An expression for the root mean square speed, v_rms, of gas molecules is derived in Topic 3D. 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 v_mean = (8RT/πM)^1/2 and vmp = (2RT/M)^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.