Question

The root mean square speed of a group of n gas molecules having speed ${v_1},{v_2},{v_3}........{v_n}:$
(A) $\dfrac{1}{n}\sqrt {{{({v_1} + {v_2} + {v_3}........{v_n})}^2}}$
(B) $\dfrac{1}{n}\sqrt {{{(v_1^2 + v_2^2 + v_3^2........v_n^2)}^2}}$
(C) $\sqrt {\dfrac{1}{n}{{(v_1^2 + v_2^2 + v_3^2........v_n^2)}^2}}$
(D) $\sqrt {\left[ {\dfrac{{{{({v_1} + {v_2} + {v_3}.....{v_n})}^2}}}{n}} \right]}$

Answer 1

Hint To find the value of the root square speed of a group of gas we have to first consider the number of particles they must be having. Once we get the number of particles, we have to find the sum of the squares of the velocities of each particle. From there we have to form the expression for mean square followed by the square root of mean square. This will give us the required answer for this question.

Complete step by step answer

Let us consider that there are n number of particles with the velocities mentioned as ${v_1},{v_2},{v_3}........{v_n}$
Now we have to find the sum of the squares of the velocities of each particle. The expression is given as:
$v_1^2 + v_2^2 + v_3^2 + ............ + v_n^2$
The mean square is expressed as:
$\dfrac{{v_1^2 + v_2^2 + v_3^2 + ............ + v_n^2}}{n}$
So, the square root of the mean square gives us the expression of RMS. Hence the expression is:
${V_{RMS}} = \sqrt {\dfrac{{v_1^2 + v_2^2 + v_3^2 + ............ + v_n^2}}{n}}$
So, the root mean square speed of a group of n gas molecules having speed ${v_1},{v_2},{v_3}........{v_n}:$ ${V_{RMS}} = \sqrt {\dfrac{{v_1^2 + v_2^2 + v_3^2 + ............ + v_n^2}}{n}}$.

Hence the correct answer is option C.

Note We should know that the root mean square speed is defined as the measurement of the speed of the particles in a specific gas. It is defined as the square root of the average velocity squared of the molecules present in that specific gas.
There are gases which consist of atoms or molecules that move at different speeds in various random directions. The root means square velocity also known as RMS velocity is a single way to find the value of the velocity for the particles.