All Problems Kinetic theory of gases. Boltzmanns Law and Maxwell distribution
Problem 2.98
What fraction of molecules in a gas at a temperature \(T\) has the kinetic energy of translational motion exceeding \(\varepsilon_{0}\) if \(\varepsilon_{0} \gg\) \(\gg k T ?\)
\frac{\Delta N}{N}=\frac{2 \pi}{(\pi k T)^{3 / 2}} \int_{\varepsilon_{0}}^{\infty} \sqrt{\varepsilon} e^{-\varepsilon / k T} d \varepsilon The principal contribution to the value of the integral is provided by the smallest values of ( arepsilon,) namely ( arepsilon approx arepsilon_{0} .) The slowly varying factor (sqrt{ arepsilon}) can be taken from under the radical sign if ascribed the constant value (sqrt{ arepsilon_{0}}). Then \Delta N / N=2 \sqrt{\varepsilon_{0} / \pi k T} \mathrm{e}^{-\varepsilon_{0} / k T}