Animated Solution for Physics - Atoms and Nuclei: The radioactive decay rate of a radioactive element is found to be $10^3$ disintegration/second at a certain time. If the half-life of the element is one second, the decay rate after one second is ....... and after three seconds is........ .

Radioactive decay is one of the most fascinating phenomena in modern physics. It is a purely statistical process, meaning we can never predict exactly when a single nucleus will decay. However, when we have a massive collection of nuclei, their collective behavior becomes incredibly predictable and follows a beautiful mathematical pattern: exponential decay.

The rate at which a radioactive sample decays is called its activity or decay rate, often denoted by $R$ or $A$. It tells us how many disintegrations are happening per second.

A crucial concept in radioactivity is the half-life ($T_{1/2}$). The half-life is the time required for exactly half of the radioactive nuclei in a sample to undergo decay. Because the activity is directly proportional to the number of undecayed nuclei ($R=\lambda N$), the activity also halves after every half-life.

This leads us to a very powerful and intuitive formula:

Here, $R_0$ is the initial activity, $R$ is the activity after a certain time, and $n$ is the number of half-lives that have elapsed. The number of half-lives can be easily calculated by dividing the total time $t$ by the half-life $T_{1/2}$:

In our specific problem, we are given a radioactive element with an initial decay rate $R_0=10^3\text{ disintegrations/second}$. We are also told that the half-life of this element is exactly $1\text{ second}$.

This means that every single second, the activity of the sample will be cut in half. Let's use our formula to find the activity at the two specific times requested.

Decay Rate After 1 Second

First, we need to find the decay rate after $t=1\text{ second}$.

Let's determine how many half-lives have passed. Since the half-life is $1\text{ second}$, a total time of $1\text{ second}$ corresponds to exactly one half-life.

Now, we substitute this into our master equation:

As expected, after exactly one half-life, the activity has halved from $1000$ to $500$.

Decay Rate After 3 Seconds

Next, we need to find the decay rate after $t=3\text{ seconds}$.

Again, we calculate the number of half-lives that have elapsed:

This means the sample has gone through three consecutive halvings. Let's plug $n=3$ into our formula:

We know that ${\left(\frac{1}{2}\right)}^3=\frac{1}{8}$. Therefore:

By simply understanding the definition of half-life and using the $R=R_0(1/2{)}^n$ formula, we can quickly solve problems involving discrete multiples of half-lives without needing to use the more complex exponential formula $R=R_0{e}^{-\lambda t}$. The activity after 1 second is $500\text{ disintegrations/second}$, and after 3 seconds, it is $125\text{ disintegrations/second}$.