Define the terms: most probable speed, average speed and root mean square speed. How are these speeds related to each other?

Explanation

i) Most probable speed
It is the speed possessed by maximum number of gas molecules at a given temperature. It is denoted by by `.alpha.`.
ii) Average speed
It is the average of the speeds of different molecules of gas at a given temperature. It is represented by `barv`. If `v_(1),v_(2),v_(3)` are the speeds of various molecules, then average speed is expressed as `barv=(v_(1)+v_(2)+v_(3))/(N)`, where, N is total number of gas molecules.
iii) Root mean square speed
Root mean square speed is defined as square root of average of the squares of the speeds of different gas molecules at a given temperature.
Let `v_(1),v_(2),v_(3)` ...... be the speeds of different gas molecules and let `n_(1),n_(2),n_(3)`....... be the number of gas molecules.
Then `U_("rms") = sqrt((n_(1)v_(1)^(2)+n_(2)v_(2)^(2) +n_(3)v_(3)^(2) + .....)/(n_(1)+n_(2)+n_(3)+ ........))`
Relation between most probable speed, average speed and root mean square speed.
These speeds are related to each other as follows:
`alpha: barv:U_("rms"):: 1:1.128:1.224`
Most probable speed is given by the expression :
`alpha=sqrt((2RT)/(M))`
Average speed is given by the expression :
`barv=sqrt((8RT)/(piM))`
Root mean square speed is given by the expression :
`U_("rms")=sqrt((3RT)/(M))`
Where, R = Universal gas constant M = Molecular mass of the gas
T = Absolute temperature