We are given that the half-life of plutonium-239 is 24,400 years, and we want to find how long it would take for 10 grams of plutonium-239 to decay to 1 gram. The decay model is given by the function:

$F(t) = A b^t$

Where:

• $F(t)$ is the remaining amount after time $t$,
• $A$ is the initial amount,
• $b$ is the decay factor per year,
• $t$ is the time in years.

Step 1: Determine the decay factor $b$

The decay factor $b$ can be determined using the half-life formula. We know that after 24,400 years, half of the substance remains, so:

$\frac{1}{2} = b^{24,400}$

Taking the natural logarithm (ln) of both sides:

$\ln\left(\frac{1}{2}\right) = 24,400 \ln(b)$

Solving for $b$:

$\ln(b) = \frac{\ln\left(\frac{1}{2}\right)}{24,400} = \frac{-\ln(2)}{24,400}$

Now, we can calculate $b$.

Step 2: Apply the decay model

Once we have $b$, we can use the equation:

$1 = 10 b^t$

where $10$ grams is the initial amount, $1$ gram is the final amount, and $t$ is the time we want to find. Solving for $t$ involves taking the logarithm again.

Let me compute this step-by-step.The decay factor $b$ is approximately $0.99997$, and the time $t$ it takes for 10 grams of plutonium-239 to decay to 1 gram is approximately 81,055 years (rounded to three significant digits).

Would you like a further breakdown of the steps or explanations?

Here are five related questions to expand on this:

1. How does the decay factor $b$ relate to the half-life of other radioactive materials?
2. What would the remaining amount be after 50,000 years, starting with 10 grams?
3. How does temperature affect the decay rate of radioactive materials like plutonium-239?
4. If the initial amount was 20 grams, how long would it take to decay to 1 gram?
5. How is the decay of radioactive material used in carbon dating?

Tip: When solving exponential decay problems, always start by determining the decay factor from the half-life!