Radioactive decay
Section: Nuclear Physics | Syllabus: Cambridge AS Level Physics 9702
Radioactive Decay Nature of Radioactive Decay Radioactive decay is the spontaneous transformation of an unstable nucleus into a more stable one. Spontaneous: Not affected by external factors like temperature, pressure, or chemical bonding.
Random: It is impossible to predict when a specific nucleus will decay, but established probabilities apply to large numbers of nuclei. Evidence for Random Decay When measuring count rate from a radioactive source using a Geiger-Müller (GM) tube, the readings fluctuate even when conditions are constant.
These fluctuations in count rate provide experimental evidence that radioactive decay is random . Activity and Count Rate Corrected Count Rate Corrected count rate = recorded count rate − background count rate Background radiation arises from natural radioactive isotopes in the ground, building materials, cosmic rays, and radon gas in the atmosphere.
The corrected count rate is proportional to the activity of the sample. Activity and Decay Constant Activity (A) The rate of decay of a radioactive sample (number of decays per unit time). SI Unit: Becquerel (Bq), where 1 Bq = 1 decay per second.
Decay Constant (λ) The probability that an individual nucleus will decay per unit time. Unit: s^-1. A = λ N A = -dN/dt Half-Life Half-Life (t_1/2) The time taken for the number of undecayed nuclei (or activity) to reduce to half its initial value.
λ = 2/t_1/2 0.693/t_1/2 Exponential Decay Equations Since decay is proportional to the number of nuclei remaining, it follows an exponential law: x = x_0 e^-λ t where x can be Number of nuclei (N), Activity (A), or Count Rate (C).
Figure 23.12: Decay Curve Graph of Activity vs Time showing exponential decay. Mark points at t_1/2 (A_0/2) and 2t_1/2 (A_0/4). Worked Examples Worked Example: Calculating Activity Question: A sample contains 2.5 × 10^20 atoms of an isotope with a half-life of 12 hours.
Calculate the initial activity. Solution 1. Convert half-life to seconds: t_1/2 = 12 × 3600 = 43200 s 2. Calculate λ: λ = 0.693/43200 = 1.60 × 10^-5 s^-1 3. Calculate Activity: A = λ N = (1.60 × 10^-5)(2.5 × 10^20) = 4.0 × 10^15 Bq Worked Example: Exponential Decay Calculation Question: The activity of a source falls from 400 Bq to 100 Bq in 24 minutes.
Calculate the half-life. Solution 400 Bq 200 Bq 100 Bq This represents two half-lives. 2 t_1/2 = 24 min t_1/2 = 12 min
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