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An ideal flow model in which all fluid elements have the same transit time. For first-order radioactive decay the outlet-to-inlet radioactive activity concentration ratio is $e^{-\lambda \tau }$.

Radiation detection makes interactions of ionizing radiation with matter into observable signals. An ionization chamber collects charge created in a gas; a scintillation detector converts deposited energy into light; a semiconductor detector measures charge carriers created in a solid. The instrument detects interactions, so its response depends on radiation type, energy and geometry.

For an ideal counting model, expected detected events satisfy $N=\epsilon Rt$, where $R$ is the relevant incident event rate, $t$ is observation time and $\epsilon$ is detection efficiency. Background adds counts, while dead time can lose them. Estimating a source's radioactive activity therefore requires calibration and a model of emissions and transport, rather than equating the displayed count rate with decay rate.

With $n$ parent atomic nuclei present, the conditional expected number of transformations in a short interval is approximately $\lambda n\Delta t$. The corresponding radioactive activity is $A=\lambda n$. Averaging over the finite-population radioactive decay model gives $\overline{A}(t)=\lambda \overline{N}(t)$. Its unit is the becquerel.^[ref]

For a finite counting interval, depletion matters. Suppose each transformation is recorded independently with detection efficiency $\epsilon$, with at most one count per transformation and no background. The expected value of the count between $t$ and $t+\Delta t$ isOnly when $\lambda \Delta t\ll 1$ does this become approximately $\epsilon \overline{A}(t)\Delta t$. A count rate therefore needs a detector model before it can be interpreted as the source's radioactive activity.

radioactive activity per volume, in $\mathrm{Bq}{\mathrm{m}}^{-3}$. This differs from mass-specific radioactive activity, usually expressed per kilogram.

Radioactive decay is the spontaneous transformation of an unstable atomic nucleus or nuclear state. Its timing can be unpredictable for an individual atomic nucleus while a large population has a reproducible average behavior. A constant nuclear decay constant $\lambda >0$ gives the familiar exponential decay law. The half-life describes how quickly the parent population falls, while radioactive activity describes its rate of transformations.^[ref]

This connection between randomness and a regular population law makes radioactive decay useful for measuring elapsed time and for understanding sources of radiogenic heating. It also distinguishes two questions in an experiment: what the source emits, and what an instrument records. Becquerel's uranium-radiation experiment used a photographic plate; the Curie ionization-current measurement measured an electrical consequence of radiation. Their readouts reveal different aspects of the same investigation.

The radioactive decay constant $\lambda$ is the probability per unit time that one surviving atomic nucleus decays in the simple constant-rate model. It gives $d\overline{N}/dt=-\lambda \overline{N}$ and hence $\overline{N}(t)=N_0{e}^{-\lambda t}$ for the mean surviving population.

The radioactive activity is $\lambda \overline{N}$, the mean lifetime is $1/\lambda$, and the half-life is $\ln 2/\lambda$. These are different quantities: a large sample can have high radioactive activity even when each atomic nucleus has a small decay rate. Counting successive time intervals tests the exponential model, but background, detection efficiency and daughter radioactive activity affect the observed counts. A constant decay constant describes stochastic individual events, not a scheduled time at which each atomic nucleus must decay.

Temporary retention of radioactive gas so that nuclear radioactive decay lowers the radioactive activity released. A guaranteed minimum residence time, an average residence time and a tank filling time are different design quantities.

Radiocarbon dating uses the loss of carbon-14 relative to stable carbon after carbon exchange stops. It can measure either an isotope ratio by accelerator mass spectrometry or radioactive activity per fixed carbon content. Interpreting an age requires radiocarbon calibration and control of contamination and reservoir differences.