Question in my book The root mean square speed of gas molecu

Question in my book?

The root mean square speed of gas molecules was derived in Section 4.10. Using the Maxwell distribution of speeds, we can also calculate the mean speed and most probable (mp) speed of a collection of molecules. The equations used to calculate these two quantities are and nu mp = (2RT/M)1/2, respectively. These values have a fixed relation to one another, (a) Place these three quantities m 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, determine the ratio of the larger values to the smallest.

Solution

a) Vav>Vmean>Vmp

b)since in all the cases, the molar mass has the same exponent of -1/2 , so when we take the ratio, it cancels off and we get the required value

c)2^1/2:(8/pi)^1/2:3^1/2

=1:1.128:1.2247