Govt. Exams
If 93.75% decays, 6.25% remains. 6.25% = 6.25/100 = 1/16 = (1/2)⁴. So 4 half-lives have passed. Time = 4 × 10 = 40 days. Correction: 6.25% = 1/16, which requires 4 half-lives = 40 days.
Compton scattering formula: Δλ = λ - λ₀ = (h/mₑc)(1 - cos θ). At θ = 90°, Δλ = (h/mₑc)(1 - 0) = h/(mₑc).
Using 1/λ = R(1/n₁² - 1/n₂²). For 3→1: 1/λ₃₋₁ = R(1 - 1/9) = 8R/9. For Lyman alpha 2→1: 1/λ₂₋₁ = R(1 - 1/4) = 3R/4. Ratio λ₃₋₁/λ₂₋₁ = (3/4)/(8/9) = 27/32. So λ₂₋₁/λ₃₋₁ = 32/27.
Half-life t₁/₂ = ln(2)/λ. Since λₐ = 2λᵦ, we have t₁/₂ₐ/t₁/₂ᵦ = λᵦ/λₐ = 1/2. Therefore, t₁/₂ₐ : t₁/₂ᵦ = 1:2.
E = hc/λ = 3.1 eV (for 400 nm). Stopping potential energy = eV₀ = 2 eV. Work function φ = E - eV₀ = 3.1 - 2 = 1.1 eV. Wait, recalculating: φ = 3.1 - 2 = 1.1 eV. Answer should be A.
Using λ = h/√(2mKE), where m = 1.67×10⁻²⁷ kg, KE = 1.6×10⁻¹⁹ J, h = 6.63×10⁻³⁴ J·s. Calculation yields λ ≈ 0.286 nm.
Pair production creates an electron-positron pair. Minimum energy = 2mₑc² = 2 × 0.511 MeV = 1.022 MeV
ΔxΔp ≥ h/4π. If Δx = 0, then Δp must be ≥ ∞ to satisfy the inequality, making momentum completely uncertain.
Characteristic X-rays arise from inner-shell electron transitions (e.g., L→K shell). Each element has specific wavelengths, hence 'characteristic'.
For hydrogen-like atoms: E = -13.6Z²/n² eV. For He⁺ (Z=2): E = -13.6×4 = -54.4 eV. For H (Z=1): E = -13.6 eV. Ratio is 4:1
