Question

Four molecules of a gas have speed 1, 2, 3 and 4 km/s. The value of the r.m.s speed of the gas molecule is
$
A:\dfrac{1}{{2\sqrt {15} }}km/s \\
B:\dfrac{1}{{2\sqrt {10} }}km/s \\
C:2.5km/s \\
D:\dfrac{{\sqrt {15} }}{2}km/s \\
 $

Answer 1

Hint:A molecule is the fundamental unit of a chemical compound which can take part in chemical reaction. Molecules are loosely packed in gases. The root mean square is defined as the square root of the mean square.

$R.m.s = \sqrt {\dfrac{1}{n}\sum\limits_i^{} {v_i^2} } $

Complete step by step solution:In the question, we are given that four molecules of a gas have speed 1, 2, 3 and 4 km/s.
\[\begin{gathered}
{v_1} = 1km/s \\
{v_2} = 2km/s \\
{v_3} = 3km/s \\
{v_4} = 4km/s \\
\end{gathered} \]
Now, we have to find the root mean square (r.m.s) of gas molecules.
As we know that the root mean square speed is the square root of the average of the square of speed that is firstly we will find the square, then average /mean of that and then finally we will find the square root of the same.

Let us find out the square of the speed in each case:
$
{v_1}^2 = {(1)^2} = 1km/s \\
{v_2}^2 = {(2)^2} = 4km/s \\
{v_3}^2 = {(3)^2} = 9km/s \\
{v_4}^2 = {(4)^2} = 16km/s \\
 $
Mean of speeds=sum of square of speeds divided by the total number of speeds
\[Mean{\text{ }}of{\text{ }}speeds = \dfrac{{sum{\text{ }}of{\text{ }}square{\text{ }}of{\text{
}}speeds}}{{total{\text{ }}number{\text{ }}of{\text{ }}speeds}}{\text{ }}\] (i)

Thus, Sum of square of speeds = ${1^2} + {2^2} + {3^2} + {4^2}$ (ii)
Total number of speeds= 4 (iii)
Put the values of equation (ii) and (iii) in equation (i) then we get

Mean of speed = $\dfrac{{({1^2} + {2^2} + {3^2} + {4^2})}}{4}$

Mean of speed=\[\dfrac{{30}}{4} = \dfrac{{15}}{2}\] (iv)

Square root of mean of speed =$\dfrac{{\sqrt {{v_1}^2 + {v_2}^2 + v_3^2 + {v_4}^2} }}{4}$

From equation (iv) we get the value of the r.m.s speed of the gas molecules.
\[Square{\text{ }}root{\text{ }}of{\text{ }}mean{\text{ }}of{\text{ }}speed(r.m.s) = \dfrac{{\sqrt {30}
}}{4} = \dfrac{{\sqrt {15} }}{2}km/s\]

The value of the root mean square speed of the gas molecules is $\dfrac{{\sqrt {15} }}{2}km/s$

Hence, the correct answer is D

Note:The average kinetic energy of the gas molecules is directly related (or proportional) to the absolute temperature. This means that all of the molecular motion is ceased in case the temperature is being reduced to absolute zero.