The half-life of a radioactive substance is 20 min. The approximate time interval (t2 - t1) between

Q: Why can't I just plug 1/3 and 2/3 directly into the decay formula?

The formula $N(t) = N_0 e^{-\lambda t}$ calculates the number of nuclei *remaining*, not the number that have decayed. If $1/3$ has decayed, $2/3$ remains! You must always use the remaining fraction.

Q: Do I need to find the decay constant $\lambda$ to solve this?

Not at all! While you *could* find $\lambda$ using $\lambda = \ln(2)/T_{1/2}$ and solve the equations, recognizing that the remaining amount halves exactly between $t_1$ and $t_2$ saves you from all that tedious math.

Q: What if the problem said 1/4 had decayed and then 3/4 had decayed?

Then at $t_1$, $3/4$ remains. At $t_2$, $1/4$ remains. The ratio is $(1/4) / (3/4) = 1/3$. Since $1/3$ is not a simple power of $1/2$, you would have to use the full logarithmic formula to find the exact time difference.

Q: Is it always true that $t_2 - t_1$ depends only on the ratio of $N_1$ and $N_2$?

Yes! Because $N_2 = N_1 e^{-\lambda(t_2 - t_1)}$, the time interval $\Delta t$ is completely determined by the ratio $N_2/N_1$, regardless of the starting amount $N_0$.

Q: I got a negative time interval. What did I do wrong?

You likely swapped $t_1$ and $t_2$. Remember, as more time passes, more of the substance decays. Since $2/3$ decayed is a larger amount than $1/3$ decayed, $t_2$ must be a later time than $t_1$.

Question

The half-life of a radioactive substance is $20 	ext{ min}$. The approximate time interval $(t_2 - t_1)$ between the time $t_2$ when $2/3$ of it has decayed and time $t_1$ when $1/3$ of it had decayed is

Options:

(A) $14 	ext{ min}$

(B) $20 	ext{ min}$

(C) $28 	ext{ min}$

(D) $7 	ext{ min}$

Sigma Percentile · Accepted Answer

### The Elegance of Half-Life: A Conceptual Shortcut

When faced with a radioactive decay problem, our first instinct is often to reach for the heavy artillery: the exponential decay formula, $N(t) = N_0 e^{-\lambda t}$. We start calculating decay constants, taking natural logarithms, and wrestling with decimals.

But sometimes, the problem is designed to test your conceptual intuition rather than your algebraic stamina. This question is a perfect example of a beautiful, elegant shortcut hiding in plain sight.

### The Trap: Decayed vs. Remaining

The most common pitfall in radioactivity problems is confusing the amount that has *decayed* with the amount that *remains*. The fundamental law of radioactive decay, $N(t) = N_0 e^{-\lambda t}$, strictly governs the number of **active, undecayed nuclei** remaining in the sample.

The problem states that at time $t_1$, $1/3$ of the sample has decayed. This means we must subtract the decayed portion from the initial amount to find what's left:

$$N_1 = N_0 - \frac{1}{3}N_0 = \frac{2}{3}N_0$$

Similarly, at a later time $t_2$, $2/3$ of the sample has decayed. The remaining active nuclei at this instant will be:

$$N_2 = N_0 - \frac{2}{3}N_0 = \frac{1}{3}N_0$$

### The "Aha!" Moment

Now, instead of plugging $N_1$ and $N_2$ into two separate exponential equations and solving for $t_1$ and $t_2$, let's take a step back and look at the relationship between the two remaining amounts.

Notice that $N_2$ is exactly half of $N_1$:

$$\frac{N_2}{N_1} = \frac{\frac{1}{3}N_0}{\frac{2}{3}N_0} = \frac{1}{2}$$

What is the physical significance of the number of active nuclei reducing to exactly half of its previous value? By definition, the time required for a radioactive substance to reduce to half of its initial value is exactly one **half-life** ($T_{1/2}$).

### The Final Conclusion

Because the sample went from $\frac{2}{3}N_0$ to $\frac{1}{3}N_0$, it halved. Therefore, the time interval between these two events, $\Delta t = t_2 - t_1$, must be exactly equal to one half-life.

We are given that the half-life of the substance is $20 	ext{ min}$.

$$	herefore t_2 - t_1 = 20 	ext{ min}$$

**The correct option is (b).**

Always keep an eye out for these simple integer ratios ($1/2$, $1/4$, $1/8$) when dealing with remaining nuclei. They are the hallmark of a conceptual shortcut that can save you precious minutes during an exam!

Animated Solution for Physics - Atoms and Nuclei: The half-life of a radioactive substance is $20 min$ . The approximate time interval $(t_{2} - t_{1})$ between the time $t_{2}$ when $2/3$ of it has decayed and time $t_{1}$ when $1/3$ of it had decayed is

Select Answer:

Visualized Solution

Visualizing Radioactive Decay

Remaining vs Decayed

Nuclei at $t_{1}$

Nuclei at $t_{2}$

The Half-Life Connection

Pro Tip

The Sigma Insight: Radioactivity

The Elegance of Half-Life

The Trap

The "Aha!" Moment

The Final Conclusion

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