Q.Define the term 'Half-life' of a radioactive substance. Two different radioactive substances have half-lives $T_1$ and $T_2$ and number of undecayed atoms at an instant, $N_1$ and $N_2$, respectively. Find the ratio of their activities at that instant.

Concept: Radioactive decay and activity

Half-life is the time required for half the nuclei in a radioactive sample to decay. It characterizes the decay rate of each substance.

The activity $A$ (disintegrations per unit time) of a radioactive sample is given by

where $\lambda$ is the decay constant and $N$ is the number of undecayed atoms at that instant.

The decay constant relates to half-life through $\lambda = \frac{\ln 2}{T}$, so for the two substances:

$\lambda_1 = \frac{\ln 2}{T_1}, \quad \lambda_2 = \frac{\ln 2}{T_2}$ …

What is Half-life?

The half-life of a radioactive substance is the time interval during which half of the initially present radioactive nuclei decay. It is a characteristic constant for each radioactive isotope, independent of the initial quantity or external conditions.

If you start with $N_0$ undecayed atoms, after one half-life $T_{1/2}$ you will have $\frac{N_0}{2}$ atoms remaining; after two half-lives, $\frac{N_0}{4}$, and so on.

Finding the Ratio of Activities

Activity measures how many disintegrations occur per unit time — it tells us the rate at which a sample is decaying right now. The key insight is that activity depends on both how many atoms you have and how quickly each type decays.

The activity $A$ of a radioactive sample is given by:

where $\lambda$ is the decay constant (probability of decay per unit time per nucleus) and $N$ is the number of undecayed atoms at that instant.

The decay constant and half-life are related through:

This comes from the exponential decay law: when $t = T_{1/2}$, exactly half the nuclei remain, which forces this relationship.

Now let's find the ratio of activities for our two substances.

1. Write the activity of substance 1: $A_1 = \lambda_1 N_1 = \frac{\ln 2}{T_1} \cdot N_1$ …