Most Probable Speed (v_mp)

Definition. The most probable speed, $v_{mp}$, is the single speed value shared by the largest number of molecules in the gas -- it is the speed at which the Maxwell speed-distribution curve (Section 9.2.8) reaches its peak. It works out to

(The full derivation of this result, and of the average-speed formula in Section 9.2.6, requires calculus applied to the Maxwell distribution function and is beyond the scope of this unit.)

Ranking the three speeds. Comparing the numerical coefficients of the three characteristic speeds, $v_{rms}=1.73\sqrt{kT/m}$, $\bar v=1.60\sqrt{kT/m}$, and $v_{mp}=1.41\sqrt{kT/m}$, shows that for any gas, at any temperature,

Expressed as a fixed ratio, $v_{rms}:\bar v:v_{mp} = \sqrt3:\sqrt{8/\pi}:\sqrt2 \approx 1.732:1.60:1.414$. The rms speed is always the greatest of the three and the most probable speed is always the least, with the plain average speed sitting in between. …