More practice questions

4. Q4. What is the angular momentum of the electron in the n = 2 orbit of hydrogen?

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   L = nh/2π = 2h/2π = h/π = 6.63 × 10⁻³⁴/π ≈ 2.11 × 10⁻³⁴ J s (NCERT Eq. (12.5), page 299).

5. Q5. The electron of a hydrogen atom is in the n = 2 state. Find its kinetic and potential energies.

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   E₂ = −13.6/4 = −3.40 eV. Since K = −E and U = 2E, K = 3.40 eV and U = −6.80 eV (reasoned from NCERT Eqs. (12.4) and (12.10), pages 296 and 299).

6. Q6. Find the ratio of the electron’s speeds in the n = 1, 2 and 3 orbits of hydrogen.

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   v_n ∝ 1/n, so v₁ : v₂ : v₃ = 1 : 1/2 : 1/3 = 6 : 3 : 2 (reasoned from NCERT Eqs. (12.5) and (12.7), page 299).

7. Q7. Find the radius of the ground-state orbit and the ground-state energy of doubly ionised lithium, Li²⁺ (Z = 3).

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   r₁ = a₀/Z = 5.3 × 10⁻¹¹/3 ≈ 1.77 × 10⁻¹¹ m, and E₁ = −13.6 × Z^² = −13.6 × 9 = −122.4 eV (beyond the NCERT text).

8. Q8. How does the electron’s frequency of revolution change with n? Find it for n = 1.

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   f = v/(2πr), with v ∝ 1/n and r ∝ n^², so f ∝ 1/n^³. For n = 1, f = (2.2 × 10⁶)/(2π × 5.3 × 10⁻¹¹) ≈ 6.6 × 10¹⁵ Hz, the frequency of NCERT Example 12.4 (reasoned from NCERT Eqs. (12.5) and (12.7), page 299).