For the molecules of a given gas at a constant temperature, arrange the most probable velocity `(c_(m))`, root mean square velocity `(c_(rms))` and average `(c_(a))` in the order of their increasing values. With the increase in temperature, will the ratio of these velocities increase, decrease or remain constant ? What will the effect of increasing temperature be on the value of `(c_(rms)-c_(m))` for a given gas?

Explanation

First part: At temperature T, the most probable velocity `(c_(m))` of the gas molecules `=sqrt((2RT)/(M))` [M=molar mass of the gas]
rms velocity `(c_(rms))=sqrt((3RT)/(M))`, average velocity `(c_(a))=sqrt((8RT)/(piM))`
`therefore c_(rms) gt c_(a) gt c_(m)`.
Second part: Since the value of each of the velocities, `c_(rms),c_(a) and c_(m)`, is proportional to `sqrt(T)`, their ratio remains unaltered with rise in temperature.
Third part: suppose,, at temperature `T_(1)K`, the root mean square velocity `(c_(rms))` and most probable velocity `(c_(m))` of the molecules of a gas are `(c_(rms)) and (c_(m))_(1)` respectively. if difference between the values of these velocities is `Deltax_(1)`. then,
`Deltax_(1)=sqrt((3RT_(1))/(M))-sqrt((2RT_(1))/(M))=sqrt((RT_(1))/(M))(sqrt(3)-sqrt(2))`
let the temperature of the gas is raised to `T_(2)K` and at this temperature the difference between the values of these two velocities is `Deltax_(2)`, then,
`Deltax_(2)=(c_(rms))_(2)-(c_(m))_(2)=sqrt((RT_(2))/(M))(sqrt(3)-sqrt(2))`
Since `T_(2) gt T_(1), Deltax_(2) gt Deltax_(1)`. hence, the difference between teh values of these two velocities is `Deltax_(2)`, then.
`Deltax_(2)=(c_(rms))_(2)-(c_(m))_(2)=sqrt((RT_(2))/(M))(sqrt(3)-sqrt(2))`
Since `T_(2) gt T_(1) , Deltax_(2) gt Deltax_(1)`. hence, the difference between the values of these velocities increases with rise iin temperature.