Question

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

Column 1 Column 2 Column 3 (P)

(I) 1s orbital (i) $\psi_{n,l,m_l} \propto (\frac{Z}{a_0})^{\frac{3}{2}} e^{-(\frac{Zr}{a_0})}$ (Q) Probability density at the nucleus $\propto \frac{1}{a_0^3}$.

(II) 2s orbital (ii) One radial node

(III) $2p_z$ orbital (iii) $\psi_{n,l,m_l} \propto (\frac{Z}{a_0})^{\frac{5}{2}} r e^{-(\frac{Zr}{a_0})} \cos \theta$ (R) Probability density is maximum at the nucleus.

(IV) $3d_z^2$ orbital (iv) $xy$-plane is a nodal plane (S) Energy needed to excite an electron from $n=2$ state to $n=4$ state is $\frac{27}{32}$ times the energy needed to excite are electron from $n=2$ state to $n=6$ state.

For $He^+$ ion, the only incorrect combination

• A. (I) (i) (S)
• B. (I) (i) (R)
• C. (I) (iii) (R)
• D. (II) (ii) (Q)

Answer 1

Answer: (C)

C

Answer 2

Let's analyze each option.

Option A: (I) (i) (S)

• (I) 1s orbital. (i) $\psi_{1s} \propto (\frac{Z}{a_0})^{\frac{3}{2}} e^{-(\frac{Zr}{a_0})}$. This is the correct form of the 1s wave function.
• (S) Energy needed to excite an electron from $n=2$ state to $n=4$ state is $\frac{27}{32}$ times the energy needed to excite an electron from $n=2$ state to $n=6$ state. For a hydrogen-like ion, $E_n = -\frac{13.6 Z^2}{n^2}$.
  • $\Delta E_{2 \to 4} = E_4 - E_2 = 13.6 Z^2 (\frac{1}{2^2} - \frac{1}{4^2}) = 13.6 Z^2 (\frac{1}{4} - \frac{1}{16}) = 13.6 Z^2 \frac{3}{16}$.
  • $\Delta E_{2 \to 6} = E_6 - E_2 = 13.6 Z^2 (\frac{1}{2^2} - \frac{1}{6^2}) = 13.6 Z^2 (\frac{1}{4} - \frac{1}{36}) = 13.6 Z^2 \frac{8}{36} = 13.6 Z^2 \frac{2}{9}$.
  • $\frac{\Delta E_{2 \to 4}}{\Delta E_{2 \to 6}} = \frac{3/16}{2/9} = \frac{3}{16} \times \frac{9}{2} = \frac{27}{32}$. Statement (S) is correct.

Combination (I) (i) (S) is correct.

Option B: (I) (i) (R)

• (I) 1s orbital. (i) $\psi_{1s} \propto (\frac{Z}{a_0})^{\frac{3}{2}} e^{-(\frac{Zr}{a_0})}$. Correct.
• (R) Probability density is maximum at the nucleus. Probability density $|\psi|^2 \propto e^{-2Zr/a_0}$. At $r=0$, $|\psi|^2 \propto 1$, which is the maximum value as $e^{-2Zr/a_0}$ decreases as $r$ increases for $r>0$. Statement (R) is correct for 1s orbital.

Combination (I) (i) (R) is correct.

Option C: (I) (iii) (R)

• (I) 1s orbital. (iii) $\psi \propto (\frac{Z}{a_0})^{\frac{5}{2}} r e^{-(\frac{Zr}{a_0})} \cos \theta$. This is not the wave function for a 1s orbital. For a 1s orbital, $l=0$, so the angular part is constant, and the radial part is proportional to $e^{-Zr/a_0}$. The given function has an angular part proportional to $\cos \theta$ (corresponding to $l=1, m_l=0$) and a radial part proportional to $r e^{-Zr/a_0}$. This radial part is not for $n=1$ or $n=2, l=1$.

Since (I) and (iii) is an incorrect combination, the entire combination is incorrect.

Let's also check (iii) and (R). For the wave function in (iii), $\psi(r=0) = 0$ because of the factor $r$. So, the probability density at the nucleus is zero. Thus, the probability density is not maximum at the nucleus. (iii) and (R) is an incorrect combination.

Therefore, combination (I) (iii) (R) is an incorrect combination.

Option D: (II) (ii) (Q)

• (II) 2s orbital. (ii) One radial node. For 2s orbital, $n=2, l=0$. Number of radial nodes = $n-l-1 = 2-0-1 = 1$. Statement (ii) is correct for 2s orbital.
• (Q) Probability density at the nucleus $\propto \frac{1}{a_0^3}$. The wave function for 2s orbital is $\psi_{2s} \propto (\frac{Z}{a_0})^{\frac{3}{2}} (2 - \frac{Zr}{a_0}) e^{-Zr/2a_0}$. At $r=0$, $\psi_{2s}(0) \propto (\frac{Z}{a_0})^{\frac{3}{2}} (2) e^0 = 2 (\frac{Z}{a_0})^{\frac{3}{2}}$. The probability density at the nucleus is $|\psi_{2s}(0)|^2 \propto (2 (\frac{Z}{a_0})^{\frac{3}{2}})^2 = 4 (\frac{Z}{a_0})^3 = 4 \frac{Z^3}{a_0^3}$. This is proportional to $\frac{1}{a_0^3}$. Statement (Q) is correct for 2s orbital.

Combination (II) (ii) (Q) is correct.

We are looking for the only incorrect combination. Option C is an incorrect combination.

Final check:

• Option A: (I) 1s, (i) 1s wave function, (S) Energy ratio. All are correct.
• Option B: (I) 1s, (i) 1s wave function, (R) Probability density max at nucleus for 1s. All are correct.
• Option C: (I) 1s, (iii) wave function which is not 1s, (R) Probability density max at nucleus. Incorrect.
• Option D: (II) 2s, (ii) One radial node for 2s, (Q) Probability density at nucleus $\propto 1/a_0^3$. All are correct.

Since the question asks for the only incorrect combination, Option C is the answer.