Animated Solution for Physics - Atoms and Nuclei: A radioactive sample disintegrates via two independent decay processes having half-lives $T_{1/2}^{(1)}$ and $T_{1/2}^{(2)}$, respectively. The effective half-life $T_{1/2}$ of the nuclei is

Imagine you are standing at a crossroads. You have a certain probability of taking the left path and a certain probability of taking the right path. Radioactive nuclei sometimes face a similar dilemma. A single unstable nucleus might have two entirely different ways to decay into a more stable state. This phenomenon is known as parallel decay or branching decay.

In our problem, we have a radioactive sample $A$ that can disintegrate into nucleus $B$ or nucleus $C$. These two processes are completely independent. The decay of one nucleus into $B$ doesn't affect the chances of another nucleus decaying into $C$.

To understand how fast the sample $A$ is disappearing overall, we need to look at the decay constant, denoted by $\lambda$. The decay constant represents the probability of decay per unit time. Because the two decay pathways are independent, the total probability of a nucleus decaying is simply the sum of the probabilities of it decaying via each individual pathway.

Therefore, the effective decay constant for the entire sample is the sum of the individual decay constants:

This is the master equation for parallel decay. It tells us that the overall rate of disappearance is faster than either of the individual rates alone.

However, the problem asks us for the effective half-life, $T_{1/2}$, not the decay constant. We must recall the fundamental relationship between these two quantities. The half-life is inversely proportional to the decay constant:

This makes intuitive sense: a larger decay constant means a higher probability of decay, which results in a shorter half-life.

Now, we substitute this relationship back into our master equation. For the overall process, we use the effective half-life $T_{1/2}$. For the individual processes, we use their respective half-lives, $T_{1/2}^{(1)}$ and $T_{1/2}^{(2)}$:

Notice the beautiful symmetry! The term $\ln (2)$ appears in every numerator. We can divide the entire equation by $\ln (2)$ to simplify it drastically:

Does this equation look familiar? It is mathematically identical to the formula for calculating the equivalent resistance of two resistors connected in parallel! Finally, we just need to perform some basic algebra to isolate $T_{1/2}$. Taking the common denominator on the right side and then taking the reciprocal yields our final answer:

This elegant result shows that the effective half-life of a sample undergoing parallel decay is always less than the shortest individual half-life. The sample is burning the candle at both ends, so it disappears much faster!