The number of nuclei, of a given radioactive nucleus, at times t=0 and t=T, are N0 and (N0/n) respectively. Obtain an expression for the half life (T1/2) of this nucleus in terms of n and T

To derive an expression for the half-life (T1/2) of a radioactive nucleus in terms of n and T, we start by understanding the concept of radioactive decay. The number of radioactive nuclei decreases over time according to a specific exponential decay law. Let's break this down step by step.

Understanding Radioactive Decay

Radioactive decay is a random process where unstable nuclei lose energy by emitting radiation. The rate of decay is characterized by the decay constant (λ), which is related to the half-life of the substance. The half-life is the time required for half of the radioactive nuclei in a sample to decay.

Initial and Final Nuclei

At time t=0, we have N0 nuclei. After a time period T, the number of remaining nuclei is given as N0/n. This indicates that the sample has undergone a certain fraction of decay. To express this mathematically, we can use the exponential decay formula:

• N(t) = N0 * e^(-λt)

In our case, at time T, we can substitute N(t) with N0/n:

• N0/n = N0 * e^(-λT)

Deriving the Decay Constant

We can simplify this equation by dividing both sides by N0:

• 1/n = e^(-λT)

Next, we take the natural logarithm of both sides to solve for λ:

• ln(1/n) = -λT
• λ = -ln(1/n) / T

Relating Decay Constant to Half-Life

The half-life (T1/2) is related to the decay constant (λ) by the following formula:

• T1/2 = ln(2) / λ

Now, substituting our expression for λ into this formula gives:

• T1/2 = ln(2) / (-ln(1/n) / T)

By simplifying this, we can express the half-life in terms of n and T:

• T1/2 = -T * ln(2) / ln(1/n)

Final Expression

Since ln(1/n) can be rewritten as -ln(n), we can further simplify the expression:

• T1/2 = T * ln(2) / ln(n)

This final expression shows how the half-life of the radioactive nucleus is related to the time period T and the factor n, which indicates how many times the original quantity has decreased after time T.

In summary, the half-life (T1/2) of the radioactive nucleus can be expressed as:

• T1/2 = T * ln(2) / ln(n)

This relationship is crucial in understanding the behavior of radioactive materials and can be applied in various fields, including nuclear physics, medicine, and environmental science.