N (N

Question

(D) The average speed $u_{avg}$ is given by the sum of velocities divided by the number of molecules $N$: $u_{avg} = \frac{1+2+3+...+N}{N} = \frac{N(N

Vedclass · Accepted Answer

ratio of rms speed to average speed of a molecule is $\sqrt{\frac{2(2N+1)}{3(N+1)}}$.

Vedclass · Answer

(D) The average speed $u_{avg}$ is given by the sum of velocities divided by the number of molecules $N$: $u_{avg} = \frac{1+2+3+...+N}{N} = \frac{N(N+1)}{2N} = \frac{N+1}{2}$.The root mean square speed $u_{rms}$ is given by the square root of the sum of squares of velocities divided by $N$: $u_{rms} = \sqrt{\frac{1^2+2^2+3^2+...+N^2}{N}} = \sqrt{\frac{N(N+1)(2N+1)}{6N}} = \sqrt{\frac{(N+1)(2N+1)}{6}}$.The ratio of rms speed to average speed is $\frac{u_{rms}}{u_{avg}} = \frac{\sqrt{\frac{(N+1)(2N+1)}{6}}}{\frac{N+1}{2}} = \sqrt{\frac{(N+1)(2N+1)}{6}} 	imes \frac{2}{N+1} = \sqrt{\frac{2^2(N+1)(2N+1)}{6(N+1)^2}} = \sqrt{\frac{4(2N+1)}{6(N+1)}} = \sqrt{\frac{2(2N+1)}{3(N+1)}}$.

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

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