Question

The root mean square speed of a group of n gas molecules having speed $v_1,v_2,v_3.......v_n$ :

• A. $\frac{1}{n}\sqrt{(v_1+v_2+v_3.......v_n{)}^2}$
• B. $\frac{1}{n}\sqrt{(v_1^2+v_2^2+v_3^2.......v_n^2)}$
• C. $\sqrt{\frac{1}{n}(v_1^2+v_2^2+v_3^2+.....+v_n^2)}$
• D. $\sqrt{\left[\frac{(v_1+v_2+v_3+.....v_n{)}^2}{n}\right]}$

Answer 1

The correct option is C $\sqrt{\frac{1}{n}(v_1^2+v_2^2+v_3^2+.....+v_n^2)}$

Let there be 'n' particles with velocities:

$v_1,v_2,v_3...v_n$

The sum of squares of the velocities of each particle is:

$v_1^2+v_2^2+v_3^2+...+v_n^2$

and the mean square is:

$\frac{v_1^2+v_2^2+v_3^2+...+v_n^2}{n}$

The square root of which is RMS:

Hence

$v_{RMS}=\sqrt{\frac{v_1^2+v_2^2+v_3^2+...+v_n^2}{n}}$

Option C is correct.