Half-life

Also called Half life

Half-life is the time it takes for half the radioactive nuclei in a sample to decay. It is fixed for a given isotope and does not depend on how much you start with, so a quarter of the original nuclei survive two half-lives whatever the sample size.

Written t1/2t_{1/2}, the half-life is the time for half of the initial number of radioactive nuclei to have spontaneously decayed (15.7.B.1.ii). It belongs to the isotope, not to the sample, so a gram and a tonne of the same material halve on the same schedule: half left after one half-life, a quarter after two, N0/2nN_0 / 2^n after nn.

Two sheet equations tie it to the decay constant λ\lambda.

λ=ln⁡2t1/2N=N0e−λt\lambda = \frac{\ln 2}{t_{1/2}} \qquad N = N_0 e^{-\lambda t}

The logarithmic form, ln⁡(N/N0)=−λt\ln(N/N_0) = -\lambda t, is labeled a derived equation in 15.7.B.2 and is not printed on the sheet, though it is the form you plot to get a straight line of gradient −λ-\lambda.

Watch the symbol. Everywhere else in this unit λ\lambda is a wavelength in meters; here it is a decay constant in inverse seconds. They share a letter and nothing else, so check which one a question means before substituting.

Half-lives run from fractions of a second to billions of years (15.7.B.3), and the Topic 15.8 boundary statement says you are not expected to know any specific isotope's value. A question that needs one gives it to you.

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