Radioactive Series Decay Example

Overview of Radioactive Series Decay

In a radioactive decay series, a parent nucleus P decays into a daughter nucleus D, which may also decay into a granddaughter nucleus G, and so on, until a stable isotope is reached. The decay of each nucleus in the chain follows its own exponential decay process.

The decay of the parent and daughter nuclei can be described by mathematical formulas involving exponential decay laws.

Mathematical Formulation

The rate of decay of the parent nucleus P is given by:

$\binom{d N_P (t)}{d t} = - λ_P N_P (t)$

The decay of the daughter nucleus D is governed by a similar equation:

$A_D (t) = \frac{λ_P A_P (0)}{λ_P - λ_D} e -^{λ_D t} - e -^{λ_P t}$

Example Calculation: Parent and Daughter Activities

Consider a radioactive substance with the following data:

Decay constant of parent nucleus λ_P = 0.0001 s^-1
Initial number of parent nuclei N_P(0) = 1 × 10⁶
Decay constant of daughter nucleus λ_D = 0.0002 s^-1

We can calculate the activity of the parent and daughter nuclei at different times. First, we'll calculate the initial activity of the parent.

Step 1: Calculate Initial Activity of Parent

The initial activity of the parent nucleus is given by:

$A_P (0) = λ_P N_P (0)$

Substituting the values:

$A_P (0) = 0.0001 (1 × 10$ ⁶

Therefore, the initial activity of the parent nucleus is:

A_P(0) = 100 disintegrations per second (Bq)

Step 2: Calculate Activity of Daughter Nucleus at Time t

The activity of the daughter nucleus at time t is given by the equation:

$A_D (t) = \frac{λ_P A_P (0)}{λ_P - λ_D} e -^{λ_D t} - e -^{λ_P t}$

For t = 1000 seconds, we substitute the known values into the equation to find the activity of the daughter nucleus.

Important Notes

- The activity of both parent and daughter nuclei changes over time and follows an exponential decay law.

- The half-life and mean lifetime of the substances can be derived from their decay constants.