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A radioactive sample contains two radioactive nucleus A and B having decay constant λ hr–1 and 2λ hr–1. Initially 20% of decay comes from A. How long (in hr) will it take before 50% of decay comes from A. [Take λ = ln 2]
- a)1
- b)2
- c)3
- d)None of these
Correct answer is option 'B'. Can you explain this answer?
Verified Answer
A radioactive sample contains two radioactive nucleus A and B having d...
∴ from (1) and (2) , eλt = 9
⇒ λt = 2ln3 ⇒ t = 2.
Most Upvoted Answer
A radioactive sample contains two radioactive nucleus A and B having d...
Understanding the Problem
Initially, the sample contains two radioactive nuclei, A and B, with decay constants λ and 2λ respectively. The problem states that initially, 20% of the decay is attributed to nucleus A.
Decay Rates
- The decay rate for nucleus A: dA/dt = -λNA
- The decay rate for nucleus B: dB/dt = -2λNB
Where NA and NB are the number of nuclei of A and B at time t.
Initial Conditions
- Initially, let the total number of nuclei be N = NA + NB.
- Since 20% of decay comes from A, we can say that:
- NA = 0.2N
- NB = 0.8N
Decay Over Time
- After a time t, the number of nuclei remaining will be:
- NA(t) = NA * e^(-λt) = 0.2N * e^(-λt)
- NB(t) = NB * e^(-2λt) = 0.8N * e^(-2λt)
- The decay rates at time t become:
- Decay from A: dA = λ * 0.2N * e^(-λt)
- Decay from B: dB = 2λ * 0.8N * e^(-2λt)
Finding the Time for 50% Decay from A
- To find when 50% of the decay comes from A, we set up the ratio:
(Decay from A) / (Total decay) = 0.5
- After simplifying, we find that this condition is met when:
0.2 * e^(−λt) = 0.5 * (0.2 * e^(−λt) + 0.8 * e^(−2λt))
- Solving this equation leads to t = 2 hours.
Conclusion
Thus, it takes 2 hours for 50% of the decay to come from nucleus A, confirming that the correct answer is option b) 2.
Initially, the sample contains two radioactive nuclei, A and B, with decay constants λ and 2λ respectively. The problem states that initially, 20% of the decay is attributed to nucleus A.
Decay Rates
- The decay rate for nucleus A: dA/dt = -λNA
- The decay rate for nucleus B: dB/dt = -2λNB
Where NA and NB are the number of nuclei of A and B at time t.
Initial Conditions
- Initially, let the total number of nuclei be N = NA + NB.
- Since 20% of decay comes from A, we can say that:
- NA = 0.2N
- NB = 0.8N
Decay Over Time
- After a time t, the number of nuclei remaining will be:
- NA(t) = NA * e^(-λt) = 0.2N * e^(-λt)
- NB(t) = NB * e^(-2λt) = 0.8N * e^(-2λt)
- The decay rates at time t become:
- Decay from A: dA = λ * 0.2N * e^(-λt)
- Decay from B: dB = 2λ * 0.8N * e^(-2λt)
Finding the Time for 50% Decay from A
- To find when 50% of the decay comes from A, we set up the ratio:
(Decay from A) / (Total decay) = 0.5
- After simplifying, we find that this condition is met when:
0.2 * e^(−λt) = 0.5 * (0.2 * e^(−λt) + 0.8 * e^(−2λt))
- Solving this equation leads to t = 2 hours.
Conclusion
Thus, it takes 2 hours for 50% of the decay to come from nucleus A, confirming that the correct answer is option b) 2.
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A radioactive sample contains two radioactive nucleus A and B having decay constant λ hr–1 and 2λ hr–1. Initially 20% of decay comes from A. How long (in hr) will it take before 50% of decay comes from A. [Take λ = ln 2]a)1b)2c)3d)None of theseCorrect answer is option 'B'. Can you explain this answer?
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A radioactive sample contains two radioactive nucleus A and B having decay constant λ hr–1 and 2λ hr–1. Initially 20% of decay comes from A. How long (in hr) will it take before 50% of decay comes from A. [Take λ = ln 2]a)1b)2c)3d)None of theseCorrect answer is option 'B'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about A radioactive sample contains two radioactive nucleus A and B having decay constant λ hr–1 and 2λ hr–1. Initially 20% of decay comes from A. How long (in hr) will it take before 50% of decay comes from A. [Take λ = ln 2]a)1b)2c)3d)None of theseCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A radioactive sample contains two radioactive nucleus A and B having decay constant λ hr–1 and 2λ hr–1. Initially 20% of decay comes from A. How long (in hr) will it take before 50% of decay comes from A. [Take λ = ln 2]a)1b)2c)3d)None of theseCorrect answer is option 'B'. Can you explain this answer?.
A radioactive sample contains two radioactive nucleus A and B having decay constant λ hr–1 and 2λ hr–1. Initially 20% of decay comes from A. How long (in hr) will it take before 50% of decay comes from A. [Take λ = ln 2]a)1b)2c)3d)None of theseCorrect answer is option 'B'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about A radioactive sample contains two radioactive nucleus A and B having decay constant λ hr–1 and 2λ hr–1. Initially 20% of decay comes from A. How long (in hr) will it take before 50% of decay comes from A. [Take λ = ln 2]a)1b)2c)3d)None of theseCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A radioactive sample contains two radioactive nucleus A and B having decay constant λ hr–1 and 2λ hr–1. Initially 20% of decay comes from A. How long (in hr) will it take before 50% of decay comes from A. [Take λ = ln 2]a)1b)2c)3d)None of theseCorrect answer is option 'B'. Can you explain this answer?.
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