How Does the Decay Constant (λ) Relate to Half-Life (T½ = ln2/λ)? (HL)

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Therefore, a larger decay constant means a shorter half-life and faster radioactive decay.

Why This Relationship Works

The HL radioactive decay law for the number of undecayed nuclei is

where $N_0$ is the initial number of undecayed nuclei and $N$ is the number remaining after time $t$. The decay constant measures the probability of decay per unit time, provided the time interval considered is sufficiently small. Its unit is inverse time, such as $\text{s}^{-1}$ or $\text{day}^{-1}$.

Physically, $\lambda$ is not the fraction of nuclei that must decay in every one-unit interval. Radioactive decay is random and spontaneous. For a sufficiently small interval $\Delta t$, the probability that one nucleus decays is approximately $\lambda\Delta t$. Across a large sample, however, the average behaviour follows the exponential law.

After one half-life, half of the original nuclei remain. Therefore, set $t=T_{1/2}$ and $N=N_0/2$:

Cancel $N_0$ and take natural logarithms:

Since $\ln(1/2)=-\ln 2$,

The inverse relationship is dimensionally consistent: if $\lambda$ is measured in $\text{s}^{-1}$, $T_{1/2}$ is in seconds. Convert units before substituting.

For example, if $\lambda=0.035\ \text{day}^{-1}$, then

After 19.8 days, the expected number of undecayed nuclei and the activity have each fallen to half their initial values. This follows because $A=\lambda N$. After two half-lives, each is one quarter; after three, one eighth.

If half-life is given instead, rearrange to $\lambda=\ln 2/T_{1/2}$. For $T_{1/2}=8.0\ \text{h}$, $\lambda=0.693/8.0=0.0866\ \text{h}^{-1}$. This confirms that the decay constant carries reciprocal-time units. Using seconds for half-life would instead produce $\lambda$ in $\text{s}^{-1}$; either choice is valid if used consistently throughout the calculation.

A common misconception is that a larger decay constant produces a longer half-life. The relationship is inverse: doubling $\lambda$ halves $T_{1/2}$.

Exam Technique

This is Additional Higher Level (AHL) content in E.3 Radioactive decay. For a Show question, include every algebraic step from $N=N_0e^{-\lambda t}$. For calculations, keep the units reciprocal and consistent.