The Distribution of Molecular Speeds — Question 9

Question 9

A ratio of speed densities can determine temperature

A Maxwell gas of molecular mass $m=4.00\times10^{-26}$ kg has measured speed-density ratio $f(600\,\mathrm{m/s})/f(300\,\mathrm{m/s})=2.00$. Use $k_B=1.381\times10^{-23}$ J/K. These are density values or equal-width narrow-bin limits, not unequal-width raw counts.

1. Infer temperature while retaining the $v^2$ prefactor in the density.

2. For these two speeds, determine the allowable ratio range for finite positive temperature and interpret a reported ratio of 5.00.

Question 9 - Solution

A ratio of speed densities can determine temperature

(a) Method and calculation

$$r=\left(\frac{600}{300}\right)^2\exp\left[-\frac{m(600^2-300^2)}{2k_BT}\right]=4e^{-A/T}.$$$$A=\frac{4.00\times10^{-26}(270000)}{2(1.381\times10^{-23})}\approx391.021\,\mathrm K,\quad T=\frac A{\ln(4/2)}\approx564.124\,\mathrm K.$$

(b) Interpretation and further calculation

$$T>0\ \Rightarrow\ 0<e^{-A/T}<1\ \Rightarrow\ 0<r<4.$$$$r=5\ \Rightarrow\ \ln(4/r)<0,$$which cannot give a positive Maxwell temperature. Such data require checking the model, detector bias or bin-width conversion.

Physical check

Omitting the $v^2$ factor would reverse the interpretation of a density that rises over the selected speed interval.