23.3 Radioactive Decay Law, Activity, and Half-Life

Apply exponential decay equations to connect nucleus count, activity, and half-life over time.

Estimated time: 44 minutes

From Random Single Events to Exponential Population Trends

Each nucleus has a constant probability per unit time to decay. That single assumption implies exponential behavior for large populations. The more nuclei present, the more decays per second you observe. As nuclei disappear, count rate falls. This is why activity curves slope downward even when microscopic decay chance for each surviving nucleus is unchanged.

$N(t)=N_0 e^{-\lambda t},\qquad A(t)=\lambda N(t)$

lambda is decay constant; activity A is the decay rate (decays per second, Bq).

Half-Life Connection and Fast Calculation Paths

$T_{1/2}=\frac{\ln 2}{\lambda},\qquad N=N_0\left(\frac12\right)^{t/T_{1/2}}$

For integral or near-integral half-life multiples, the power-of-one-half form is fastest.

Use the half-life power form when time is quoted as a clear multiple of (T_{1/2}). Use the exponential form when non-integer ratios or decay constants are given directly. Both are equivalent; choosing the cleaner form reduces arithmetic mistakes and keeps trend interpretation transparent.

Activity follows the same fractional decrease as nucleus count for one isotope because (A=lambda N) and (lambda) is constant. That means if nuclei drop to one quarter, activity also drops to one quarter. The common confusion is to treat half-life as a fixed number of nuclei decaying each interval; it is always a fixed fraction of what remains.

Background Counts and Measurement Interpretation

In practical counting, measured rate includes source plus background. To model source decay correctly, subtract background before fitting or estimating half-life from data. If you ignore this offset, late-time activity appears to flatten too early and inferred half-life becomes inaccurate.

Simulation: Decay Curves, Half-Life, and Activity

Vary initial nuclei, half-life, and elapsed time to observe exponential decay, remaining nuclei, and changing activity.

Link nucleus composition, binding-energy trends, decay statistics, and strong-force evidence in one chapter workspace.

Initial nuclei1.600 x10^12Half-life6.000 hElapsed time18.000 h

Remaining nuclei

2.00e+11

Decayed nuclei

1.40e+12

Activity A

2.31e+10 h^-1 nuclei

P(decay in next hour)

0.1091

Test Yourself

A sample starts with 1.6 x 10^12 nuclei and has half-life 6.0 h. Enter the number of nuclei remaining after 18 h.

Hint: 18 h is three half-lives, so multiply by (1/2)^3.

Your numerical answer (nuclei)

23.2 Radioactive Decay Modes and Radiation Properties

23.4 Nuclear Radii, Strong-Force Evidence, and Closest Approach