Animated Solution for Physics - Atoms and Nuclei: At a given instant there are 25% undecayed radioactive nuclei in a sample. After 10 s the number of undecayed nuclei reduces to 12.5%. Calculate (a) mean life of the nuclei, (b) the time in which the number of undecayed nuclei will further reduce to 6.25% of the reduced number.

Radioactive decay is one of the most fascinating phenomena in modern physics. It is a purely quantum mechanical process, meaning we can never predict exactly when a single nucleus will decay. However, when we have billions of them, their collective behavior follows a beautifully predictable mathematical law. Let's dive deep into this problem and unravel the mysteries of half-life and mean life.

The problem presents us with a snapshot of a radioactive sample. At a certain instant, $25\%$ of the nuclei are undecayed. We start our stopwatch. Exactly $10\text{ s}$ later, we check again, and the number of undecayed nuclei has dropped to $12.5\%$.

Notice the numbers carefully. What is the relationship between $25$ and $12.5$?

The amount of the substance has exactly halved! By definition, the time it takes for a radioactive sample to reduce to half of its initial amount is called its half-life ($t_{1/2}$). Since this halving occurred over a span of $10\text{ s}$, we can immediately conclude without any complex integration that:

Part (a) of the question asks for the mean life ($\tau$) of the nuclei. While the half-life tells us when 50% of the sample is gone, the mean life represents the average lifespan of a single nucleus in the sample.

Mathematically, the mean life is the reciprocal of the decay constant ($\lambda$), and it is related to the half-life by a very standard formula:

We know that the natural logarithm of 2 ($\ln 2$) is approximately $0.693$. Substituting our known half-life into the equation gives:

This tells us that, on average, a nucleus in this sample will survive for about $14.43\text{ s}$ before decaying.

Now, let's tackle part (b). The phrasing here is crucial: we need to find the time in which the number of undecayed nuclei will further reduce to $6.25\%$ of the reduced number.

Our "reduced number" is the amount we had at the end of our first observation, which was $12.5\%$ of the original sample. Let's call this amount $N_2$. We want to find the time it takes to reach a new amount, $N_3$, such that:

Let's convert that percentage into a fraction to see the physics more clearly:

So, we want the sample to reduce to $\frac{1}{16}$ of its current size. How does this relate to half-lives? Every half-life reduces the sample by a factor of $2$. Therefore, $n$ half-lives will reduce the sample by a factor of $2^n$.

Since $16=2^4$, we can write:

This beautifully reveals that the sample must undergo exactly 4 half-lives to reduce to $6.25\%$ of its value.

Since each half-life is $10\text{ s}$, the total time required is simply:

And there we have it! By understanding the physical meaning behind the percentages, we bypassed the heavy exponential equations and solved the problem using pure logic and intuition.