Question

Four molecules of a gas have speeds of $1$, $2$, $3$, $4cm{s^{ - 1}}$ respectively. The root mean square velocity is:
A.$\sqrt {7.5}$
B.$\sqrt {30}$
C.$30$
D.$15$

Answer 1

Initially, this question gives us the knowledge about the root mean square velocity. Root mean square velocity is generally the square root of mean values of the squares of the velocities of the distinct gaseous molecules.
Formula used: The formula used to determine the root mean square velocity is as follows:
${v_{rms}} = \sqrt {\dfrac{{v_1^2 + v_2^2 + v_3^2 + v_4^2}}{4}}$
Where ${v_{rms}}$ is the abbreviation for root mean square velocity and ${v_n}$is the speed of the gas.

Complete step by step answer:
The root mean square velocity of the various gases using the formula of root mean square velocity as follows:
$\Rightarrow {v_{rms}} = \sqrt {\dfrac{{v_1^2 + v_2^2 + v_3^2 + v_4^2}}{4}}$
Substitute ${v_1}$ as $1$, ${v_2}$ as $2$, ${v_3}$ as $3$ and ${v_4}$ as $4$ in the above formula as follows:
$\Rightarrow {v_{rms}} = \sqrt {\dfrac{{{{\left( 1 \right)}^2} + {{\left( 2 \right)}^2} + {{\left( 3 \right)}^2} + {{\left( 4 \right)}^2}}}{4}}$
On simplifying, we get
$\Rightarrow {v_{rms}} = \sqrt {\dfrac{{1 + 4 + 9 + 16}}{4}}$
On further simplifying, we get
$\Rightarrow {v_{rms}} = \sqrt {\dfrac{{30}}{4}}$
On simplifying the above equation we get the root mean square velocity as
$\Rightarrow {v_{rms}} = \sqrt {7.5}$
Therefore, the root mean square velocity is $\sqrt {7.5} cm/s$.

Hence, option $A$ is the correct option.

Note:
Root mean square velocity is generally the square root of mean values of the squares of the velocities of the distinct gaseous molecules. The SI unit of root mean square velocity is $cm/\sec$. The two factors that are considered by root mean square velocities are temperature and molecular weight. Root mean square velocity is directly proportional to the temperature. The root mean square is basically the measure of the velocity of the gaseous molecules.