitude + v two magnitude.
+ up to v-n magnitude. divided by n. obviously as we are taking the average of magnitudes
it can never be zero. like the very first speed we have. seen that was the average velocity.
so velocity include the direction also that’s when we calculate it the average of all velocities
it was zero because. due to directions all opposite directions, canceled each other.
in this situation we are just calculating the, average of magnitudes that is the mean
speed. and its final expression, can be obtained , again by maxwellian velocity distribution.
so we can say. by maxwell’s velocity distribution function . its expression. is. this mean,
speed. is given in expressional form as root of, eight r t by, pie, m. the derivation of
this result is out of the scope of. this article so right now you just need to, keep this result
in your mind, as well as the way. how it is calculated if , speed of an individual molecule
is given to us. this mean speed is use to calculate the mean free path that we have
studied. which is the distance travelled by the particle mean distance travelled by the
particle , between two successive collisions denoted by lambda m, which can be given as
v-mean multiplied by , the relaxation time that is time between two successive collisions.
so mean speed is quite useful in calculation of mean free path. for a given gas. the last
speed which we need to study is the most probable speed of gas molecules. about most probable
speed which can be by its name which can be , understood as the probability. a particular
speed for which the probability for a gas molecule to have, is maximum is called most
probable speed. or in another way we can write . this is the speed . which is. common. in.
maximum number of molecules. of a gas. and it is given as. again we want get into the
derivation of the most probable. speed for a gas molecules. we just need to keep the
result in your mind it is root of two r t by m. this is the most probable speed which
we use. and we can also analyze about the, important speed we can see this most probable
speed is the least. and mean speed is more then , this most probable speed and r m s
speed is, further more then the mean speed. this you need to keep in mind in the very
next article will study about the distribution of molecule speed. the basic phenomena related
to , the molecular speed distribution that is maxwellian distribution curve . with which
he’ll be clear with how. the. speed of gas molecules are distributed among all the molecules.
again we’ll not get into the depth of, that maxwellian function. as it is out of the scope
of this lectures.