Animated Solution for Physics - Atoms and Nuclei: After 280 days, the activity of a radioactive sample is 6000 dps. The activity reduces to 3000 dps after another 140 days. The initial activity of the sample in dps is

Radioactivity is a fascinating phenomenon where unstable atomic nuclei spontaneously disintegrate, releasing energy and particles. One of the most elegant ways to understand this process is through the concept of half-life ($T_{1/2}$), which is the time required for exactly half of the radioactive nuclei in a sample to decay.

Let's embark on a journey to solve a classic problem that tests our intuition about half-lives and exponential decay.

Imagine a timeline. We are given a snapshot of a radioactive sample at a specific moment. The problem states that after $280$ days, the activity of the sample is $6000\text{ dps}$ (disintegrations per second).

Then, we are given another crucial piece of information: after another $140$ days, the activity reduces to $3000\text{ dps}$.

Finding the Half-Life

Look closely at the numbers. The activity drops from $6000\text{ dps}$ to $3000\text{ dps}$. It has exactly halved!

By definition, the time it takes for the activity to reduce to half its value is the half-life of the substance. Since this halving occurred over a span of $140$ days, we can immediately deduce:

Our goal is to find the initial activity, $A_0$, which is the activity at $t=0$.

We know the activity at $t=280$ days is $6000\text{ dps}$. The time elapsed from the beginning ($t=0$) to this point is exactly $280$ days.

How many half-lives fit into this $280$-day window? We can find this by dividing the total time elapsed by the half-life:

So, exactly two half-lives have passed since the initial moment.

The fundamental law of radioactive decay tells us that after $n$ half-lives, the remaining activity $A$ is related to the initial activity $A_0$ by the equation:

Substituting the values we have discovered:

To isolate $A_0$, we simply multiply both sides by $4$:

And there we have it! The initial activity of the sample was $24000\text{ dps}$. This elegant problem demonstrates how powerful the concept of half-life can be when dealing with exponential decay processes.