All Problems Kinetic theory of gases. Boltzmanns Law and Maxwell distribution
Problem 2.111
The potential energy of gas molecules in a certain central field depends on the distance \(r\) from the field's centre as \(U(r)=a r^{2}\) where \(a\) is a positive constant. The gas temperature is \(T,\) the concentration of molecules at the centre of the field is \(n_{0} .\) Find: (a) the number of molecules located at the distances between \(r\) and \(r+d r\) from the centre of the field; (b) the most probable distance separating the molecules from the centre of the field; (c) the fraction of molecules located in the spherical layer between \(r\) and \(r+d r\) (d) how many times the concentration of molecules. in the centre of the field will change if the temperature decreases \(\eta\) times.
2.111. (a) \(d N=n_{0} e^{-a r^{2} / k T} 4 \pi r^{2} d r\) (b) \(r_{p r}=\sqrt{k T / a}\) (c) \(d N / N=\) \(=(a / \pi k T)^{3 / 2} \mathrm{e}^{-a r 2 / k T} 4 \pi r^{2} d r ;\) (d) Will increase \(\eta^{3 / 2}\) -fold.