Revise Radioactive decay for Cambridge 9702 (23.2)

Radioactive decay isn't something we can control or predict for a single nucleus. It possesses two crucial characteristics: it's spontaneous and random. This means it occurs without any external influence, such as changes in temperature or pressure, and we can't tell when a specific nucleus will decay. Evidence for this randomness comes from observing the count rate of a radioactive source with a Geiger-Muller tube; the readings fluctuate unpredictably around an average value, confirming that the decays are individual, random events.

While individual decays are random, for a large sample, the overall rate of decay can be statistically predicted. The decay constant (symbol \lambda, pronounced 'lambda') helps us quantify this. It represents the probability that any single nucleus will decay per unit of time. A larger \lambda means a higher probability of decay and thus a less stable nucleus, leading to a shorter half-life. Conversely, a very small decay constant indicates a very stable isotope that decays slowly over a long period.

The activity of a radioactive sample measures how many nuclei are decaying per second. It's essentially the 'strength' of the radiation from a source. Since it's directly related to the decay constant and the number of undecayed nuclei, activity also decreases as the sample decays. While the SI unit is the Becquerel (Bq), an older, non-SI unit, the Curie (Ci), is sometimes encountered, where 1 Ci = 3.7 x 10¹⁰ Bq.

Activity, $A = \lambda N$ Where: $A$ = activity (Bq) $\lambda$ = decay constant (s^{-1}) $N$ = number of undecayed nuclei

The number of undecayed nuclei, and consequently the activity, in a radioactive sample decreases exponentially over time. This means that in equal time intervals, the same fraction of remaining nuclei will decay, not the same absolute number. When plotted on a graph of N or A against time, this relationship produces a characteristic exponential decay curve that starts at N₀ (or A₀) and asymptotically approaches zero but never quite reaches it. These exponential relationships are critical for calculations.

Number of undecayed nuclei: $N = N_0 e^{-\lambda t}$ Activity: $A = A_0 e^{-\lambda t}$ Where: $N_0$ = initial number of nuclei $A_0$ = initial activity $N$ = number of nuclei at time $t$ $A$ = activity at time $t$ $e$ = Euler's number (approx. 2.718)

The half-life is a unique characteristic for each radioactive isotope. It's the time it takes for exactly half of the radioactive nuclei in a sample to decay. After one half-life, 50% of the original sample remains. After two half-lives, 25% (half of 50%) remains, and so on. Importantly, the half-life remains constant, regardless of the initial amount of the substance or its physical conditions.

Half-life and decay constant: $t_{1/2} = \frac{\ln 2}{\lambda}$ Where: $t_{1/2}$ = half-life (s, min, hr, etc.) $\lambda$ = decay constant (s^{-1}, min^{-1}, hr^{-1}, etc.) $\ln 2 \approx 0.693$

The relationship between half-life and the decay constant can be derived directly from the exponential decay equation. By definition, at the moment the time elapsed equals one half-life ($t = t_{1/2}$), the number of remaining nuclei ($N$) will be exactly half of the initial number ($N = N_0 / 2$). By substituting these conditions into the decay equation, we can isolate and solve for $t_{1/2}$.

Start with the decay equation: $N = N_0 e^{-\lambda t}$ Substitute $N = N_0/2$ and $t = t_{1/2}$: $\frac{N_0}{2} = N_0 e^{-\lambda t_{1/2}}$ Divide by $N_0$: $\frac{1}{2} = e^{-\lambda t_{1/2}}$ Take the natural logarithm (ln) of both sides: $\ln(\frac{1}{2}) = \ln(e^{-\lambda t_{1/2}})$ Using logarithm rules, $\ln(1/x) = -\ln(x)$ and $\ln(e^y) = y$: $-\ln(2) = -\lambda t_{1/2}$ Finally, rearrange for $t_{1/2}$: $t_{1/2} = \frac{\ln 2}{\lambda}$