Part A

Consider a monatomic gas of particles each with mass $m.$ What is $v_{x,rms}=\sqrt[\phantom{2}]{?},$ the root mean square $($rms$)$ of the $x$ component of velocity of the gas particles if the gas is at an absolute temperature $T?$

Express your answer in terms of $T,k_B,m,$ and other given quantities.

View Available Hint$($s$)$

$v_{x,rms}=\sqrt[\phantom{2}]{(\text{:}{v}_x^2\text{:})}=$

Part B

Now consider the same system: a monatomic gas of particles of mass $m,$ except in three dimensions. Find $v_{rms},$ the rms speed if the gas is at an absolute temperature $T.$

Express your answer in terms of $T,k_B,m,$ and other given quantities.

$v_{rms}=\sqrt[\phantom{2}]{?}=$

Part C

What is the rms speed $v_0$ of molecules in air at $0C?$ Air is composed mostly of $N_2$ molecules, so you may assume that it has molecules of average mass $28\mathrm{.}01\mathrm{.}661{10}^{-27}kg=4\mathrm{.}65{10}^{-26}kg.$

Express your answer in meters per

Now consider a rigid dumbbell with two masses, each of mass $m,$ spaced a distance $d$ apart.

Part D

Find $\sqrt[\phantom{2}]{({}_x^2{)}_{\text{a}\text{v}\text{g}}},$ the rms angular speed of the dumbbell about a single axis $($taken to be the $x$ axis$),$ assuming that the dumbbell is lined up on the $z$ axis and is in equilibrium at temperature $T.$

Express the rms angular speed in terms of $T,k_B,m,d,$ and other given quantities.

Part E

What is the typical rotational frequency $f_{\text{r}\text{o}\text{t}}$ for a molecule like $N_2$ at room temperature $(25C)?$ Assume that $d$ for this molecule is $1={10}^{-10}m.$ Take the total mass of an $N_2$ molecule to be $m_{N_2}=4\mathrm{.}65{10}^{-26}kg.$ You will need to account for rotations around two axes $($not just one$)$ to find the correct frequency.$]$