Animated Solution for Physics - Atoms and Nuclei: A radioactive sample consists of two distinct species having equal number of atoms initially. The mean life of one species is $\tau$ and that of the other is $5\tau$. The decay products in both cases are stable. A plot is made of the total number of radioactive nuclei as a function of time. Which of the following figure best represents the form of this plot?

The phenomenon of radioactive decay is one of the most fascinating and predictable processes in quantum physics. While we can never predict exactly when a single unstable nucleus will decay, the collective behavior of a large number of such nuclei follows a beautifully precise mathematical law.

In this problem, we are presented with a fascinating scenario: a radioactive sample that doesn't just contain one, but two distinct radioactive species.

Imagine you are in a laboratory, observing a freshly prepared radioactive sample. The problem states that initially, both species have an equal number of atoms. Let us denote this initial number as $N_0$. Therefore, at the very beginning (at $t=0$), the total number of radioactive nuclei in our sample is simply $N_0+N_0=2N_0$.

We are also given a crucial piece of information about their decay rates. The first species has a mean life of $\tau$, while the second species has a mean life of $5\tau$. The mean life is inversely proportional to the decay constant ($\lambda =\frac{1}{\tau }$). This means the first species decays five times faster than the second species.

Another vital constraint provided is that the decay products in both cases are stable. This ensures that as the original nuclei decay, they do not create new radioactive isotopes that would complicate our total count. The total number of radioactive nuclei will only depend on the decay of the original two species.

According to the radioactive decay law, the number of undecayed nuclei of a specific species at any time $t$ is given by the exponential function:

Let's write down the equations for our two specific species. For the first species, the number of remaining nuclei is:

For the second species, with its longer mean life, the number of remaining nuclei is:

The question asks us to find the plot for the total number of radioactive nuclei as a function of time. This total number, let's call it $N_{\text{total}}(t)$, is simply the algebraic sum of the two individual populations:

Now, let's analyze the mathematical properties of this combined function.

First, let's check the initial condition. At $t=0$, $e^0=1$. The graph must start at a positive value on the y-axis.

Next, let's look at the behavior over time. Both $e^{-t/\tau }$ and $e^{-t/5\tau }$ are strictly decreasing functions for all $t>0$. When you add two strictly decreasing functions together, their sum must also be a strictly decreasing function.

Physically, this makes perfect sense. Both species are continuously decaying into stable products. There is no mechanism in this system to create new radioactive nuclei. Therefore, the total count of radioactive nuclei must continuously drop. It can never pause and remain constant, nor can it ever reverse direction and increase.

Finally, let's look at the long-term behavior. As $t\to \infty$, both exponential terms approach zero. The graph must asymptotically approach the time axis.

Armed with this mathematical and physical understanding, let's evaluate the given graphical options.

Graph (a) shows the number of nuclei decreasing initially, but then it flattens out, implying the decay process has stopped and the number of radioactive nuclei remains constant. As we established, radioactive decay is a continuous process that only stops when all nuclei are depleted. This graph is incorrect.

Graph (b) shows the number of nuclei decreasing to a minimum and then increasing. An increase would imply that new radioactive nuclei are being synthesized, which contradicts the premise that the decay products are stable. This graph is physically impossible for this system.

Graph (c) shows a complex behavior of decreasing, increasing, and then decreasing again. For the same reasons as graph (b), any region of increase is impossible.

Graph (d) shows a curve that starts at a positive value and monotonically decreases, asymptotically approaching zero. This perfectly matches our mathematical derivation and physical intuition. The total number of radioactive nuclei continuously dwindles as both species decay away.

This problem beautifully illustrates how complex systems can often be understood by breaking them down into their fundamental components. By recognizing that the total population is just the sum of two independently decaying populations, and by understanding the strict mathematical properties of exponential decay, we can confidently identify the correct graphical representation without needing to plot a single data point. The elegance of physics lies in this logical consistency.