Question

If ${V_H},\,{V_N}\,and\,{V_O}$ denotes the root mean square velocities of molecules of hydrogen, nitrogen and oxygen respectively at a given temperature then:

(A) ${V_H} > {V_N} > {V_O}$
(B) ${V_H} = {V_N} = {V_O}$
(C) ${V_O} > {V_H} > {V_N}$
(D) ${V_N} > {V_O} > {V_H}$

Answer 1

Hint: Let start with finding the relation between the root mean square velocity and the masses of the molecules of Hydrogen, Nitrogen and oxygen. Then on the basis of the relation between their masses get the relation between the root mean square velocity of the molecules given.

Formula Used:
 ${V_{rms}}$ is given by:
${V_{rms}} = \sqrt {\dfrac{{3RT}}{M}} $
Where, R is constant
T is temperature
And M is the molecular mass of the particle.

Complete answer:
First start with the given information in the question:
Let ${V_{rms}}$ be the root mean square velocity.
Now, the root mean square velocity of Hydrogen be ${V_H}$ ,
the root mean square velocity of Nitrogen be ${V_N}$
and the root mean square velocity of Oxygen be ${V_O}$

Now, we know that ${V_{rms}}$ is given by:
${V_{rms}} = \sqrt {\dfrac{{3RT}}{M}} $

So, from the above equation we can find that root mean square velocity is inversely proportional to the mass of the molecule.
${V_{rms}} \propto \dfrac{1}{\surd{M}}$
Now, we know that;
${M_H} = $ , ${M_N} = 28$ and ${M_0} = 32$
Means, ${M_H} < {M_N} < {M_O}$
Therefore, ${V_H} > {V_N} > {V_O}$

Hence the correct answer is Option(A).



Note: Here we have used the formula of root mean square velocity in terms of the universal gas constant, temperature of the gas and the mass of the gas molecule. As all the other quantities were constant hence we get the required relation for the root mean square velocity and the mass of the gas.