Question

What is the ratio of mean speed of an ${O_2}$ molecule to the rms speed of an ${O_2}$ molecule at same T?
a.) ${\left( {3\pi /7} \right)^{1/2}}$
b.) ${\left( {8/3\pi } \right)^{1/2}}$
c.) $\sqrt {\dfrac{{8RT}}{{\pi M}}}$ ${\left( {3\pi } \right)^{1/2}}$
d.) ${\left( {4\pi /9} \right)^{1/2}}$

Answer 1

Hint: In this given question the rms speed of a molecule is given which means the root-mean-square-speed which is used to measure the speed of particles present in a gas. The Rms can be taken as both molecular weight and temperature.

Complete answer:
According to this question;
The man speed $V = \sqrt {\dfrac{{8RT}}{{\pi M}}}$
The rms speed U=$\sqrt {\dfrac{{3RT}}{M}}$
Here, M= molar mass.
T= temperature.
R= universal gas constant.
Now average speed =$\sqrt {\dfrac{{8RT}}{{\pi M}}}$=$\sqrt {\dfrac{{3RT}}{M}}$
Combine the both ratios we get, $\dfrac{V}{U}$= a $\dfrac{{\sqrt {\dfrac{{8RT}}{{\pi M}}} }}{{\sqrt {\dfrac{{3RT}}{M}} }}$
$\Rightarrow$ And here we are left with $\dfrac{{\sqrt {\dfrac{8}{\pi }} }}{{\sqrt 3 }}$
The answer we get is $\sqrt {\dfrac{8}{{3\pi }}}$=${\left( {8/3\pi } \right)^{1/2}}$
To solve this question first get the man speed $V = \sqrt {\dfrac{{8RT}}{{\pi M}}}$ and the rms speed U=$\sqrt {\dfrac{{3RT}}{M}}$. However we know that the molecules of the speed in gas particles is directly proportional to the temperature and inversely proportional to the molar mass of gas. Helium has the highest molecular speed. The behaviour of helium gas in balloons is an example of molecular effusion. The molecular effusion is gas molecules in which the random and the tiny-tiny rapid motion of gas molecules are present in it and Xenon has the lowest molecular speed. Then combine the ratio and cancel the similar terms and you will get the answer ${\left( {8/3\pi } \right)^{1/2}}$.

Therefore, from the above option (B) ${\left( {8/3\pi } \right)^{1/2}}$ is the correct answer.

Note: Molecular effusion can travel through small openings. Molecular effusion is a chemical term and it is a property gas property. Moreover the term rms is used to express an AC quantity of voltage or current with respect to function with equivalent to DC.