The wave function, ψ n , l , m l is a mathematical function whose value depends upon spherical polar…

The wave function, ${\mathrm{\psi }}_{\mathrm{n},\mathrm{ }\mathrm{l},\mathrm{ }{\mathrm{m}}_{\mathrm{l}}}$ is a mathematical function whose value depends upon spherical polar coordinates $\left(\mathrm{r},\mathrm{ }\mathrm{\theta },\mathrm{ }\mathrm{\varphi }\right)$ of the electron and characterized by the quantum numbers $\mathrm{n},\mathrm{ }\mathrm{l}$ and ${\mathrm{m}}_{\mathrm{l}}$ . Here $\mathrm{r}$ is distance from nucleus, $\mathrm{\theta }$ is colatitude and $\mathrm{\varphi }$ is azimuth. In the mathematical functions given in the Table, Z is atomic number and ${\mathrm{a}}_{\mathrm{o}}$ is Bohr radius.
 

Column – 1	Column – 2	Column – 3
(I) Is orbital	(i) ${\mathrm{\psi }}_{\mathrm{n},\mathrm{ }\mathrm{l},\mathrm{ }{\mathrm{m}}_{\mathrm{l}}}\mathrm{ }\propto {\left(\frac{\mathrm{Z}}{{\mathrm{a}}_{\mathrm{o}}}\right)}^{\frac{3}{2}}\mathrm{ }{\mathrm{e}}^{-\left(\frac{\mathrm{Zr}}{{\mathrm{e}}_{\mathrm{o}}}\right)}$	(P)
(II) 2s orbital	(ii) One radial node	(Q) Probability density at nucleus $\propto \frac{1}{a_o^3}$
(III) $2{\mathrm{p}}_{\mathrm{z}}$ orbital	(iii) ${\mathrm{\psi }}_{\mathrm{n},\mathrm{ }\mathrm{l},\mathrm{ }{\mathrm{m}}_{\mathrm{l}}}\mathrm{ }\propto {\left(\frac{\mathrm{Z}}{{\mathrm{a}}_{\mathrm{o}}}\right)}^{\frac{5}{2}}{\mathrm{re}}^{-\left(\frac{\mathrm{Zr}}{2{\mathrm{a}}_{\mathrm{o}}}\right)}\cos \mathrm{\theta }$	(R) Probability density is maximum at nucleus
(IV) $3{\mathrm{d}}_{\mathrm{z}}^2$ orbital	(iv) $xy$ -plane is a nodal plane	(S) Energy needed to excite an electron from $\mathrm{n}=2$ state to $\mathrm{n}=4$ state is $\frac{27}{32}$ times the energy needed to excite electron from $\mathrm{n}=2$ state to $\mathrm{n}=6$ state.