Radioactive Decay and Half-Life: Quick Revision

A brief note on Radioactive Decay and Half-Life. Read it before a test. Try each question at the end before you open its answer.

The three decays

• Alpha: ${}^{A}_{Z}\text{X} \to {}^{A-4}_{Z-2}\text{Y} + {}^{4}_{2}\text{He}$. The alpha particle takes the larger share of the kinetic energy, $K_\alpha = Q\,\dfrac{A - 4}{A}$.
• Beta-minus: $n \to p + e^- + \bar{\nu}$. The mass number stays and $Z$ rises by $1$.
• Beta-plus: $p \to n + e^+ + \nu$. The mass number stays and $Z$ falls by $1$.
• Gamma: an excited nucleus sends out a photon. $A$ and $Z$ stay.
• Conserved: the total mass number, the total charge, the energy and the momentum.
• Energy: $Q = \Delta m \times 931.494\ \text{MeV/u}$ with atomic masses. For beta-plus, subtract $2m_e = 1.022$ MeV.

The decay law

• Law: $\dfrac{dN}{dt} = -\lambda N$, so $N = N_0e^{-\lambda t} = N_0\,2^{-t/T}$.
• Half-life: $T = \dfrac{\ln 2}{\lambda} = \dfrac{0.693}{\lambda}$.
• Mean life: $\tau = \dfrac{1}{\lambda} = 1.44\,T$. At $t = \tau$, $\dfrac{N}{N_0} = \dfrac{1}{e}$.
• Two ways to decay: $\lambda = \lambda_1 + \lambda_2$, and $\dfrac{1}{T} = \dfrac{1}{T_1} + \dfrac{1}{T_2}$.
• Activity: $R = \lambda N = R_0\,2^{-t/T}$. $1\ \text{Bq} = 1\ \text{s}^{-1}$ and $1\ \text{Ci} = 3.7 \times 10^{10}\ \text{Bq}$.

Series and dating

• Series: an alpha decay lowers $A$ by $4$, a beta decay does not change it. So $A$ changes by multiples of $4$ along a series.
• Radiocarbon age: $t = T\log_2\dfrac{R_0}{R}$, with $T = 5730$ y.
• Uranium-lead age: $t = \dfrac{T}{\ln 2}\ln\left(1 + \dfrac{N_{Pb}}{N_U}\right)$, with numbers of atoms.

A worked example

A source has an activity of $1200$ Bq and a half-life of $15$ min. Find its activity after $1.0$ h, and the time in which it falls to $75$ Bq.

1. After 1.0 h. The time is $60$ min, which is $4$ half-lives. $R = 1200\ \text{Bq} \times \dfrac{1}{16} = 75$ Bq.
2. Down to 75 Bq. The ratio is $\dfrac{1200}{75} = 16 = 2^4$. So $4$ half-lives are needed, which is $60$ min. The two parts agree.

Practise with new numbers

Check yourself

1. Complete: ${}^{238}_{92}\text{U} \to {}^{A}_{Z}\text{Th} + {}^{4}_{2}\text{He}$. What are $A$ and $Z$?

2. A nucleus has a decay constant of $0.0231\ \text{min}^{-1}$. What is its half-life?

3. The carbon-14 activity of a bone is a quarter of that of a living bone. How old is it?

Sources

• NCERT Physics Part II (Class 12, rationalised edition), Chapter 13 Nuclei.
• OpenStax University Physics Volume 3, Chapter 10 Nuclear Physics.
• OpenStax Chemistry 2e, Chapter 21 Nuclear Chemistry.