The half-life of 131 I is 8 days. Given a sample of 131 I at time t = 0, we can assert that
(a) no n

Q: Why can't we say all nuclei will decay in 16 days (two half-lives)?

After one half-life (8 days), 50% of the sample remains. After two half-lives (16 days), 50% of that remaining 50% decays, leaving 25% of the original sample still undecayed. It theoretically takes infinite time for all nuclei to decay.

Q: Doesn't a half-life of 8 days mean it takes 8 days for a nucleus to decay?

No, half-life is a macroscopic property. It means in 8 days, half of a large sample will decay. An individual nucleus can decay at any random moment, whether it's 1 second or 100 years from now.

Q: Is it possible for a nucleus to decay at exactly $t=0$?

Decay is a continuous process starting from $t=0$. A nucleus can decay infinitesimally close to $t=0$. There is no 'waiting period' before decay begins.

Q: What determines exactly when a specific nucleus will decay?

According to quantum mechanics, it is fundamentally random. There is no hidden variable or internal timer inside the nucleus that dictates when it will decay.

Q: If the decay is random, how can we have a precise formula like $N = N_0 e^{-\lambda t}$?

The formula applies to a very large number of nuclei (macroscopic scale) where statistical averages become highly predictable, even though individual events are completely random. It's like flipping a coin: you can't predict one flip, but you know roughly 50% will be heads if you flip it a million times.

Question

The half-life of $^{131}	ext{I}$ is 8 days. Given a sample of $^{131}	ext{I}$ at time $t = 0$, we can assert that

Options:

(A) no nucleus will decay before $t = 4$ days

(B) no nucleus will decay before $t = 8$ days

(C) all nuclei will decay before $t = 16$ days

(D) a given nucleus may decay at any time after $t = 0$

Sigma Percentile · Accepted Answer

### The Quantum Casino: Understanding Radioactive Decay

Imagine you are holding a sample of Iodine-131. You know its half-life is 8 days. But what does that actually mean for a single, solitary nucleus inside that sample? Does it have a tiny internal clock ticking down to exactly 8 days before it decides to pop?

Absolutely not. The world of the very small is governed by the bizarre and fascinating rules of quantum mechanics, where certainty is replaced by probability.

### The Exponential Law

Radioactive decay is a completely spontaneous and random process. We cannot predict exactly when a specific nucleus will decay. However, when we have a massive number of nuclei (like in any macroscopic sample), their collective behavior becomes highly predictable.

The law of radioactive decay states that the rate of decay is proportional to the number of undecayed nuclei present:

$$\frac{dN}{dt} = -\lambda N$$

Solving this differential equation gives us the famous exponential decay formula:

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

Notice how the curve smoothly goes down. It never truly reaches zero in a finite amount of time. It theoretically takes infinite time ($t 	o \infty$) for *all* nuclei to decay. This immediately tells us that option (c), which claims all nuclei will decay in 16 days, is fundamentally wrong. After 16 days (two half-lives), 25% of the sample will still be happily undecayed.

### The Misconception of Half-Life

The half-life of 8 days simply means that after 8 days, statistically half of the sample will have decayed. It is a macroscopic average.

It absolutely does not mean that the nuclei wait for 4 or 8 days to start decaying. Decay is a continuous process that begins the moment the sample is created (at $t=0$). Therefore, options (a) and (b) are incorrect because nuclei are decaying constantly, even before 4 or 8 days have passed.

### The Final Verdict

Because the process is continuous and probabilistic, a specific nucleus could decay in the very next second, or it might survive for a thousand years. There is no 'safe period' where a nucleus is guaranteed not to decay.

Therefore, the only logically sound assertion is that **a given nucleus may decay at any time after $t = 0$**.

Animated Solution for Physics - Atoms and Nuclei: The half-life of $^{131} I$ is 8 days. Given a sample of $^{131} I$ at time $t = 0$ , we can assert that

Select Answer:

Visualized Solution

$Nature of Radioactive Decay$

$N (t) = N_{0} e^{- λ t}$

$Understanding Half-Life (T_{1/2})$

$Probability of Decay$

The Sigma Insight: Radioactivity

The Quantum Casino

The Exponential Law

The Misconception of Half-Life

The Final Verdict

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Animated Solution for Physics - Atoms and Nuclei: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The half-life of 131I is 8 days. Given a sample of 131I at time t=0, we can assert that