Understanding Radioactive Isotope Decay: Calculating Half-Life From Activity Changes

In the study of radioactive isotopes, understanding the decay process is crucial. One essential aspect is the calculation of the half-life, which indicates the time it takes for an isotope to decay to half its original quantity. This article provides a detailed step-by-step guide on how to calculate the half-life of a radioactive isotope when given its activity change over a specific period.

Given Data and Initial Assumptions

Consider an example where the activity of a radioactive isotope decreases from 80,000 to 10,000 over 60 days. Let's break down the steps to calculate the half-life in years.

Step 1: Determine the Decay Constant k

Using the relationship between the remaining quantity N and the initial quantity N0, we have:

[ N N0 e^{-kt} ]

Where:

N0 initial activity (80,000) N remaining activity (10,000) t time (60 days) k decay constant

Substituting the values, we get:

[ frac{N}{N_0} e^{-kt} ]

Rearranging the equation to solve for k:

[ lnleft(frac{N}{N_0}right) -kt ]

Substitute the values:

[ lnleft(frac{10,000}{80,000}right) -k cdot 60 ]

Simplify:

[ lnleft(frac{1}{8}right) -60k ]

Since:

[ lnleft(frac{1}{8}right) -3 ln(2) approx -3 times 0.693 -2.079 ]

Thus:

[ -2.079 -60k ]

Therefore:

[ k frac{2.079}{60} approx 0.03465 text{ days}^{-1} ]

Step 2: Calculate the Half-Life t1/2

The half-life of the isotope is given by:

[ t_{1/2} frac{ln(2)}{k} ]

Substituting the value of k:

[ t_{1/2} approx frac{0.693}{0.03465} approx 20.0 text{ days} ]

Step 3: Convert Half-Life to Years

To convert days to years:

[ t_{1/2} approx frac{20.0 text{ days}}{365} approx 0.0548 text{ years} ]

Thus, the half-life of the isotope is approximately 0.055 years.

Conclusion

Understanding the decay process and the calculation of half-life is essential in the study of radioactive isotopes. By following the steps outlined in this article, researchers and students can accurately determine the half-life of a radioactive isotope, which is crucial for various applications in science and technology.