Animated Solution for Physics - Atoms and Nuclei: Two radioactive materials $A$ and $B$ have decay constants $10\lambda$ and $\lambda$, respectively. If initially they have the same number of nuclei, then the ratio of the number of nuclei of $A$ to that of $B$ will be $1/e$ after a time

Imagine two radioactive samples, $A$ and $B$, sitting side-by-side on a laboratory bench. They both start their journey with the exact same number of active, unstable nuclei, which we will denote as $N_0$. However, they are fundamentally different in their nature. Material $A$ is highly unstable and is decaying at a frantic pace—ten times faster than material $B$. Mathematically, we express this by saying its decay constant is ${\lambda }_A=10\lambda$, while $B$'s decay constant is simply ${\lambda }_B=\lambda$.

Because $A$ is decaying so much faster, its population of surviving nuclei will plummet rapidly compared to $B$. The question asks us to find the exact moment in time, $t$, when the ratio of the remaining nuclei of $A$ to the remaining nuclei of $B$ drops to exactly $\frac{1}{e}$.

Radioactive Decay Law

To track how these populations change over time, we rely on the fundamental law of radioactive decay. This law states that the number of undecayed nuclei $N(t)$ at any given time $t$ follows an exponential decay curve:

This beautiful equation tells us that the survival of nuclei is governed by the natural exponential function, scaled by the initial amount $N_0$ and the specific decay constant $\lambda$ of the material.

We are given a specific snapshot in time where the ratio of the populations is $\frac{1}{e}$. Let's write down this condition mathematically:

Now, we substitute our decay expressions for both materials into this ratio. For material $A$, we plug in $10\lambda$ for the decay constant, and for material $B$, we use $\lambda$. Also, recall that $\frac{1}{e}$ can be elegantly written as $e^{-1}$.

Look closely at the left side of our equation. Since both samples started with the identical initial amount $N_0$, this term beautifully cancels out from the numerator and the denominator. This is a profound physical insight: the time it takes to reach a specific ratio depends only on the decay constants, not on how much material we started with!

After canceling $N_0$, we are left with:

Now, we invoke the laws of exponents. When dividing terms with the same base, we subtract the exponent of the denominator from the exponent of the numerator:

We have reached the final stretch. We have an equation where both sides share the same base, $e$. For this equality to hold true, their exponents must be absolutely identical. By equating the exponents, we get a simple linear equation:

The negative signs cancel out on both sides, leaving us with:

Solving for $t$, we find our final answer:

This is the exact moment when the frantic decay of material $A$ leaves it with exactly $\frac{1}{e}$ times the nuclei of the slower-decaying material $B$.