`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 is 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)//6`
D. ratio of rms speed to average speed of molecules is `sqrt(((2n+1))/(6(N+1)))`

Answer

D

Explanation

`V_("rms")=sqrt((V_(1)^(2)+V_(2)^(2)+.......V_(N)^(2))/(N))=sqrt((1^(2)+2^(2)+.......+ N^(2))/(N))`
`=sqrt((N(N+1)(2N+1))/(6N))rArr V_("rms")=sqrt(((N+1)(2N+1))/(6))`
`V_("avg")=(V_(1)+V_(2)+......+V_(N))/(N)`
`=(1+2+......+N)/(N)=(N(N+1))/(2N)=(N+1)/(2)`
`(V_("rms"))/(V_("avg"))=2 sqrt(((2N+1))/(6(N+1)))`.