Structure of Atom

2.55 eV

Step 1: Energy of emitted photon = E₄ – E₂ (energy of higher orbit minus lower orbit). Step 2: E₄ = –13.6/16 = –0.85 eV; E₂ = –13.6/4 = –3.4 eV. Step 3: ΔE = E₄ – E₂ = –0.85 – (–3.4) = 2.55 eV. Step 4: This transition (n=4 to n=2) belongs to the Balmer series and falls in the visible region. Step 5: Common errors include subtracting in the wrong order (giving negative energy) or using n=4 to n=1 (which gives a Lyman series photon of ~12.75 eV).

It predicted that revolving electrons would lose energy, spiral inward, and the atom would collapse – contradicting atomic stability.

Step 1: In Rutherford's model, electrons revolve around the nucleus in circular orbits. Step 2: A revolving electron is an accelerating charge (centripetal acceleration). Step 3: According to classical electromagnetic theory, any accelerating charge radiates energy continuously. Step 4: As electrons lose energy, their orbit radius decreases and they spiral toward the nucleus – the atom would collapse in ~10⁻⁸ seconds. Step 5: Also, as they spiral in, they would emit radiation of continuously changing frequency (continuous spectrum), contradicting the observed discrete line spectra of atoms.

3.16 × 10⁻³⁴ kg m² s⁻¹

Step 1: Bohr's second postulate states that angular momentum L = mvr = nh/2π. Step 2: For the 3rd orbit, n = 3. Step 3: L₃ = 3h/2π = 3 × 6.62 × 10⁻³⁴ / (2 × 3.14). Step 4: L₃ = 19.86 × 10⁻³⁴ / 6.28 = 3.16 × 10⁻³⁴ kg m² s⁻¹. Step 5: Note that h/2π ≈ 1.055 × 10⁻³⁴ J·s is called ħ (h-bar). The angular momentum is simply nħ. Option (a) equals h/2π (n=1), option (c) is just h (forgetting the 2π), and option (d) = 3h (another common error).

2.48 × 10⁻¹¹ m

Step 1: Duane-Hunt law: eV = hc/λ_min, so λ_min = hc/eV. Step 2: V = 50 kV = 50 × 10³ V = 5 × 10⁴ V. Step 3: Numerator: hc = 6.62 × 10⁻³⁴ × 3 × 10⁸ = 19.86 × 10⁻²⁶ J·m. Step 4: Denominator: eV = 1.6 × 10⁻¹⁹ × 5 × 10⁴ = 8 × 10⁻¹⁵ J. Step 5: λ_min = 19.86 × 10⁻²⁶ / 8 × 10⁻¹⁵ = 2.48 × 10⁻¹¹ m. This corresponds to X-rays in the hard X-ray region. Doubling the voltage halves λ_min.