define activity and decay constant, and recall and use A = λN

Published by Patrick Mutisya · 14 days ago

Cambridge A-Level Physics 9702 – Radioactive Decay

Radioactive Decay

Key Definitions

When a nucleus undergoes radioactive decay it transforms into a different nucleus, emitting particles or radiation. Two fundamental quantities used to describe a radioactive sample are the activity and the decay constant.

  • Activity (A): the number of decays occurring per unit time. It is measured in becquerels (Bq), where 1 Bq = 1 decay s⁻¹.

  • Decay constant (λ): the probability per unit time that a given nucleus will decay. It has units of s⁻¹.

Mathematical Relationship

The activity of a sample containing N radioactive nuclei is given by

\$A = \lambda N\$

where:

  • \$A\$ is the activity (Bq),

  • \$\lambda\$ is the decay constant (s⁻¹),

  • \$N\$ is the number of undecayed nuclei.

Derivation from Exponential Decay

The number of nuclei remaining after a time \$t\$ follows the exponential law

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

Differentiating with respect to time gives the rate of change of \$N\$:

\$\frac{dN}{dt} = -\lambda N(t)\$

The negative sign indicates a decrease in \$N\$. The magnitude of this rate is the activity:

\$A(t) = -\frac{dN}{dt} = \lambda N(t)\$

Half‑Life and Decay Constant

The half‑life \$t_{1/2}\$ is the time required for half of the original nuclei to decay. It is related to the decay constant by

\$t_{1/2} = \frac{\ln 2}{\lambda}\$

Thus, knowing either \$t_{1/2}\$ or \$\lambda\$ allows you to calculate the other.

Sample Calculations

  1. Given a sample with \$N = 2.0 \times 10^{20}\$ nuclei and \$\lambda = 5.0 \times 10^{-4}\,\text{s}^{-1}\$, find its activity.

  2. Given an activity of \$3.0 \times 10^{6}\,\text{Bq}\$ and a decay constant of \$2.0 \times 10^{-3}\,\text{s}^{-1}\$, determine the number of undecayed nuclei.

Answers

QuestionSolution
1

\$A = \lambda N = (5.0 \times 10^{-4}\,\text{s}^{-1})(2.0 \times 10^{20}) = 1.0 \times 10^{17}\,\text{Bq}\$

2

\$N = \frac{A}{\lambda} = \frac{3.0 \times 10^{6}\,\text{Bq}}{2.0 \times 10^{-3}\,\text{s}^{-1}} = 1.5 \times 10^{9}\,\text{nuclei}\$

Units Summary

QuantitySymbolSI UnitTypical Symbol
Activity\$A\$becquerel (Bq)\$\text{s}^{-1}\$
Decay constant\$\lambda\$second⁻¹ (s⁻¹)\$\text{s}^{-1}\$
Number of nuclei\$N\$dimensionless (count)–
Half‑life\$t_{1/2}\$second (s)–

Suggested diagram: Decay chain showing a parent nucleus, emitted radiation, and the resulting daughter nucleus, with arrows indicating the direction of decay and a label for the decay constant λ.

Key Points to Remember

  • The activity is directly proportional to both the decay constant and the number of undecayed nuclei.

  • A larger decay constant means a faster‑decaying (more radioactive) sample.

  • Half‑life and decay constant are inversely related through \$\ln 2\$.

  • When solving problems, always keep track of units; convert to seconds if necessary.