λ₁ = 350 nm, λ₂ = 450 nm; maximum velocities differ by factor 2 (v₁ = 2v₂). Find work function φ.

Einstein’s equations:

hc/λ₁ = φ + ½mᵉv₁² ...(1)

hc/λ₂ = φ + ½mᵉv₂² ...(2)

Substitute v₁ = 2v₂ ⇒ v₁² = 4v₂² and subtract (2) from (1):

hc(1/λ₁ − 1/λ₂) = (3/2)mᵉv₂²

Calculate left side:

1/350nm − 1/450nm = 100/(350×450) nm⁻¹ = 6.349×10⁵ m⁻¹

hc × 6.349×10⁵ = 1.262×10⁻¹⁹ J

Solve for v₂²:

v₂² = (2/3) × 1.262×10⁻¹⁹ / 9.1×10⁻³¹ = 9.245×10¹⁰ m²/s²

Substitute into equation (2):

hc/λ₂ = (6.626×10⁻³⁴×3×10⁸) / (450×10⁻⁹) = 4.417×10⁻¹⁹ J

½mᵉv₂² = 0.5 × 9.1×10⁻³¹ × 9.245×10¹⁰ = 4.206×10⁻²⁰ J

φ = 4.417×10⁻¹⁹ − 0.4206×10⁻¹⁹ ≈ 3.996×10⁻¹⁹ J ≈ 2.5 eV

Answer: Work function φ ≈ 2.5 eV