Animated Solution for Physics - Atoms and Nuclei: If $N_0$ is the original mass of the substance of half-life period $t_{1/2}=5$ yr, then the amount of substance left after 15 yr, is

Radioactive decay is one of the most fascinating phenomena in physics, where a substance literally transforms over time. In this problem, we are given a radioactive substance with an initial mass of $N_0$. We are told that its half-life, denoted by $t_{1/2}$, is $5\text{ yr}$. Our goal is to determine the exact amount of this substance that remains after a total time of $t=15\text{ yr}$.

Before we jump into the math, let's understand what half-life actually means. The half-life of a radioactive isotope is the time required for exactly half of the active nuclei in a sample to decay. This means that after one half-life, you are left with half of what you started with. After two half-lives, you are left with half of that half, which is a quarter, and so on.

This is an exponential decay process. The mass doesn't decrease by a fixed amount every year; rather, it decreases by a fixed fraction over a specific time interval.

To formalize this mathematically, we use the standard decay formula expressed in terms of half-lives:

Here, $N$ represents the amount of substance left after time $t$, $N_0$ is the initial amount, and $n$ is the number of half-lives that have elapsed.

To find $n$, we simply divide the total elapsed time by the duration of one half-life:

Now, let's substitute the values given in our problem. The total time $t$ is $15\text{ yr}$, and the half-life $t_{1/2}$ is $5\text{ yr}$.

This tells us that exactly $3$ half-lives have passed. Now, we bring this value of $n$ back into our master equation:

Calculating the cube of one-half:

Therefore, the amount of substance left after 15 years is $\frac{N_0}{8}$. This elegant result shows how rapidly exponential decay reduces the mass of a radioactive sample.