Energy and radius of first Bohr orbit of He+ and Li2+ are [Given RH = 2.18 × 10–18 J, a0 = 52.9 pm]

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Question

Energy and radius of first Bohr orbit of He+ and Li2+ are [Given RH = 2.18 × 10–18 J, a0 = 52.9 pm]
(1) En(Li2+) = –8.72 × 10–16 J; rn(Li2+) = 17.6 pm; En(He+) = –19.62 × 10–16 J; rn(He+) = 17.6 pm
(2) En(Li2+) = –19.62 × 10–18 J; rn(Li2+) = 17.6 pm; En(He+) = –8.72 × 10–18 J; rn(He+) = 26.4 pm
(3) En(Li2+) = –8.72 × 10–18 J; rn(Li2+) = 26.4 pm; En(He+) = –19.62 × 10–18 J; rn(He+) = 17.6 pm
(4) En(Li2+) = –19.62 × 10–16 J; rn(Li2+) = 17.6 pm; En(He+) = –8.72 × 10–16 J; rn(He+) = 26.4 pm

Accepted Answer

Correct Answer: B — Option 2Step-by-step Solution:For hydrogen-like species (n=1, ground state):Energy: E₁ = −RH × Z²/n² = −2.18×10⁻¹⁸ × Z² JRadius: r₁ = a₀ × n²/Z = 52.9/Z pmFor He⁺ (Z=2):E₁(He⁺) = −2.18×10⁻¹⁸ × 4 = −8.72×10⁻¹⁸ Jr₁(He⁺) = 52.9/2 = 26.45 ≈ 26.4 pmFor Li²⁺ (Z=3):E₁(Li²⁺) = −2.18×10⁻¹⁸ × 9 = −19.62×10⁻¹⁸ Jr₁(Li²⁺) = 52.9/3 = 17.63 ≈ 17.6 pmOption 2 matches exactly. Options 1 and 4 have wrong exponent (10⁻¹⁶); Option 3 swaps He⁺ and Li²⁺ values.Key Concepts:Energy of nth orbit: Eₙ = −RHZ²/n² (more negative = more stable; Li²⁺ has lower energy than He⁺)Radius of nth orbit: rₙ = a₀n²/Z (smaller Z → larger radius; He⁺ has larger radius than Li²⁺)As Z increases: orbital shrinks (smaller radius) and electron is more tightly bound (lower, more negative energy).

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