The wave function ψn.ℓ,mℓ is a mathematical function whose value depends upon spherical polar coordinates (r,θ,φ) of the electron and characterized by the quantum numbers n,ℓ and mℓ. Here r is distance from nucleus, θ is colatitude and φ is azimuth. In the mathematical functions given in the Table, Z is atomic number and a0 is Bohr radius.Column 1Column 2Column 3(I) 1 s orbital (II ) 2s orbital(ii) One radial node(Q) Probability density at nuvleus (III) 2pz orbital (R) Probability density is maximun at nucleus(IV) 3dz2 orbital(iv) xy- plane is a nodal plane(S) Energy needed to excite electron from n=2 state to n=4 state is times the energy needed to excite electron from n=2 state

The wave function ψ_n.ℓ,mℓ is a mathematical function whose value depends upon spherical polar coordinates (r,θ,φ) of the electron and characterized by the quantum numbers n,ℓ and m_ℓ. Here r is distance from nucleus, θ is colatitude and φ is azimuth. In the mathematical functions given in the Table, Z is atomic number and a₀ is Bohr radius.

Column 1	Column 2	Column 3
(I) 1 s orbital	$(i) Ψ_{n, ℓ, m_{ℓ}} \propto {(\frac{Z}{a_{0}})}^{\frac{3}{2}} e^{- (\frac{Zr}{a_{0}})}$
(II ) 2s orbital	(ii) One radial node	(Q) Probability density at nuvleus $\propto \frac{1}{a_{0}^{3}}$
(III) 2p_z orbital	$(iii) Ψ \propto {(\frac{Z}{a_{0}})}^{\frac{5}{2}} {re}^{- (\frac{Zr}{{za}_{0}})} cosθ$	(R) Probability density is maximun at nucleus
(IV) 3d_z² orbital	(iv) xy- plane is a nodal plane	(S) Energy needed to excite electron from n=2 state to n=4 state is times the energy needed to excite electron from n=2 state

Linked Question 1

For He⁺ ion, the only INCORRECT combination is

(II) (ii) (Q)

(I) (i) (R)

(I) (i) (S)

(I) (iii) (R)

For He⁺ ion

(A) For 1s-orbital, $ψ_{n, ℓ, m_{ℓ}} \propto {(\frac{Z}{a_{0}})}^{\frac{3}{2}} . e^{- (\frac{Zr}{a_{0}})}$

and probability density = ${| ψ_{n, ℓ, m_{ℓ}} |}^{2}$

s-orbital has no angular node.

$∴$ (I) (i) (R) is correct combination

(B) For 2s-orbital. There us one radial node & $ψ_{n, ℓ, m_{ℓ}} \propto {(\frac{Z}{a_{0}})}^{\frac{3}{2}} . e^{- (\frac{Zr}{2 a_{0}})}$

$∴$ Probability density $\propto \frac{1}{a_{0}^{3}}$

$∴$ (II) (ii) (Q) is correct combination

${({ΔE}_{2 \to 4})}_{{He}^{+}}$ = (–0.85 + 3.4) × 2² eV

${({ΔE}_{2 \to 4})}_{{He}^{+}} = 13 .6 \times (\frac{1}{4} - \frac{1}{36}) \times 2^{2} eV$ = (2.55 × 2² )eV

= 3.022 × 2² eV

$\Rightarrow \frac{27}{32}$ × 3.022 × 2² = 2.55 × 2²

$∴$ (I) (i) (S) is correct combination

(D) For 1s-orbital : $ψ_{n, ℓ, m_{ℓ}} \propto {(\frac{Z}{a_{0}})}^{\frac{3}{2}} . e^{- (\frac{Zr}{a_{0}})}$

$∴$ (I) (iii) are incorrect combination

$∴$ Answer (D)

Linked Question 2

For the given orbital in column 1, the only CORRECT combination for any hydrogen-like species is

(I) (ii) (S)

(III) (iii) (P)

(II) (ii) (P)

(IV) (iv) (R)

For 2s-orbital :

Number of radial node = 1

& probability density at nucleus = ${| ψ_{n, ℓ, m_{ℓ}} |}^{2}$

i.e. probability density at nucleus $\propto \frac{1}{a_{0}^{3}}$

$∴$ (II) (ii) (P) is incorrect combination

$∴$ Answer (D)

Linked Question 3

For hydrogen atom, the only CORRECT combination is

(I) (i) (S)

(II) (i) (Q)

(I) (iv) (R)

(I) (i) (P)

(D) For 1s-orbital : $ψ_{n, ℓ, m_{ℓ}} \propto {(\frac{Z}{a_{0}})}^{\frac{3}{2}} . e^{- (\frac{Zr}{a_{0}})}$

( $Δ$ E_2→4)_H = (–0.85 + 3.4) eV = 2.55 eV

${({ΔE}_{2 \to 6})}_{H} = 13 .6 (\frac{1}{4} - \frac{1}{36}) eV$ = 3.022

But, 3.022 × $\frac{27}{32}$ = 2.55

$∴$ (I) (i) (S) is correct combination

$∴$ Answer (D)