Question

$N$(<100) molecules of a gas have velocities 1,2,3 ....$N/km/s$ respectively. Then:

• 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 a molecule is $2/\sqrt{6}x\sqrt{(2N+1)/(N+1)}$

Answer 1

The correct option is D ratio of $rms$ speed to average speed of a molecule is $2/\sqrt{6}x\sqrt{(2N+1)/(N+1)}$

As we know, sum of the first $n$ natural numbers is $n[n+1]/2$ and sum of the squares of first $n$ natural numbers is $n[n+1][2n+1]/6$.
So $u_{rms}=\sqrt{n[n+1][2n+1]/6n}$ and $u_{avg}=n[n+1]/2n$.
So, ratio of $rms$ speed to average speed of a molecule is $2/\sqrt{6}x\sqrt{(2N+1)/(N+1)}$.