Question

The wave function, ${\Psi }_{n,l,m_l}$ is a methematical function whose value depends upon spherical polar coordinates (r, θ, Ф) of the electron and characterized by the quantum numbers n, l and m. Here r is distance from nucleus, θ is colatitude and Ф is azimuth. In the mathematical functions given in the Table, Z is atomic number and $a_0$ is Bohr radius.

Column 1	Column 2	Column 3
(i) 1s orbital	(i) ${\Psi }_{n,l,m_l}\propto {\left(\frac{Z}{a_0}\right)}^{\frac{3}{2}}{e}^{\left(\frac{-Zr}{a_0}\right)}$	(P)
(ii) 2s orbital	(ii) One radial node	(Q) Probability density at nucleus $\propto \frac{1}{a_0^3}$
(iii) 2 $p_z$ orbital	(iii) ${\Psi }_{n,l,m_l}\propto {\left(\frac{Z}{a_0}\right)}^{\frac{5}{2}}r{e}^{\left(\frac{-Zr}{2a_0}\right)}cos\theta$	(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 state

For $H{e}^{+}$ion, the only INCORRECT combination is:

• A. (l) (i) (P)
• B. (l) (iv) (R)
• C. (ll) (i) (Q)
• D. (l) (i) (S)

Answer 1

The correct option is D (l) (i) (S)

For 1s orbital Ψ should be independent of ፀ, also it does not contain any radial node.

$\frac{E_4-E_2}{E_6-E_2}=\frac{\frac{E_1}{16}-\frac{E_1}{4}}{\frac{E_1}{36}-\frac{E_1}{4}}=\frac{\frac{3E_1}{16}}{\frac{8E_1}{36}}=\frac{3\times 36}{8\times 16}=\frac{27}{32}$