Quantum mechanical model of the atom
Question

The wave function ${\psi }_{n,l,{m}_{l}}$ is a mathematical function whose value depends upon spherical polar coordinates (r,$\theta ,\varphi$) of the electron and characterized by the quantum numbers n, l and ${m}_{l}$. Here r is distance from nucleus, $\theta$  is colatitude and $\varphi$ is azimuth. In the mathematical functions given in the Table, Z is atomic number and ${a}_{0}$ is Bohr radius.Column 1Column 2Column3(I) Is orbital (i)  (P)  (II) 2S  orbital(ii) one radial node (Q) Probability density  at nucleus$\alpha \frac{1}{{{a}_{0}}^{3}}$ (III) $2{P}_{z}$ orbital(iii) ${\psi }_{n,l,{m}_{j}}\alpha {\left(\frac{Z}{{a}_{0}}\right)}^{\frac{5}{2}}r{e}^{-\left(\frac{Zr}{2{a}_{0}}\right)}\mathrm{COS}\theta$(R) Probability density is maximum at nucleus(IV) $3{{d}_{z}}^{2}$ orbital(iv) xy-plane is a nodal plane(S) Energy needed to excite electron from n=2 state to n= 4 state is $\frac{27}{32}$ times the energy needed to excite electron from n=2 state to n=6 stateFor the given orbital in column I, the only CORRECT combination for any hydrogen-like species is

Difficult
Solution

1s ®  0    0 radial  nodes , no angular  component (Þ no cos q)2s ®  0   1 radial node, , no angular  component (Þ no cos q)$2{p}_{z}\to$ 0 radial node, angular component is present (Þ no cos q)$3{d}_{z}^{2}\to$  0 radial node, angular component is present(Þ no cos q)a) 1s ® 0   radial node Þ wrong as (ii),(i)  1 radial node  b) $3{d}_{z}^{2}⇒d{z}^{2}\text{\hspace{0.17em}\hspace{0.17em}}\text{'}R\text{'}\text{\hspace{0.17em}}is\text{\hspace{0.17em}\hspace{0.17em}}wrong\text{\hspace{0.17em}\hspace{0.17em}}po\mathrm{int}$c)for $2{p}_{z}graph\text{\hspace{0.17em}\hspace{0.17em}}\text{'}p\text{'}iswrong$d) correct

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