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: Understanding Orbital Energy Comparisons
- The energy levels of hydrogen-like atoms depend on the principal quantum number and nuclear charge.
- For hydrogen (\(H\)), energy levels follow: \[ E_n = -\frac{13.6}{n^2} \text{ eV} \]
- For helium (\(He^+\)), the energy levels are modified by the nuclear charge \(Z\): \[ E_n = -\frac{13.6 Z^2}{n^2} \text{ eV} \] Step 2: Evaluating the Given Statements
- (I) \(E_2(H) = E_2(H)\): This is trivially true.
- (II) \(E_3(H) = E_3(He)\): Incorrect, as energy levels differ for different atomic numbers.
- (III) \(E_{2s}(H)<E_{2s}(He)\): Incorrect, as higher nuclear charge in \(He\) causes lower energy.
- (IV) \(E_{3s}(He)<E_{3s}(H)\): True, since helium has a stronger nuclear charge, lowering energy levels. Thus, the correct answer is \( \mathbf{(4)} \) I, IV only.