The relation between root mean square speed $(v_{rms})$ and most probable speed $(v_{p})$ for the molar mass $M$ of oxygen gas molecule at the temperature of $300\, K$ is:

$A$ certain amount of an ideal gas is taken from its initial state $1$ to final state $4$ through the path $1 \rightarrow 2 \rightarrow 3 \rightarrow 4$ as shown in the figure. $AB, CD, EF$ are all isotherms. If $v_p$ is the most probable speed of the molecules, then:

The $rms$ speed of hydrogen molecules at room temperature $(300 \ K)$ will be:

$A$ 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

What is $rms$ value? What is $v_{rms}$? Derive the equation of $v_{rms}$ in terms of pressure.

The root mean square speed of smoke particles of mass $5 \times 10^{-17} \, kg$ in their Brownian motion in air at $NTP$ is approximately $....... \, mm \, s^{-1}$. [Given $k = 1.38 \times 10^{-23} \, J \, K^{-1}$ and $T = 293 \, K$]