Define (a) RMS (b) average and (c) most probable speeds of gas molecules. Give their interrelationship,

Explanation

a) RMS velocity : It is defined as the square root of the mean of the squares of the velocities of all the molecules present in the gas.
Let there be `n_(1)` molecules with `V_(1)` velocity, `n_(2)` molecules with `V_(2)` velocity, `n_(3)` molecules with `V_(3)` velocity and so on .
Then , RMS velocity
`=sqrt((n_(1)V_(1)^(2)+n_(2)V_(2)^(2)+n_(3)V_(3)^(2)+.....)/(n_(1)+n_(2)+n_(3)))`
RMS velocity (C )
`=sqrt((3RT)/(M))=sqrt((3PV)/(M))=sqrt((3P)/(d))`
T= absolute temperature, M=Molecular weight, `R=8.314xx10^(7) "erg"K^(-1)"mol"^(-1)`
b) Average velocity : It is the arithmetic mean of velocities of gas molecules. Let there be `n_(1)` molecules with velocity `V_(1), n_(2)` molecules with velocity `V_(2),n_(3)` molecules with velocity `V_(3)` and so on.
Then Average velocity
`(bar(C))=(n_(1)V_(1)+n_(2)V_(2)+n_(3)V_(3)+....)/(n_(1)+n_(2)+n_(3)+....)`
Average Velocity `(bar(C))=sqrt((8RT)/(piM))`
c) Most probable velocity : The velocity possessed by maximum number of molecules is called most probable velocity. It is denoted by `C_(p).C_(p)=sqrt((2RT)/(M))`
Relation between different velocities :
`C_(p):bar(C):C=sqrt((2RT)/(M)):sqrt((8RT)/(pi)):sqrt((3RT)/(M))`
= `sqrt(2):sqrt((8)/(pi)):sqrt(3)`
= `1:1.128:1.224`