Understanding Half-Life and Activity in Radioactive Decay

Radioactive decay is a fundamental process in nuclear physics where unstable atomic nuclei lose energy by emitting radiation. The number of decays per second is known as activity, and it plays a crucial role in determining the characteristics of a radioactive substance. In this article, we will explore the relationship between activity, number of atoms, and half-life. We will also delve into practical applications, such as how to achieve a specific activity rate, and discuss the implications of different half-lives.

Activity and Its Relation to Nuclei

Activity, denoted by A, is defined as the product of the number of atoms N and the decay constant. The decay constant is often expressed in terms of the half-life or the natural logarithm of 2. These relationships are given by the following equations:

A N(frac{ln2}{half-life})

A Nf

By rearranging these equations, we can solve for the half-life, which is a key parameter in understanding the decay process:

Half-life (N(frac{ln2}{A}))

Decaying Thousand Times Per Second

When a radioactive substance decays a thousand times per second, this activity is significant in various applications, from nuclear physics to medical imaging. To find the required half-life for such an activity, we use the formula derived above:

Half-life (N(frac{ln2}{1000}))

For example, if we have a mole of the sample:

Half-life (6.023 times 10^{23})(frac{0.000693147}{1000}) Half-life 4.17 times 10^{20} seconds

Conversely, with just a thousand atoms, the half-life would be much shorter:

Half-life (1000)(frac{0.000693147}{1000}) Half-life 0.693147 seconds

Implications of Different Half-Lives

Any half-life can be used, but the quantity of material needed to achieve a specific activity rate varies. Longer half-lives mean fewer nuclei are involved in the decay per second, while shorter half-lives require a higher concentration of nuclei. For instance, tritium has a half-life of about 12 years. This means that a sample would have a fairly constant number of decays per second for the first year or so, after which the decays would decrease.

For practical applications, such as measuring 1000 decays per second, we can calculate the number of atoms needed:

Proportionality factor: (1000 / (6.023 times 10^{23} times 1.77 times 10^{-9})) (approx 500 times 10^9 text{ atoms})

Practical Example: Tritium Decays

Let’s consider the decay of tritium, a radioactive isotope with a half-life of 12 years. To measure about one thousand decays per second, we can use the following calculations:

(text{Probability of decay per second} frac{ln2}{text{half-life}} frac{0.693147}{388 times 10^6}) ( 1.77 times 10^{-9})

Thus, a sample of (500 times 10^9) atoms (or (500 times 10^9 / 6.023 times 10^{23} text{ moles})) would produce the desired activity rate.

Conclusion

The relationship between activity, number of atoms, and half-life is fundamental to understanding radioactive decay. Whether you need a long or short half-life, the amount of material required to achieve a specific activity rate is directly proportional to the decay constant and inversely proportional to the half-life. By grasping these concepts, you can better predict and control radioactive decay processes in diverse applications.