The speed of molecules according to kinetic theory is given by the Boltzmann distribution function \(f\). Thus \(f(c) d c\) represents the probability that \(c\) lies between \(c\) and \(c+d c\) and the distribution function is

\[f(c)=\frac{4}{\sqrt{\pi}} \alpha^{3 / 2} c^{2} \exp \left(-\alpha c^{2}\right)\]

where \(\alpha=m /(2 \kappa T)\).

Show that the area under the distribution function is unity as expected for any probability distribution function.
Find the mean speed defined as

\[\bar{c}=\int_{0}^{\infty} c f(c) d c\]

Find the mean kinetic energy defined as

\[\mathrm{KE}=\frac{1}{2} m \int_{0}^{\infty} c^{2} f(c) d c\]

Verify that the expressions given in the text are valid by doing these integrations. Use of a table of integrals or other software such as Mathematica or MAPLE is useful to find the integrals.