This is a problem about determining the half-life of a radioactive element, using exponential decay models.

Given Information:

• Initial sample: $2 \, \text{grams}$
• After 10 days, $0.1 \, \text{grams}$ remain.
• We are asked to find the half-life.

Exponential Decay Formula:

The general formula for radioactive decay is: $y = A e^{-kt}$ Where:

• $y$ is the amount of substance remaining after time $t$,
• $A$ is the initial amount (here $A = 2 \, \text{grams}$),
• $k$ is the decay constant,
• $t$ is the time elapsed,
• $e$ is Euler's number (approximately 2.718).

Step-by-Step Solution:

1. Using the decay formula: We know that after 10 days, the remaining amount $y = 0.1 \, \text{grams}$. Plugging into the formula: $0.1 = 2 e^{-k \cdot 10}$ Simplifying this: $\frac{0.1}{2} = e^{-10k}$ $0.05 = e^{-10k}$

2. Taking the natural logarithm (ln) of both sides: $\ln(0.05) = -10k$ $k = \frac{-\ln(0.05)}{10}$ Using a calculator, $\ln(0.05) \approx -2.9957$, so: $k \approx \frac{2.9957}{10} = 0.29957$

3. Half-Life Formula: The half-life $T_{\frac{1}{2}}$ is related to $k$ by: $T_{\frac{1}{2}} = \frac{\ln(2)}{k}$ Using $\ln(2) \approx 0.6931$: $T_{\frac{1}{2}} \approx \frac{0.6931}{0.29957} \approx 2.314 \, \text{days}$

Thus, the half-life of the radioactive element is approximately 2.31 days (rounded to three significant digits).

Let me know if you need further clarification or any additional details.

Related Questions:

1. What happens to the decay constant if the half-life is doubled?
2. How would the solution change if the remaining mass after 10 days was 0.5 grams instead of 0.1 grams?
3. Can we use this method for other decay processes such as carbon dating?
4. What is the relationship between the decay constant $k$ and the half-life $T_{\frac{1}{2}}$?
5. How does the decay model change if the element decays into a stable product with a different mass?

Tip:

Always use natural logarithms when working with exponential decay models involving radioactive substances!