$3\mathrm{.}16$ An expression for the root mean square speed, $v_{rms},$ of gas molecules is derived in Topic $3$D$.$ Using the Maxwell distribution of speeds, the mean speed and the most probable $(mp)$ speed of a collection of molecules can be calculated and are

$()mean$ and $v_{mp}=(2R\frac{T}{M}{)}^{\frac{1}{2}}.$ These values have a fixed relation to each other. $($a$)$ Place these three quantities in order of increasing magnitude. $($b$)$ Show that the relative magnitudes are independent of the molar mass of the gas. $($c$)$ Using the smallest speed as the reference point, rank them by magnitude and determine the ratios of the larger values to the smallest.