Question

If the r.m.s velocity of Hydrogen gas at a certain temperature is $c$, then the r.m.s velocity of Oxygen at the same temperature is
$\text{A}\text{. }\dfrac{c}{8}$
$\text{B}\text{. }\dfrac{c}{10}$
$\text{C}\text{. }\dfrac{c}{4}$
$\text{D}\text{. }\dfrac{c}{2}$

Answer 1

Hint: For a given temperature, the root mean square (rms) velocity is directly proportional to the square root of molecular mass. We will use the formula of r.m.s velocity to find the relation between r.m.s velocity of Hydrogen and Oxygen at a given temperature.

Formula used:
${{v}_{rms}}=\sqrt{\dfrac{3RT}{M}}$

Complete step by step answer:
Root mean square velocity of any substance is defined as the square root of the average of the square of the velocity or we can say that r.m.s velocity is the square root of the mean square velocity.
Expression for Root mean square velocity
${{v}_{rms}}=\sqrt{{{v}_{1}}^{2}+{{v}_{2}}^{2}+{{v}_{3}}^{2}+...{{v}_{n}}^{2}}$
where ${{v}_{1}},{{v}_{2}},{{v}_{3}}....{{v}_{n}}$ are the individual velocities of the particles
Formula for Root mean square velocity of gases:
${{v}_{rms}}=\sqrt{\dfrac{3RT}{M}}$
where $R$ is gas constant, $T$ is the temperature and $M$ is the molecular mass of gas
We are given that the rms velocity of Hydrogen at certain temperature is $c$
It means that ${{v}_{rmsH}}=\sqrt{\dfrac{3RT}{M}}=\sqrt{3RT}=c$
Molecular mass of Hydrogen is 1.
For oxygen,
${{v}_{rmsO}}=\sqrt{\dfrac{3RT}{M}}=\sqrt{\dfrac{3RT}{16}}=\dfrac{c}{4}$
We get, ${{v}_{rmsO}}=\dfrac{{{v}_{rmsH}}}{4}=\dfrac{c}{4}$
Root mean square velocity of Oxygen at that temperature is $\dfrac{c}{4}$
Hence, the correct option is C.

Additional information:
Apart from Root mean square velocity, we have Average velocity and Most probable velocity of a gas. Average velocity is defined as the average of the individual velocities of the particles. Average velocity of an ideal gas molecule is zero, because the motion of particles is totally random, and the vector sum of all the velocities becomes zero. Most probable velocity is defined as the velocity acquired by the maximum fraction of the total number of the molecules.

Note: While calculating the value of velocity, keep in mind whether you have to take the molecular mass or the atomic mass of the particle. In the above question, we took the atomic mass of Hydrogen and Oxygen for calculating the ratio of their r.m.s velocities.