A radioactive decay series is the sequence of radioactive transformations that an unstable parent nucleus undergoes until a stable (non-radioactive) isotope is formed. Each step in the series involves the emission of alpha (α), beta (β), or gamma (γ) radiation, producing a new element known as a daughter nucleus.

For example, uranium-238 (U-238) decays through a series of alpha and beta emissions until it forms lead-206 (Pb-206), which is stable. This process is known as the uranium series.

Other decay series include:

• Thorium series – starting with thorium-232 (Th-232) and ending with lead-208 (Pb-208).
• Actinium series – starting with uranium-235 (U-235) and ending with lead-207 (Pb-207).
• Neptunium series – starting with neptunium-237 (Np-237) and ending with bismuth-209 (Bi-209).

The rate of radioactive decay is the number of nuclei that disintegrate per unit time. It is a random process, but for a large number of atoms, it follows a definite exponential law.

The rate of decay at any time is proportional to the number of undecayed nuclei present:

$$ \frac{dN}{dt} = -\lambda N $$

Where:

• \( N \) = number of undecayed atoms at time \( t \)
• \( \lambda \) = decay constant (probability of decay per unit time)
• \( \frac{dN}{dt} \) = rate of decay

Integrating gives the radioactive decay law:

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

Where \( N_0 \) is the initial number of radioactive atoms.

The half-life of a radioactive element is the time required for half of the original number of radioactive atoms to decay into a different element or isotope. It is a constant property of each radioactive isotope and is independent of temperature, pressure, and chemical state.

Mathematically, the relationship between half-life and decay constant is:

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

After \( n \) half-lives, the fraction of radioactive atoms remaining is given by:

$$ N = N_0 \left(\frac{1}{2}\right)^n $$

Example 1: The half-life of a certain radioactive isotope is 10 hours. If the initial activity is 800 counts per minute, what will the activity be after 30 hours?

$$ n = \frac{30}{10} = 3 $$ $$ A = A_0 \left(\frac{1}{2}\right)^3 $$ $$= 800 \times \frac{1}{8}$$ $$= 100 \text{ counts per minute} $$

Example 2: A sample contains \( 1.6 \times 10^{20} \) radioactive atoms. After 3 half-lives, how many remain undecayed?

$$ N = N_0 \left(\frac{1}{2}\right)^3 $$ $$= 1.6 \times 10^{20} \times \frac{1}{8} $$ $$= 2.0 \times 10^{19} \text{ atoms} $$

Example 3: The decay constant of a radioactive isotope is \( 2.31 \times 10^{-5} \, s^{-1} \). Calculate its half-life.

$$ T_{1/2} = \frac{0.693}{\lambda} $$ $$ = \frac{0.693}{2.31 \times 10^{-5}} $$ $$= 3.0 \times 10^4 \, \text{seconds} $$

$$ T_{1/2} = 30,000 \, \text{seconds} = 8.33 \, \text{hours} $$

Example 4: A radioactive substance decays from 400 g to 50 g in 24 hours. Find its half-life.

$$ \frac{N}{N_0} = \frac{50}{400} $$ $$ = \frac{1}{8} = \left(\frac{1}{2}\right)^3 $$ Hence, the number of half-lives \( n = 3 \).

$$ T_{1/2} = \frac{24}{3} = 8 \, \text{hours} $$

Example 5: A radioactive isotope has a half-life of 5 days. What fraction of it will remain after 20 days?

$$ n = \frac{20}{5} = 4 $$ $$ \text{Fraction remaining} = \left(\frac{1}{2}\right)^4 $$ $$ = \frac{1}{16} = 0.0625 $$

Example 6: Given that the half-life of a radioisotope is 2500 seconds, determine the decay constant of the radioisotope.

From the relation: $$ T_{1/2} = \frac{0.693}{\lambda} $$ Rearranging gives: $$ \lambda = \frac{0.693}{T_{1/2}} $$

Substituting: $$ \lambda = \frac{0.693}{2500} $$ $$ = 2.77 \times 10^{-4} \, s^{-1} $$