Plutonium 239 is used as fuel for some nuclear reactors, and also as the fissionable material in atomic bombs. It has a half-life of 24400 years. How long would it take 10 grams of plutonium 239 to decay to 1 gram? (round answer to three significant digits) Hint use F(t)=Ab^t, where t is the number of years elapsed

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!

Plutonium 239 is used as fuel for some nuclear reactors, and also as the fissionable material in atomic bombs. It has a half-life of 24400 years. How long would it take 10 grams of plutonium 239 to decay to 1 gram? (round answer to three significant digits) Hint use F(t)=Ab^t, where t is the number of years elapsed.

Discover how to calculate the time for 10 grams of plutonium-239 to decay to 1 gram using the exponential decay formula. Learn about the decay factor, half-life, and logarithmic equations, with a focus on plutonium's half-life of 24,400 years. Perfect for high school students in Grades 10-12.

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