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Modern Physics in the JEE. Half-life of a radioactive substance is a vital area that lays the groundwork for other concepts. Calculation of half-life of a radioactive substance is not a very tedious task and questions are often picked from these areas in various competitive exams. We discuss some numerical on how to calculate the wavelength, mass and half-life of a radioactive decay.
Q1. r1 and r2 are the radii of atomic nuclei of mass numbers 64 and 27 respectively. The ratio r1/r2 is
(A) 64/27 (B) 27/64
(C) 4/3 (D) 1
Solution: Since r1 and r2 are given to be 64 and 27 so the ratio r1/r2 is 64/27. Hence the correct option is (A).
Q2. Two radioactive materials. X1 and X2 have decay constants 10 and λ respectively. If initially they have the same number of nuclei, then the ratio of the number of nuclei of X1 to that of X2. will be 1/e after a time
(A) 1/10 λ (B) 1/11λ
(C) 11/10λ (D) 1/9λ
Solution: Let N0 be the initial number of nuclei in both X1 and X2.
Let N1 be the number of nuclei in X1 and N2 be the number of nuclei in X2 now.
Then N1 = N0 e–10λt
N2 = N0 e–λt
We need to find N1/N2.
Dividing the above two equations we get,
N1/N2 = e–9λt
This ratio is given to be 1/e and we need to calculate the time‘t’ in that case.
e–9λt = 1/e.
=> 9λt = 1, or t = 1/9λ. This gives (D) as the correct option.
Q3.The wave length of the characteristic X-ray Kα line emitted by a hydrogen like atom is 0.32 Å. The wave length of Kp line emitted by the same element is:
(A) 0.18Å (B) 0.48Å
(B) 0.27Å (D) 0.38A
Solution: The wavelength of x-rays emitted for Kα line and Kp line are given by
λa = hc / Ex – EL and λp = hc / EK – EM
Hence, (C) is the correct option.
Q4. The ratio of the mean life of a radionuclide to its half-life is
(A) 1 (B) 1.44
(C) 2 (D) 3.12
Solution: We are required to find the ratio of mean-life and half-life.