Question

Five gas molecules chosen at random are found to have speeds of 500, 600, 700, 800 and $900\,{\text{m/s}}$. Find the rms speed. Is it the same as the average speed?

Answer 1

Use the formulae for the root mean square speed of the gas molecules and average speed of the gas molecules. Rewrite these formulae for the root mean square speed and average speed of the five gas molecules and calculate the values of these two speeds. Then compare them to check if the values of these two speeds are the same or not.

Formulae used:
The root mean square speed ${v_{rms}}$ of the N number of gas molecules is given by
${v_{rms}} = \sqrt {\dfrac{{v_1^2 + v_2^2 + v_3^2 + - - - + v_N^2}}{N}}$ …… (1)
Here, ${v_1}$ is the speed of the first gas molecule, ${v_2}$ is the speed of second gas molecule, ${v_3}$ is the speed of third gas molecule and ${v_N}$ is the speed of Nth gas molecule.
The average speed ty ${v_{avg}}$ of the N number of gas molecules is given by
${v_{avg}} = \dfrac{{{v_1} + {v_2} + {v_3} + - - - + {v_N}}}{N}$ …… (2)
Here, ${v_1}$ is the speed of the first gas molecule, ${v_2}$ is the speed of second gas molecule, ${v_3}$ is the speed of third gas molecule and ${v_N}$ is the speed of Nth gas molecule.

Complete step by step answer:
We have given the speeds of the five randomly chosen gas molecules as 500, 600, 700, 800 and $900\,{\text{m/s}}$.
${v_1} = 500\,{\text{m/s}}$
$\Rightarrow{v_2} = 600\,{\text{m/s}}$
$\Rightarrow{v_3} = 700\,{\text{m/s}}$
$\Rightarrow{v_4} = 800\,{\text{m/s}}$
$\Rightarrow{v_5} = 900\,{\text{m/s}}$
Let us first calculate the root mean square velocity of the five gas molecules.

Rewrite equation (1) for the root mean square speed of these five gas molecules.
${v_{rms}} = \sqrt {\dfrac{{v_1^2 + v_2^2 + v_3^2 + v_4^2 + v_5^2}}{5}}$
Substitute $500\,{\text{m/s}}$ for ${v_1}$, $600\,{\text{m/s}}$ for ${v_2}$, $700\,{\text{m/s}}$ for ${v_3}$, $800\,{\text{m/s}}$ for ${v_4}$ and $900\,{\text{m/s}}$ for ${v_5}$ in the above equation.
${v_{rms}} = \sqrt {\dfrac{{{{\left( {500\,{\text{m/s}}} \right)}^2} + {{\left( {600\,{\text{m/s}}} \right)}^2} + {{\left( {700\,{\text{m/s}}} \right)}^2} + {{\left( {800\,{\text{m/s}}} \right)}^2} + {{\left( {900\,{\text{m/s}}} \right)}^2}}}{5}}$
$\Rightarrow {v_{rms}} = 714\,{\text{m/s}}$
Hence, the root mean square speed of the five molecules is $714\,{\text{m/s}}$.

Let us now calculate the average speed of the five gas molecules.Rewrite equation (1) for the average speed of these five gas molecules.
${v_{avg}} = \dfrac{{{v_1} + {v_2} + {v_3} + {v_4} + {v_5}}}{5}$
Substitute $500\,{\text{m/s}}$ for ${v_1}$, $600\,{\text{m/s}}$ for ${v_2}$, $700\,{\text{m/s}}$ for ${v_3}$, $800\,{\text{m/s}}$ for ${v_4}$ and $900\,{\text{m/s}}$ for ${v_5}$ in the above equation.
${v_{avg}} = \dfrac{{{{\left( {500\,{\text{m/s}}} \right)}^2} + {{\left( {600\,{\text{m/s}}} \right)}^2} + {{\left( {700\,{\text{m/s}}} \right)}^2} + {{\left( {800\,{\text{m/s}}} \right)}^2} + {{\left( {900\,{\text{m/s}}} \right)}^2}}}{5}$
$\therefore {v_{avg}} = 700\,{\text{m/s}}$
Hence, the average speed of the five molecules is $700\,{\text{m/s}}$.

Hence, the root mean square speed of the five gas molecules is $714\,{\text{m/s}}$ and it is not equal to the average speed of these molecules.

Note: The students should be careful while using the formulae for the root mean square speed and average speed of the gas molecules. If one gets confused in these two formulae then we will end with the same conclusion that these two speeds of the gas molecules are not the same but the value of rms speed will be incorrect.