Radioactive plutonium-239 (t 1/2 = 2.41 × 10 5 yr) is used in nuclear reactors and atomic bombs. If there are 5.70 × 10 2 g of plutonium isotope in a small atomic bomb, how long will it take for the substance to decay to 3.00 × 10 2 g?

We're told that radioactive plutonium 239 has a half life of 2.41 times 10 to the 5 years, and it's used in nuclear reactors and atomic bombs. Now if there are 5.70 times 10 to the 2 grams of plutonium isotope in a small atomic bomb, how long will it take for the substance substance to decay to 3.00 times 10 to the 2 grams? Alright. So first of all, it's radioactive, so that tells us that this is a first order rate law process. And because of that, we know that half life for first order is ln2 over k. But this question isn't asking us about the half life, it's asking us how long will it take. So it's actually asking us to find time, t. That we find in the first order integrated rate law equation, which is l n of our final concentration equals negative kt+ln of our initial concentration. So we need to solve for this t here. Now we have this many grams initially and then they're saying it's decaying to this final grams at the end. So that's our final that's our initial and our final respectively. So ln of 3.00 times 10 to the 2 is our final, and then plus ln of 5.70 times 10 to the 2 grams over here as our initial. Now, here we can't solve for t yet because we also need to know what k is. Well, that's where the half life equation comes into play. So here, we'd rearrange this, multiply both sides by k, so k times half life equals ln2. Divide both sides by half life, so k equals ln2 over half life. So here ln2 divided by 2.41 times 10 to the 5 years. When we do that that's gonna give me my k which is 2.876 times 10 to the negative 6, years inverse. Take that and plug it in, so we're gonna do negative 2.8 76 times 10 negative 6 years inverse. We're going to do what? We're going to now subtract ln of 5.70 times 10 to the 2 from both sides. So when I do that here, I'm gonna get here negative 0.64185 equals negative 2.876 times 10 to the negative 6 times t. Divide both sides now by negative 2.876 times 10 to the negative 6. Almost done. So you can see that this is a long process. When we do that we get t, which is time, equals 2.23 times 10 to the 5 years. So this would be our final answer in how long it would take it to decay to 3.00 times 10 to the 2 grams.

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1. Intro to General Chemistry3h 48m
2. Atoms & Elements4h 16m
3. Chemical Reactions4h 10m
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23. Chemistry of the Nonmetals2h 39m
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15. Chemical Kinetics

Half-Life

General Chemistry

15. Chemical Kinetics

Half-Life

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Multiple Choice

Radioactive plutonium-239 (t_1/2 = 2.41 × 10⁵ yr) is used in nuclear reactors and atomic bombs. If there are 5.70 × 10² g of plutonium isotope in a small atomic bomb, how long will it take for the substance to decay to 3.00 × 10² g?

2.23×10⁵ yr

2.60×10⁵ yr

517 yr

600 yr

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Verified step by step guidance

Understand the concept of half-life, which is the time required for half of the radioactive substance to decay. For plutonium-239, the half-life is given as 2.41 × 10^5 years.

Use the formula for radioactive decay: \( N = N_0 \times \left( \frac{1}{2} \right)^{\frac{t}{t_{1/2}}} \), where \( N \) is the final quantity, \( N_0 \) is the initial quantity, \( t \) is the time elapsed, and \( t_{1/2} \) is the half-life.

Substitute the given values into the formula: \( N_0 = 5.70 \times 10^2 \) g, \( N = 3.00 \times 10^2 \) g, and \( t_{1/2} = 2.41 \times 10^5 \) years.

Rearrange the formula to solve for \( t \): \( t = t_{1/2} \times \frac{\log(N/N_0)}{\log(1/2)} \).

Calculate the time \( t \) using the rearranged formula, which will give you the duration it takes for the plutonium-239 to decay from 5.70 × 10^2 g to 3.00 × 10^2 g.