Measures of the Maxwell Distribution of Speeds

We can consider individual distributions of different gases (or even a single gas) at different temperatures, for instance. But there are certain measures of the distribution of speeds for a particular gas that we can consider in order to make quick comparisons. We consider these next.

• Most probable speed, $\nu_{mp}$

  
  

  $$\frac{d \ F(\nu)}{d \ \nu} \ = \ \frac{d}{d \ \nu} \left ( 4 \ \pi \ \left ( \frac{m}{2 \ \pi \ k_B \ T} \right )^{3/2} \ \nu^2 \ e^{-m \ nu^2 / 2\ k_B \ T} \right )$$

  $$= 4 \ \pi \ \left ( \frac{m}{2 \ \pi \ k_B \ T} \right )^{3/2} \frac{d}{d \ \nu} \ \left ( \nu^2 \ e^{-m \ nu^2 / 2\ k_B \ T} \right )$$

  $$= 4 \ \pi \ \left ( \frac{m}{2 \ \pi \ k_B \ T} \right )^{3/2} \ e^{-m \ nu^2 / 2\ k_B \ T} \left [ 2 \ \nu - \frac{m \nu^3}{k_B \ T} \right ]$$

  
  

  The maximum probability occurs when the derivative is identically $0$; we thus obtain:

  
  

  $$2 \ \nu_{mp} \ - \ \frac{m \ \nu^3_{mp}}{k_B \ T} \ = \ 0 \$$

  $$\nu_{mp} \ = \ \sqrt{ \frac{2 \ k_B \ T}{m}}$$

  
  

• Average Speed

  
  

  $$< \nu> \ = \ \nu_{ave} \ = \ \displaystyle\int_{0}^{\infty} \ \nu \ F(\nu) \ d \nu$$

  $$= 4 \ \pi \ \left ( \frac{m}{2 \ \pi \ k_B \ T} \right )^{3/2} \ \displaystyle\int_{0}^{\infty} \ \nu^3 e^{-m \ \nu^2 / 2\ k_B \ T} \ d \nu$$

  $$= \left ( \frac{8 \ k_B \ T \ }{\pi \ m} \right)^{1/2} \ = \ \left ( \frac{8 \ R \ T \ }{\pi \ M} \right)^{1/2}$$

  
  

• Root-Mean-Square Speed

  
  

  $$\nu_{rms} \ = \ \left [ \left < \nu^2 \right > \right ]^{1/2} \ = \ \left( \frac{3 k_B \ T}{m} \right)^{1/2}$$

  
  

• Using the previous expressions for average speed and root-mean-square speed, we obtain the following relation between the two measures of the speed distribution:

  
  

  $$\left [ \left < \nu^2 \right > \right ]^{1/2} \ = \ \sqrt{ \frac{3 \ \pi}{8} } \ \left < \nu \right >$$

  
  

• $\nu_{rms} > \nu_{ave} > \nu_{mp}$

• All measures are proportional to $T^{1/2}$ and $M^{-1/2}$.