23.2 Radioactive decay
- Syllabus
- 9702–2028–2029
- Topic
- 23.2
- Level
- A2
Learning objectives
23.2.1Fluctuations in count rate provide evidence for the random nature of• understand that fluctuations in count rate provide evidence for the random nature of radioactive decay23.2.2Radioactive decay is both spontaneous and random• understand that radioactive decay is both spontaneous and random23.2.3Activity and decay constant, and recall and use A = λN• define activity and decay constant, and recall and use A = λN23.2.4Half-life• define half-life23.2.5Λ = 0.693 / t 1/2• use λ = 0.693 / t 1/223.2.6The exponential nature of radioactive decay, and sketch and use the• understand the exponential nature of radioactive decay, and sketch and use the relationship x = x 0e–λt, where x could represent activity, number of undecayed nuclei or received count rate
Count-rate fluctuations reveal the random timing of nuclear decays
Measure counts in many equal time intervals under unchanged conditions: the results fluctuate above and below a mean or smooth decay trend rather than repeating exactly.
These irregular fluctuations are evidence that individual decay events occur at unpredictable times. Only the expected behavior of a large population is stable.
Longer counting intervals usually collect more events, so the fractional fluctuation is smaller even though decay remains random.
Scatter is not automatically evidence that λ or the detector changes. Compare repeated equal intervals and allow for background count before interpreting the trend.
Radioactive decay is random and spontaneous—but the terms mean different things
Random: it is impossible to predict which particular unstable nucleus will decay next or exactly when that nucleus will decay.
Spontaneous: decay occurs without being triggered and its probability is unaffected by external or environmental factors such as temperature, pressure and chemical state.
Although one event is unpredictable, each nucleus of the same isotope has the same constant decay probability per unit time, so a large sample has a predictable statistical law.
Spontaneous does not mean immediate, and random does not mean there is no stable probability or no predictable average behavior.
Activity is the sample decay rate; decay constant is probability per unit time
Activity A is the number of nuclear disintegrations per unit time. Its unit is the becquerel: 1 Bq=1 s⁻¹.
Decay constant λ is the probability per unit time that one undecayed nucleus decays. Its unit is reciprocal time, such as s⁻¹ or min⁻¹.
A=λN=−dN/dt
If λ=2.0×10⁻⁴ s⁻¹ and N=3.0×10¹², then A=6.0×10⁸ Bq. As N falls, A falls in the same proportion.
Activity is the source disintegration rate; received count rate can be smaller because detector efficiency and geometry are not 100%. Match the time unit of λ to the required rate unit.
Half-life is the constant time for a decay quantity to halve
Half-life t½ is the time taken for the number of undecayed nuclei—or equivalently the activity—to decrease to half its current value.
afternhalf−lives:x=x0(1/2)n1→1/2→1/4→1/8→…
For a particular isotope, each halving interval has the same duration because the decay constant is constant; the absolute amount lost in each interval becomes smaller.
Half-life is not the time for every nucleus to decay and not the lifetime of one particular nucleus. For measured count rate, subtract constant background before halving.
Decay constant and half-life are reciprocal through ln 2
λ=ln2/t½=0.693/t½t½=0.693/λ
A larger λ means a greater decay probability per unit time and therefore a shorter half-life. The unit of λ is the reciprocal of the time unit used for t½.
For t½=110 min=(110)(60)=6600 s, λ=0.693/6600=1.05×10⁻⁴ s⁻¹.
If λ=0.048 min⁻¹, t½=0.693/0.048=14 min.
Do not mix minutes with an s⁻¹ answer. The numerator is dimensionless, so λ and t½ must have reciprocal units.
A constant decay probability produces exponential decay
dx/dt=−λx⇒x=x0e(−λt)xmaybeN,activityAorbackground−correctedreceivedcountrate.
Because A=λN, the rate of decrease is proportional to the amount still undecayed. As x becomes smaller, the magnitude of the negative gradient also becomes smaller.
Sketch: start at (0,x₀), decrease continuously with a steepest negative gradient initially, curve upward as the slope becomes less negative, and approach zero asymptotically without crossing it.
For x₀=180 Bq, λ=0.048 min⁻¹ and t=8.4 min: x=180e^(−0.048×8.4)=120 Bq.
lnx=lnx0−λtAgraphoflnxagainsttisastraightlineofgradient−λandinterceptlnx0.
If measured count C includes constant background B, apply the decay law to C−B, not C. Add B back only if the question asks for the measured count.