`N (lt 100)` molecules of a gas have velocities 1,2,3….N km/s respectively. Then

Options

A. rms speed and average speed of molecules are same
B. Ratio of rms speed to average speed is `sqrt((2N+1)(N+1))/(6N)`
C. Ratio of rms speed to average speed is `sqrt((2N+1)(N+1))/(6N)`
D. Ratio of rms speed to average speed of a molecule `(2)/(sqrt(6)) sqrt(((2N+1))/((N+1)))`

Answer

D

Explanation

Average speed
`v_("avg") = (1+2+ … +N)/(N) = (N(N+1))/(2.N) = (N+1)/(2)`
rms speed
`v_("rms") = sqrt((1^(2) + 2^(2) + 3^(2) + …… +N^(2))/(N))`
= `sqrt((N(N+1)(2N+1))/(6N)) = sqrt(((N+1)(2N+1))/(N))`
`therefore (v_("rms"))/(v_("avg")) = sqrt(((N+1)(2N+1))/(N)) xx (2)/((N+1)) = (2)/(sqrt(6)) sqrt(((2N+1)/(N+1)))`