For a given gas at temperature $T$,the root mean square velocity is $\nu_{rms}$,the average velocity is $\nu_{av}$,and the most probable velocity is $\nu_{mp}$. Which of the following relations is correct?

$A$ mixture of $2\, moles$ of helium gas (atomic mass $= 4\, u$) and $1\, mole$ of argon gas (atomic mass $= 40\, u$) is kept at $300\, K$ in a container. The ratio of their rms speeds $\left[ \frac{V_{rms}(\text{helium})}{V_{rms}(\text{argon})} \right]$ is close to:

At a temperature of $27^{\circ}C$ and a pressure of $1.0 \times 10^5 \, N/m^2$,the $rms$ speed of a gas is $200 \, m/s$. What is the $rms$ speed at a temperature of $127^{\circ}C$ and a pressure of $0.5 \times 10^5 \, N/m^2$?

When the temperature of a gas is raised from $27^{\circ} C$ to $90^{\circ} C$,the increase in the rms velocity of the gas molecules is: (in $\%$)

$A$ vessel is partitioned into two equal halves by a fixed diathermic separator. Two different ideal gases are filled in the left $(L)$ and right $(R)$ halves. The $rms$ speed of the molecules in the $L$ part is equal to the mean speed of the molecules in the $R$ part. Then the ratio of the mass of a molecule in the $L$ part to that of a molecule in the $R$ part is

The root mean square (rms) velocity of an ideal gas at temperature $T$ is $v$. If the temperature is increased to $4 T$,the rms velocity of the gas is