If f denotes the ratio of the number of nuclei decayed (Nd) to the number of nuclei at t=0 (N0), the

Q: Why is the number of decayed nuclei $N_0 - N$?

Because the total number of nuclei is conserved. If you start with $N_0$ nuclei and $N$ are left, the difference must be the ones that have decayed.

Q: Why does the derivative of $e^{-\lambda t}$ have a negative sign?

By the chain rule of calculus, the derivative of $e^{u}$ is $e^{u} \cdot \frac{du}{dt}$. Here $u = -\lambda t$, so its derivative is $-\lambda$.

Q: What does the fraction $f$ represent physically?

It represents the probability that any single nucleus has decayed by time $t$, or equivalently, the fraction of the entire sample that has decayed.

Q: Why is the rate of change of $f$ positive?

Because the number of decayed nuclei is constantly increasing as time goes on. A positive derivative means the function is growing.

Q: Will $f$ ever reach exactly 1?

Mathematically, $f$ approaches 1 asymptotically as $t \to \infty$. Physically, once all discrete nuclei have decayed, $f$ becomes exactly 1.

Question

If $f$ denotes the ratio of the number of nuclei decayed ($N_d$) to the number of nuclei at $t=0$ ($N_0$), then for a collection of radioactive nuclei, the rate of change of $f$ with respect to time is given as [$\lambda$ is the radioactive decay constant]

Options:

(A) $-\lambda(1 - e^{-\lambda t})$

(B) $\lambda(1 - e^{-\lambda t})$

(C) $\lambda e^{-\lambda t}$

(D) $-\lambda e^{-\lambda t}$

Sigma Percentile · Accepted Answer

### The Mathematics of Radioactive Decay

Radioactive decay is a fascinating natural phenomenon where unstable atomic nuclei lose energy by emitting radiation. But how do we quantify this process?

The fundamental law of radioactive decay states that the rate of decay is proportional to the number of undecayed nuclei present. This gives rise to the famous exponential decay formula:

$$N(t) = N_0 e^{-\lambda t}$$

where $N(t)$ is the number of active nuclei at time $t$, $N_0$ is the initial number of nuclei, and $\lambda$ is the decay constant.

### Tracking the Decayed Nuclei

While $N(t)$ tells us what remains, we are often interested in what has already decayed.

By the principle of conservation of mass, the number of decayed nuclei $N_d(t)$ is simply the initial amount minus the remaining amount:

$$N_d(t) = N_0 - N(t) = N_0(1 - e^{-\lambda t})$$

This equation shows that as time progresses, the number of decayed nuclei grows, asymptotically approaching the initial number $N_0$.

### The Fraction of Decay

The problem introduces a variable $f$, defined as the ratio of decayed nuclei to the initial number of nuclei.

This is a dimensionless fraction that tells us the progress of the decay process:

$$f = \frac{N_d}{N_0} = 1 - e^{-\lambda t}$$

At $t=0$, $f=0$ because nothing has decayed yet. As $t 	o \infty$, $f 	o 1$ meaning everything has decayed.

### Finding the Rate of Change

To find how fast this fraction is changing, we need to differentiate $f$ with respect to time $t$.

This requires a basic application of calculus, specifically the chain rule for exponential functions:

$$\frac{df}{dt} = \frac{d}{dt} (1 - e^{-\lambda t})$$

The derivative of the constant $1$ is $0$. For the exponential term, the derivative of $e^{-\lambda t}$ is $-\lambda e^{-\lambda t}$.

Carefully managing the negative signs, we get:

$$\frac{df}{dt} = 0 - (-\lambda e^{-\lambda t}) = \lambda e^{-\lambda t}$$

This result is profoundly elegant. It tells us that the rate at which the decayed fraction grows is directly proportional to the fraction of nuclei that are **still active**.

As the active pool depletes, the rate of new decays slows down. The final answer is **$\lambda e^{-\lambda t}$**.

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Visualized Solution

Radioactive Decay Law for $N (t)$

Expression for $N_{d} (t)$

Defining the Ratio $f$

Rate of Change $\frac{df}{d t}$

Final Result for $\frac{df}{d t}$

Physical Significance of $\frac{df}{d t}$

The Sigma Insight: Radioactivity

The Mathematics of Radioactive Decay

Tracking the Decayed Nuclei

The Fraction of Decay

Finding the Rate of Change

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Visualized Solution

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

Expression for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Nd​(t)

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