Half-Life and Mean-Life Relationship in Radioactive Decay

TITLE: Half-Life and Mean-Life Relationship in Radioactive Decay DESCRIPTION: Derive and understand the relationship between half-life ($T_{1/2}$) and mean-life ($\tau$) for radioactive decay, a key concept in Modern Physics.

Concept Overview

This question tests the understanding of radioactive decay kinetics, specifically the relationship between two fundamental time parameters: half-life ($T_{1/2}$) and mean-life ($\tau$). It requires deriving this relationship from the basic exponential decay law, demonstrating an understanding of the probabilistic nature of nuclear decay and the definitions of these time constants. The derivation involves calculus and the definition of expected values.

Step 1: Recall the law of radioactive decay. The number of radioactive nuclei $N$ at time $t$ is given by:

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

where $N_0$ is the initial number of nuclei at $t=0$, and $\lambda$ is the decay constant. This equation describes the exponential decrease in the number of radioactive atoms over time.

Step 2: Define half-life ($T_{1/2}$). Half-life is the time required for half of the radioactive nuclei in a sample to decay. At $t = T_{1/2}$, $N(T_{1/2}) = N_0/2$. Substituting this into the decay law:

$$\frac{N_0}{2} = N_0 e^{-\lambda T_{1/2}}$$

Dividing both sides by $N_0$ gives:

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

Taking the natural logarithm of both sides:

$$\ln\left(\frac{1}{2}\right) = -\lambda T_{1/2}$$ $$-\ln(2) = -\lambda T_{1/2}$$ $$T_{1/2} = \frac{\ln(2)}{\lambda}$$

This equation shows that the half-life is inversely proportional to the decay constant.

Step 3: Define mean-life ($\tau$). The mean-life (or average life) of a radioactive nucleus is the average time for which a nucleus exists before decaying. It is calculated as the expectation value of the time $t$ for decay, weighted by the probability of decay at time $t$. The probability of a nucleus decaying between time $t$ and $t + dt$ is given by $\lambda e^{-\lambda t} dt$. The mean-life $\tau$ is then:

$$\tau = \int_0^\infty t \cdot (\lambda e^{-\lambda t}) dt$$

This integral represents the weighted average of decay times.

Step 4: Evaluate the integral for mean-life. We can solve this integral using integration by parts, where $u = t$ and $dv = \lambda e^{-\lambda t} dt$. Then $du = dt$ and $v = \int \lambda e^{-\lambda t} dt = -e^{-\lambda t}$.

$$\tau = \left[ -t e^{-\lambda t} \right]_0^\infty - \int_0^\infty (-e^{-\lambda t}) dt$$

Evaluating the first term:

$$\lim_{t \to \infty} (-t e^{-\lambda t}) - (-0 \cdot e^0) = 0 - 0 = 0$$

(since $e^{-\lambda t}$ goes to zero faster than $t$ goes to infinity for $\lambda > 0$). The integral becomes:

$$\tau = \int_0^\infty e^{-\lambda t} dt$$ $$\tau = \left[ -\frac{1}{\lambda} e^{-\lambda t} \right]_0^\infty$$ $$\tau = \left( \lim_{t \to \infty} -\frac{1}{\lambda} e^{-\lambda t} \right) - \left( -\frac{1}{\lambda} e^0 \right)$$ $$\tau = 0 - \left( -\frac{1}{\lambda} \right) = \frac{1}{\lambda}$$

Thus, the mean-life is the reciprocal of the decay constant.

Step 5: Establish the relationship between $T_{1/2}$ and $\tau$. From Step 2, we have $T_{1/2} = \frac{\ln(2)}{\lambda}$. From Step 4, we have $\tau = \frac{1}{\lambda}$. We can substitute $\frac{1}{\lambda}$ from the expression for $\tau$ into the expression for $T_{1/2}$:

$$T_{1/2} = \ln(2) \cdot \left(\frac{1}{\lambda}\right)$$ $$T_{1/2} = \tau \ln(2)$$

This is the fundamental relationship between the half-life and the mean-life of a radioactive substance. It shows that the half-life is approximately 0.693 times the mean-life.

Key Takeaways:

• Radioactive decay follows an exponential law, $N(t) = N_0 e^{-\lambda t}$.
• Half-life ($T_{1/2}$) is the time for half the sample to decay, related by $T_{1/2} = \frac{\ln(2)}{\lambda}$.
• Mean-life ($\tau$) is the average lifetime of a nucleus, given by $\tau = \frac{1}{\lambda}$.
• The relationship is $T_{1/2} = \tau \ln(2)$, meaning $T_{1/2} \approx 0.693 \tau$.

Answer: $T_{1/2} = \tau \ln(2)$