Calculating Half-Life from Average Lifetime

The question asks us to find the half-life of a radioactive substance given its average lifetime.

For any radioactive substance, there are two related time periods that describe its decay:

• Average Lifetime ($\tau$): This is the average time a nucleus of the substance exists before decaying. It is the reciprocal of the decay constant ($\lambda$). Mathematically, $\tau = \frac{1}{\lambda}$.
• Half-Life ($T_{1/2}$): This is the time it takes for half of the radioactive nuclei in a sample to decay.

There is a specific relationship between the average lifetime ($\tau$) and the half-life ($T_{1/2}$). This relationship is given by the formula:

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

where $\ln(2)$ is the natural logarithm of 2, which is approximately 0.693.

Given in the question, the average lifetime ($\tau$) of the radioactive substance is 2000 years.

Now, we can use the formula to calculate the half-life ($T_{1/2}$):

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

Substitute the given value of $\tau$:

$$T_{1/2} = 2000 \text{ years} \times \ln(2)$$

Using the approximate value $\ln(2) \approx 0.693$:

$$T_{1/2} \approx 2000 \text{ years} \times 0.693$$

Calculating the product:

$$T_{1/2} \approx 1386 \text{ years}$$

Thus, the half-life of the radioactive substance is approximately 1386 years.

Let's compare this result with the given options:

• 1386 years
• 1000 years
• 386 years
• 4386 years

Our calculated value matches the first option.

The half-life of the radioactive substance with an average lifetime of 2000 years is 1386 years.