Chapter 7.8, Problem 60ES

The average speed, $\bar{\upsilon },$ of the molecules of an ideal gas is given by

$\bar{\upsilon }=\frac{4}{\sqrt{\pi }}{\left(\frac{M}{2RT}\right)}^{3/2}{\displaystyle {\int }_0^{+\infty }{\upsilon }^{3}e}{}^{-M\upsilon /(2RT)}d\upsilon$

and the root-mean-square speed, ${\upsilon }_{rms},$ by

${\bar{\upsilon }}^2{}_{rms}=\frac{4}{\sqrt{\pi }}{\left(\frac{M}{2RT}\right)}^{3/2}{\displaystyle {\int }_0^{+\infty }{\upsilon }^{4}e}{}^{-M{\upsilon }^2/(2RT)}d\upsilon$

Where $\upsilon$ is the molecular speed, T is the gas temperature, M is the molecular weight of the gas, and R is the gas constant.

(a) Use a CAS to show that

${\displaystyle {\int }_0^{+\infty }{x}^3{e}^{-a^2{x}^2}}dx=\frac{1}{2a^4},a>0$

and use this result to show that $\bar{\upsilon }=\sqrt{8RT/\left(\pi M\right).}$

(b) Use a CAS to show that

${\displaystyle {\int }_0^{+\infty }{x}^{4}e{}^{-a^2{x}^2}}dx=\frac{3\sqrt{\pi }}{8a^5},a>0$

and use this result to show that ${\upsilon }_{rms}=\sqrt{3RT/M.}$