Question

The following orbital energies (\(E\)) are compared. Identify the correct sets 

(I) E₂(H) = E₂(H)
(II) E₃(H) = E₃(He)
(III) E₂s(H) < E₂s(He)
(IV) E₃s(He) < E₃s(H)

• A. I, II only
• B. I, II, III, IV
• C. III, IV only
• D. I, IV only

Hint:

For hydrogen-like atoms, the energy of an orbital is given by: \[ E_n = -\frac{13.6 Z^2}{n^2} \text{ eV} \] where \(Z\) is the atomic number. A higher nuclear charge (\(Z\)) lowers orbital energy.

Answer 1

The Correct Option is D

Step 1: Energy levels of hydrogen-like species

\[ E_n = -\frac{13.6}{n^2} \text{ eV} \quad (\text{for H}) \]

\[ E_n = -\frac{13.6 Z^2}{n^2} \text{ eV} \quad (\text{for hydrogen-like ions}) \]

\[ Z(\text{He}^+) = 2 \Rightarrow E_n(\text{He}^+) = -\frac{54.4}{n^2} \text{ eV} \]

Step 2: Compare statements

(I) \( E_2(H) = E_2(H) \) → True

(II) \( E_3(H) = E_3(He^+) \) → False

(III) \( E_{2s}(H) < E_{2s}(He^+) \) → False

(IV) \( E_{3s}(He^+) < E_{3s}(H) \) → True

Final Answer: I and IV only