Half-lives of two radioactive elements A and B are 20 min and 40 min, respectively. Initially, the s

Q: Why can't we just take the ratio of their half-lives to find the answer?

Radioactive decay is an exponential process, not a linear one. The number of nuclei doesn't decrease at a constant rate, so simple linear ratios of time or half-lives will give you the wrong answer.

Q: I got 1:4 as my answer. What did I do wrong?

You fell into a classic trap! You calculated the ratio of the *remaining* nuclei ($N_A / N_B$). The question specifically asked for the ratio of the *decayed* nuclei. Always double-check what the question is asking for.

Q: What does $n$ represent in the formula $N = N_0 / 2^n$?

$n$ represents the number of half-lives that have elapsed. It is calculated by dividing the total elapsed time $t$ by the half-life $T_{1/2}$ of the substance.

Q: Does the initial number of nuclei $N_0$ matter? What if they were different?

In this specific problem, since both elements start with the same $N_0$ and we are finding a ratio, $N_0$ cancels out algebraically. However, if the problem stated they started with different amounts, you would absolutely need to keep track of $N_{A,0}$ and $N_{B,0}$.

Q: Why do we subtract the remaining nuclei from $N_0$?

The standard decay formula $N = N_0 / 2^n$ calculates what is *left over* (undecayed). To find out what has *disappeared* (decayed), you must subtract what is left from the total amount you started with.

Question

Half-lives of two radioactive elements $A$ and $B$ are $20	ext{ min}$ and $40	ext{ min}$, respectively. Initially, the samples have equal number of nuclei. After $80	ext{ min}$, the ratio of decayed numbers of $A$ and $B$ nuclei will be

Options:

(A) $1 : 16$

(B) $4 : 1$

(C) $1 : 4$

(D) $5 : 4$

Sigma Percentile · Accepted Answer

### Setting the Stage

Imagine a race against time, but instead of runners, we have two radioactive elements, $A$ and $B$. They both start at the exact same starting line with an equal number of initial nuclei, which we will call $N_0$.

However, they decay at different speeds. Element $A$ is the faster one, with a half-life ($T_{1/2}$) of just $20	ext{ min}$. Element $B$ is more sluggish, taking $40	ext{ min}$ to halve its population. The question asks us to freeze time at exactly $80	ext{ min}$ and compare how much of each element has **decayed**.

### The Mathematics of Decay

To solve this, we need our trusty radioactive decay formula. The number of nuclei remaining after a certain time $t$ is given by:

$$N = \frac{N_0}{2^n}$$

Here, $n$ represents the number of half-lives that have elapsed. We can easily find $n$ by dividing the total time $t$ by the half-life $T_{1/2}$ of the element.

### Analyzing Element A

Let's focus on the fast-decaying element $A$. In the $80	ext{ min}$ window, how many half-lives does it go through?

$$n_A = \frac{80	ext{ min}}{20	ext{ min}} = 4$$

Element $A$ undergoes $4$ complete half-lives. Plugging this into our formula, the number of nuclei *remaining* is:

$$N_A = \frac{N_0}{2^4} = \frac{N_0}{16}$$

But wait! The question doesn't care about what's left; it cares about what's gone. To find the **decayed** nuclei, we must subtract the remaining amount from the initial amount:

$$	ext{Decayed}_A = N_0 - \frac{N_0}{16} = \frac{15N_0}{16}$$

Almost the entire sample of $A$ has decayed!

### Analyzing Element B

Now, let's look at the slower element $B$. In the same $80	ext{ min}$, how many half-lives does it experience?

$$n_B = \frac{80	ext{ min}}{40	ext{ min}} = 2$$

Element $B$ only goes through $2$ half-lives. The number of nuclei *remaining* is:

$$N_B = \frac{N_0}{2^2} = \frac{N_0}{4}$$

Again, we need the **decayed** amount:

$$	ext{Decayed}_B = N_0 - \frac{N_0}{4} = \frac{3N_0}{4}$$

### The Final Showdown

We have the decayed amounts for both elements. The final step is to find their ratio:

$$	ext{Ratio} = \frac{	ext{Decayed}_A}{	ext{Decayed}_B} = \frac{\frac{15N_0}{16}}{\frac{3N_0}{4}}$$

The initial amount $N_0$ beautifully cancels out, leaving us with a simple fraction division:

$$	ext{Ratio} = \frac{15}{16} 	imes \frac{4}{3} = \frac{5}{4}$$

The ratio of decayed nuclei of $A$ to $B$ is **$5:4$**.

### The Trap of the Remaining Nuclei

This question contains a classic examiner's trap. If you were rushing and calculated the ratio of the *remaining* nuclei instead, you would get:

$$\frac{N_A}{N_B} = \frac{\frac{N_0}{16}}{\frac{N_0}{4}} = \frac{1}{4}$$

Notice that $1:4$ is sitting right there in the options as choice (c)! Always read the question carefully to ensure you are answering exactly what is being asked. In radioactivity problems, the distinction between "decayed" and "remaining" is everything.

Animated Solution for Physics - Atoms and Nuclei: Half-lives of two radioactive elements $A$ and $B$ are $20 min$ and $40 min$ , respectively. Initially, the samples have equal number of nuclei. After $80 min$ , the ratio of decayed numbers of $A$ and $B$ nuclei will be

Select Answer:

Visualized Solution

$Initial Setup$

$Radioactive Decay Law$

$Half-lives of Element A$

$Decayed Nuclei of A$

$Half-lives of Element B$

$Decayed Nuclei of B$

$Ratio of Decayed Nuclei$

$The Way Forward$

The Sigma Insight: Radioactivity

Setting the Stage

The Mathematics of Decay

Analyzing Element A

Analyzing Element B

The Final Showdown

The Trap of the Remaining Nuclei

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Visualized Solution

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Initial Setup

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Radioactive Decay Law

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Half-lives of Element A

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Decayed Nuclei of A

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Half-lives of Element B

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Decayed Nuclei of B

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Ratio of Decayed Nuclei

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The Way Forward