Dual Nature of Radiation and Matter

5.16 × 10¹⁴ Hz

Step 1: At threshold frequency, the photon energy exactly equals the work function. hν₀ = φ₀ Step 2: Convert work function to joules. φ₀ = 2.14 × 1.6 × 10⁻¹⁹ = 3.424 × 10⁻¹⁹ J Step 3: Solve for threshold frequency. ν₀ = φ₀/h = 3.424 × 10⁻¹⁹ / 6.626 × 10⁻³⁴ Step 4: Calculate. ν₀ = 5.16 × 10¹⁴ Hz Why wrong options are wrong: - 3.24 × 10¹⁴ Hz: Results from forgetting to convert eV to joules. - 8.20 × 10¹⁴ Hz: Results from using incorrect conversion factor. - 2.58 × 10¹⁴ Hz: Results from dividing by 2h instead of h.

Electrons exhibit wave nature and undergo diffraction similar to X-rays.

Step 1: In the Davisson-Germer experiment, an electron beam was directed at a nickel crystal and the scattered intensity was measured at different angles. Step 2: A distinct peak (maximum) in intensity was observed at 50°, indicating constructive interference — a characteristic property of waves. Step 3: The experimentally calculated wavelength (1.65 Å) matched the de Broglie wavelength (1.67 Å) calculated for 54 V electrons. Step 4: Diffraction is a wave phenomenon. The fact that electrons showed diffraction conclusively proves the wave nature of matter, confirming de Broglie's hypothesis.

1.325 × 10⁻²⁷ kg m/s

Step 1: Use the photon momentum formula. p = h/λ Step 2: Convert wavelength to metres. λ = 500 nm = 500 × 10⁻⁹ m = 5 × 10⁻⁷ m Step 3: Calculate momentum. p = 6.626 × 10⁻³⁴ / 5 × 10⁻⁷ Step 4: Result. p = 1.325 × 10⁻²⁷ kg m/s Why wrong options are wrong: - 2.65 × 10⁻²⁷ kg m/s: Results from using λ = 250 nm (half the given value). - 6.626 × 10⁻³⁴ kg m/s: This is Planck's constant itself, not the momentum. - 3.97 × 10⁻¹⁹ kg m/s: This is the photon energy in joules, not momentum.

Wave theory predicts a large time lag before emission, but experiments show instantaneous emission (less than 10⁻⁹ s).

Step 1: According to wave theory, energy of light is spread uniformly over the entire wavefront. Step 2: Each electron absorbs energy continuously and slowly. Calculations show it would take approximately 79 days (for cesium) before an electron accumulates enough energy to overcome the work function. Step 3: But experiments show photoelectric emission is almost instantaneous — within 10⁻⁹ s of light being incident. Step 4: This huge contradiction (days vs nanoseconds) is one of the key failures of wave theory. Wave theory also fails on threshold frequency (it predicts emission at any freque