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Two radioactive substances A and B have decay constants 5λ and λ, respectively. At t = 0, a sample has the same number of the two nuclei. The time taken for the ratio of the number of nuclei to become (1/e)2 will be:
- a)2/λ
- b)1/4λ
- c)1/2λ
- d)1/λ
Correct answer is option 'C'. Can you explain this answer?
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Two radioactive substances A and B have decay constants 5λ and&...
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Two radioactive substances A and B have decay constants 5λ and&...
Understanding Radioactive Decay
At t = 0, let N0 be the number of nuclei for both substances A and B. The decay of a radioactive substance can be expressed by the formula:
N(t) = N0 * e^(-λt)
For substances A and B:
- Substance A:
- Decay constant: 5λ
- N_A(t) = N0 * e^(-5λt)
- Substance B:
- Decay constant: λ
- N_B(t) = N0 * e^(-λt)
Finding the Ratio of Nuclei
We need to find the time when the ratio of the number of nuclei becomes (1/e)². This means we want:
N_A(t) / N_B(t) = (1/e)²
Substituting the expressions for N_A(t) and N_B(t):
(N0 * e^(-5λt)) / (N0 * e^(-λt)) = (1/e)²
This simplifies to:
e^(-5λt + λt) = (1/e)²
Which can be rewritten as:
e^(-4λt) = e^(-2)
Simplifying the Equation
Taking the natural logarithm on both sides gives:
-4λt = -2
Solving for t:
t = 2 / (4λ) = 1 / (2λ)
Conclusion
Thus, the time taken for the ratio of the number of nuclei to become (1/e)² is:
t = 1/(2λ)
The correct answer is option 'C'.
At t = 0, let N0 be the number of nuclei for both substances A and B. The decay of a radioactive substance can be expressed by the formula:
N(t) = N0 * e^(-λt)
For substances A and B:
- Substance A:
- Decay constant: 5λ
- N_A(t) = N0 * e^(-5λt)
- Substance B:
- Decay constant: λ
- N_B(t) = N0 * e^(-λt)
Finding the Ratio of Nuclei
We need to find the time when the ratio of the number of nuclei becomes (1/e)². This means we want:
N_A(t) / N_B(t) = (1/e)²
Substituting the expressions for N_A(t) and N_B(t):
(N0 * e^(-5λt)) / (N0 * e^(-λt)) = (1/e)²
This simplifies to:
e^(-5λt + λt) = (1/e)²
Which can be rewritten as:
e^(-4λt) = e^(-2)
Simplifying the Equation
Taking the natural logarithm on both sides gives:
-4λt = -2
Solving for t:
t = 2 / (4λ) = 1 / (2λ)
Conclusion
Thus, the time taken for the ratio of the number of nuclei to become (1/e)² is:
t = 1/(2λ)
The correct answer is option 'C'.
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Two radioactive substances A and B have decay constants 5λ andλ,respectively. At t = 0, a sample has the same number of the two nuclei. The time taken for the ratio of the number of nuclei to become (1/e)2will be:a)2/λb)1/4λc)1/2λd)1/λCorrect answer is option 'C'. Can you explain this answer?
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