Solved Examples on Half-Life

Radioactivity is an important topic of 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 N_{1 =}  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 K_α line emitted by a hydrogen like atom is 0.32 Å. The wave length of K_p 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 K_α line and K_p line are given by   

 λ_a = hc / E_x – E_L and λ_p = hc / E_K – E_M        

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. 

We know the formulae

Mean-life (t)_mean = 1/λ and half-life (t)_1/2 = 0.693 / λ

Hence, (B) is the correct option.

Q5. The half-life of radioactive substance is 48 hr. How much time it will take to disintegrate to its 1/16th part?

(A) 12 hr.                                       (B) 16 hr.

(C) 48 hr.                                       (D) 192 hr.

Solution: We need to find the time taken by the substance to disintegrate to its 1/16^th part.

So, the required formula is

This gives (D) as the correct option.

Watch this Video for more reference 

Q6. Radioactive substance emits

(A) α-rays                                        (B) β-rays

(C) γ-rays                                        (D) all of these.

Solution:

It is a known fact that a radioactive substance emits alpha, beta as well as gamma rays. When an alpha particle is emitted the mass (A) is reduced by 4 and atomic number (Z) is reduced by 2. Therefore for new nucleus, mass no. is (A – 4) and atomic number is (Z – 2). Hence, (D) is the correct option.

Q7. A radioactive substance has a half-life of 1 year. The fraction of this material that would remain after 5 years will be

(A) 1/32                                             (B) 1/5

(C) 1/2                                               (D) 4/5

Solution:

The substance has half-life of 1 year. Let N

₀ be the initial quantity of material and N be the new quantity.

Then N/N₀ = (1/2)ⁿ 

Hence, N = 1/32 × N₀ 

= 1/32. Hence, (A) is the correct option.

Q8. Half-life of a substance is 20 minutes. What is the time between 33% decay and 67% decay?

(A) 40 minutes                                    (B) 20 minutes

(C) 30 minutes                                    (D) 25 minutes

Solution:

Let N₀ be the number of nuclei at the beginning.

Number of undecayed nuclei after 33% decay = 0.67N₀

Number of undecayed nuclei after 67% decay = 0.33N₀

Also 0.33N₀  >> 0.67N₀ / 2.

And in one half-life the number of undecayed nuclei becomes half.

Hence, (B) is the correct option.

Q9. The half-life of a radioactive substance is 3.6 days. How much of 20 mg of that radioactive substance will remain after 40 days?

(A) 2.68 × 10³ mg                              (B) 4.31 × 10^–2 mg

(C) 6.20 × 10^–3 mg                             (D) 9.76 × 10^–3 mg

Solution:

The half-life of the radioactive substance is given to be 3.6 days.

So, after 40 days, the quantity of substance remaining is given by

Hence, this gives (D) as the correct option.

Q10. A substance reduces to 1/16th of its original mass in 2 hours. The half-life period of the substance will be:

(A) 15 min                                          (B) 30 min

(C) 60 min                                          (D) 120 min

Solution:

Let N0 be the original quantity and N be the new quantity then,

N / N₀ = 1/2^t/T = 1/16 = 1/2⁴ 

t/T = 4 => T = t/4 = 120 / 4 = 30 min

This gives (B) as the correct option.

Q11. Which of the following processes represents a gamma-decay?

(A) ^AX_Z+r —> ^AX_Z–1 + a + b           (B) ^AX_Z —> ¹n₀ —> ³Z_Z–2 + C

(C) ^AX_Z —> ^AX_Z + f                       (D) ^AX_Z + e_–1  —> ^AX_Z–1 + g

Solution:

The gamma rays do not cause any change neither in the atomic number nor the mass number. Hence,

(C)is the correct option.

Q12. The half-life of ²¹⁵At is 100 ms. The time taken for the radioactivity of a sample of 215At to decay to (1/16) th of its initial value is

(A) 400 μs                                       (B) 6.3 μs

(C) 40 μs                                         (D) 300 μs

Solution:

The substance decays to

(1/16) th of its original mass.

Hence, the number of half-lives is 4.

Total time = 4 × t1/2 = 4 × 100 μs = 400 μs.

Hence the correct answer is (A).

The concepts of half-life and average life for radioactive material are quite related. askIITians offers comprehensive study material with solved problems on half-life and mean life of radioactive substance. It is vital to master these areas to remain competitive in the JEE.
 

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