The decay constant of a radioactive sample is lambda. The half-life and mean-life of the sample are

Q: Why do we use natural logarithm ($\ln$) instead of base-10 logarithm ($\log_{10}$) in the half-life formula?

The radioactive decay law naturally arises from a differential equation $\frac{dN}{dt} = -\lambda N$, whose solution involves the exponential function $e^x$. Therefore, taking the inverse operation requires the natural logarithm (base $e$).

Q: Is mean-life longer or shorter than half-life?

Mean-life is longer. Since $\tau = \frac{1}{\lambda}$ and $T_{1/2} = \frac{\ln 2}{\lambda} \approx \frac{0.693}{\lambda}$, the mean-life is about $1.44$ times the half-life.

Q: Does the decay constant $\lambda$ depend on temperature or pressure?

No, radioactive decay is a nuclear process. It is completely unaffected by macroscopic physical conditions like temperature, pressure, or chemical state.

Q: What happens to the sample after one mean-life?

After one mean-life ($t = \tau = 1/\lambda$), the number of remaining nuclei is $N(\tau) = N_0 e^{-1} \approx 0.37 N_0$. So, about $37\%$ of the sample remains undecayed.

Q: Can we predict exactly when a specific nucleus will decay?

No, radioactive decay is a fundamentally quantum mechanical and probabilistic process. We can only predict the statistical behavior of a large number of nuclei, not the exact fate of an individual one.

Question

The decay constant of a radioactive sample is $\lambda$. The half-life and mean-life of the sample are respectively given by

Options:

(A) $\frac{1}{\lambda}$ and $\frac{\ln 2}{\lambda}$

(B) $\frac{\ln 2}{\lambda}$ and $\frac{1}{\lambda}$

(C) $\lambda(\ln 2)$ and $\frac{1}{\lambda}$

(D) $\frac{\lambda}{\ln 2}$ and $\frac{1}{\lambda}$

Sigma Percentile · Accepted Answer

### Understanding Radioactive Decay

Radioactivity is a fascinating phenomenon where unstable atomic nuclei release energy by emitting radiation. To quantify how quickly a radioactive substance decays, we use a parameter called the **decay constant**, denoted by $\lambda$. This constant represents the probability of decay per unit time for a single nucleus.

### The Concept of Half-Life

Imagine you have a large sample of radioactive atoms. You might wonder, "How long will it take for exactly half of these atoms to decay?" This specific duration is known as the **half-life** ($T_{1/2}$).

Mathematically, the radioactive decay law is given by:
$$N(t) = N_0 e^{-\lambda t}$$
where $N_0$ is the initial number of nuclei and $N(t)$ is the number of nuclei remaining at time $t$.

To find the half-life, we set $N(t) = \frac{N_0}{2}$:
$$\frac{N_0}{2} = N_0 e^{-\lambda T_{1/2}}$$

$$\frac{1}{2} = e^{-\lambda T_{1/2}}$$

Taking the natural logarithm on both sides:
$$\ln\left(\frac{1}{2}ight) = -\lambda T_{1/2}$$

$$-\ln 2 = -\lambda T_{1/2}$$

$$T_{1/2} = \frac{\ln 2}{\lambda}$$

This formula beautifully connects the macroscopic observable (half-life) with the microscopic probability (decay constant).

### The Concept of Mean-Life

While half-life is incredibly useful, physicists also use another measure called the **mean-life** ($	au$). Think of it as the average lifespan of all the radioactive nuclei in the sample. Some nuclei will decay almost immediately, while others might survive for a very long time. If you were to average the lifetimes of every single nucleus, you would get the mean-life.

Through integration of the decay probability over all time from zero to infinity, the mean-life is calculated as the reciprocal of the decay constant:
$$	au = \frac{1}{\lambda}$$

### Conclusion

By comparing the two derived expressions, we can clearly see the relationship between them. The half-life is $\frac{\ln 2}{\lambda}$ and the mean-life is $\frac{1}{\lambda}$. Since $\ln 2 \approx 0.693$, the half-life is approximately $69.3\%$ of the mean-life.

Returning to our question, we are asked for the half-life and mean-life respectively. The correct pair is $\frac{\ln 2}{\lambda}$ and $\frac{1}{\lambda}$, which corresponds to option (b).

Animated Solution for Physics - Atoms and Nuclei: The decay constant of a radioactive sample is $λ$ . The half-life and mean-life of the sample are respectively given by

Select Answer:

Visualized Solution

$T_{1/2}$

$τ$

$Conclusion$

The Sigma Insight: Radioactivity

Understanding Radioactive Decay

The Concept of Half-Life

The Concept of Mean-Life

Conclusion

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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 decay constant of a radioactive sample is λ. The half-life and mean-life of the sample are respectively given by