Problem 16
Question
An artifact originally had 16 grams of carbon- 14 present. The decay model \(A=16 e^{-0.0001211}\) describes the amount of carbonI 4 present, \(A,\) in grams, after \(t\) years. The half-life of the radioactive element plutonium-239 is \(25,000\) years. If 16 grams of plutonium- 239 are initially present how many grams are present after \(25,000\) years? \(50,000\) years? \(75,000\) years? \(100,000\) years? \(125,000\) years?
Step-by-Step Solution
Verified Answer
The amount of plutonium-239 left after 25,000 years is 8 grams, after 50,000 years it is 4 grams, after 75,000 years it is 2 grams, after 100,000 years it is 1 gram and after 125,000 years it is 0.5 grams.
1Step 1: Understand the half-life property
The half-life of a substance is the time it takes for half of it to decay. In this case, plutonium-239 has a half-life of 25,000 years, meaning every 25,000 years, half of it decays.
2Step 2: Calculate the amount of plutonium-239 after 25,000 years
Since the half-life is 25,000 years, half of the originally present plutonium-239 will decay after these years. Therefore, there will be \(16/2 = 8\) grams of plutonium-239 after 25,000 years.
3Step 3: Calculate the amount of plutonium-239 after 50,000 years
After another 25,000 years (total of 50,000 years), again half of the remaining plutonium-239 will decay. So, there will be \(8/2 = 4\) grams of plutonium-239 after 50,000 years.
4Step 4: Calculate the amount of plutonium-239 after 75,000 years
Proceeding similarly for 75,000 years, we have \(4/2 = 2\) grams of plutonium-239.
5Step 5: Calculate the amount of plutonium-239 after 100,000 years
For 100,000 years, half of the remaining amount will decay again, leaving \(2/2 = 1\) gram.
6Step 6: Calculate the amount of plutonium-239 after 125,000 years
Lastly, for 125,000 years, we will have \(1/2 = 0.5\) grams of plutonium-239 remaining.
Key Concepts
Carbon-14 DecayHalf-LifeDecay ModelPlutonium-239
Carbon-14 Decay
Carbon-14 is a naturally occurring radioactive isotope of carbon. It's used in dating archaeological and geological samples because of its predictable decay over time.
When living organisms die, they stop absorbing carbon, and the amount of carbon-14 within them starts to decrease as it decays. The decay can be mathematically represented using the decay model. For example, a model like \(A = 16 e^{-0.0001211t}\) describes how the amount of carbon-14 decreases over time.
When living organisms die, they stop absorbing carbon, and the amount of carbon-14 within them starts to decrease as it decays. The decay can be mathematically represented using the decay model. For example, a model like \(A = 16 e^{-0.0001211t}\) describes how the amount of carbon-14 decreases over time.
- \(A\) is the remaining amount of carbon-14 in grams.
- \(t\) represents time in years.
- \(e^{-0.0001211t}\) reflects the decay process, where \(e\) is the base of the natural logarithm.
Half-Life
Half-life is a crucial concept in understanding radioactive decay. It tells us how long it takes for half of a given amount of a radioactive substance to decay.
For instance, if you start with 16 grams of a substance, after one half-life period, you'll have 8 grams left. After another half-life, you'll have 4 grams, and so on.
In the case of plutonium-239, the half-life is 25,000 years. This means:
For instance, if you start with 16 grams of a substance, after one half-life period, you'll have 8 grams left. After another half-life, you'll have 4 grams, and so on.
In the case of plutonium-239, the half-life is 25,000 years. This means:
- After 25,000 years, 8 grams remain from an initial 16 grams.
- Continuing for another 25,000 years (total 50,000 years), you'll have 4 grams.
- This process repeats every half-life period, effectively halving the substance each time.
Decay Model
A decay model provides a mathematical representation of how a substance decreases over time. In radioactive decay, such a model relies on exponential functions to describe the decay process.
An exponential decay model usually takes the form of \(A = A_0 e^{-kt}\), where:
An exponential decay model usually takes the form of \(A = A_0 e^{-kt}\), where:
- \(A\) is the amount of substance remaining at time \(t\).
- \(A_0\) is the initial amount of the substance.
- \(k\) is the decay constant, which indicates the rate of decay.
- \(t\) represents the time elapsed.
Plutonium-239
Plutonium-239 is a radioactive isotope commonly used in nuclear reactions. It is a significant material, especially in the context of nuclear energy and weaponry.
Plutonium-239 has a relatively long half-life of 25,000 years, meaning it remains radioactive for a substantial period.
Starting with 16 grams, the decay of plutonium-239 can be predicted through successive halving:
Plutonium-239 has a relatively long half-life of 25,000 years, meaning it remains radioactive for a substantial period.
Starting with 16 grams, the decay of plutonium-239 can be predicted through successive halving:
- After the first 25,000 years, 8 grams will remain.
- 50,000 years in, 4 grams will be left.
- After 75,000 years, only 2 grams will endure.
- At 100,000 years, 1 gram remains.
- Eventually, after 125,000 years, just 0.5 grams persist.
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