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

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

Question	Solution
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

Quantity	Symbol	SI Unit	Typical 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)	–

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.