Question

Which of the following is/are correct regarding root mean square speed$\left({\mathrm{U}}_{\mathrm{rms}}\right)$ & average translation K.E.$\left({\mathrm{E}}_{\mathrm{av}}\right)$of molecule in gas at equilibrium?

• A. $\left({\mathrm{E}}_{\mathrm{av}}\right)$ is doubled when its temperature is increased 4 times
• B. $\left({\mathrm{U}}_{\mathrm{rms}}\right)$is inversely proportional to the square root of its molecular mass
• C. $\left({\mathrm{E}}_{\mathrm{av}}\right)$at a given temperature doesn’t depend on its molecular mass
• D. $\left({\mathrm{U}}_{\mathrm{rms}}\right)$ is doubled when its temperature is increased 4 times

Answer 1

Answer: (B), (C), (D)

$\left({\mathrm{U}}_{\mathrm{rms}}\right)$ is doubled when its temperature is increased 4 times

The explanation for the correct answer

(B)$\left({\mathrm{U}}_{\mathrm{rms}}\right)$is inversely proportional to the square root of its molecular mass

1. $\left({\mathrm{U}}_{\mathrm{rms}}\right)$ & average translation K.E .$\left({\mathrm{E}}_{\mathrm{av}}\right)$of molecule has the following relationship ${\mathrm{E}}_{\mathrm{av}}=\frac{3}{2}\mathrm{RT},{\mathrm{U}}_{\mathrm{rms}}=\sqrt{\frac{3\mathrm{RT}}{\mathrm{M}}},{\mathrm{U}}_{\mathrm{rms}}\mathrm{\alpha }\frac{1}{\sqrt{\mathrm{M}}}$
2. So, by the above relationship we can say that $\left({\mathrm{U}}_{\mathrm{rms}}\right)$is inversely proportional to the square root of its molecular mass.
3. Thus, it is a correct answer.

(C) $\left({\mathrm{E}}_{\mathrm{av}}\right)$at a given temperature doesn’t depend on its molecular mass

1. Through the above relationship we came to know that $\left({\mathrm{E}}_{\mathrm{av}}\right)$does not have any relationship with molecular mass.
2. $\left({\mathrm{E}}_{\mathrm{av}}\right)$ does not have any relationship with molecular mass and hence does not depend on it.
3. Thus, it is also a correct answer.

(D)$\left({\mathrm{U}}_{\mathrm{rms}}\right)$ is doubled when its temperature is increased 4 times

1. Through the above relationship we came to know that $\left({\mathrm{U}}_{\mathrm{rms}}\right)$ is doubled when its temperature is increased 4 times. ${\mathrm{U}}_{\mathrm{rms}}=\sqrt{\frac{3\mathrm{RT}}{\mathrm{M}}}$
2. Thus, it is a correct answer.

The explanation for the incorrect answer

(A)$\left({\mathrm{E}}_{\mathrm{av}}\right)$ is doubled when its temperature is increased 4 times

1. According to the relationship${\mathrm{E}}_{\mathrm{av}}=\frac{3}{2}\mathrm{RT}$, when we increase the temperature 2 times then $\left({\mathrm{E}}_{\mathrm{av}}\right)$ it doubled.
2. So, this is an incorrect answer.

Thus, Options B, C, and D are correct.