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.