use λ = 0.693 / t

Radioactive Decay

Learning Objective

By the end of this lesson you should be able to calculate the decay constant (λ) of a radionuclide using the relationship

\$\lambda = \frac{0.693}{t_{1/2}}\$

where \$t_{1/2}\$ is the half‑life of the nuclide.

1. Introduction to Radioactivity

Radioactivity is the spontaneous transformation of an unstable nucleus into a more stable configuration. The process is random for individual nuclei but follows a predictable statistical law for a large collection.

2. The Decay Law

The number of undecayed nuclei, \$N\$, at any time \$t\$ is given by the exponential decay law:

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

• \$N_0\$ – initial number of nuclei at \$t=0\$
• \$\lambda\$ – decay constant (s\(^{-1}\))
• \$t\$ – elapsed time (s)

3. Half‑Life and the Decay Constant

The half‑life, \$t{1/2}\$, is the time required for half of the original nuclei to decay. Setting \$N(t{1/2}) = \tfrac{1}{2}N_0\$ in the decay law gives:

\$\frac{1}{2}N0 = N0 e^{-\lambda t{1/2}} \;\;\Rightarrow\;\; e^{-\lambda t{1/2}} = \frac{1}{2}\$

Taking natural logarithms leads to the useful relationship:

\$\lambda = \frac{\ln 2}{t{1/2}} = \frac{0.693}{t{1/2}}\$

4. Types of Radioactive Decay

5. Practical Calculations

1. Determine the half‑life \$t_{1/2}\$ of the radionuclide (from tables or experiment).
2. Calculate the decay constant using \$\lambda = 0.693/t_{1/2}\$.
3. Use \$N(t)=N_0 e^{-\lambda t}\$ to find the remaining nuclei after a given time.
4. Alternatively, calculate the activity \$A = \lambda N\$ (in becquerels, Bq).

6. Example Problem

Problem: A sample contains \$2.0 \times 10^{20}\$ atoms of \$^{60}\$Co, whose half‑life is \$5.27\$ years. Find the activity after \$10\$ years.

Solution:

• Convert half‑life to seconds: \$t_{1/2}=5.27\;\text{yr}\times 3.156\times10^{7}\;\text{s yr}^{-1}=1.66\times10^{8}\;\text{s}\$.
• Decay constant: \$\displaystyle \lambda = \frac{0.693}{1.66\times10^{8}\;\text{s}} = 4.18\times10^{-9}\;\text{s}^{-1}\$.
• Number of atoms after \$10\$ yr (\$t = 3.156\times10^{8}\;\text{s}\$):

  \$N = N_0 e^{-\lambda t}=2.0\times10^{20} e^{-4.18\times10^{-9}\times3.156\times10^{8}} \approx 1.0\times10^{20}.\$

• Activity: \$A = \lambda N = 4.18\times10^{-9}\times1.0\times10^{20}=4.2\times10^{11}\;\text{Bq}\$.

7. Summary

• The decay constant \$\lambda\$ quantifies the probability per unit time that a nucleus will decay.
• It is directly related to the half‑life by \$\lambda = 0.693/t_{1/2}\$.
• Radioactive decay follows an exponential law, allowing prediction of remaining nuclei and activity.
• Understanding decay modes helps interpret the changes in atomic number and mass number.