Nuclei — Questions & Answers

Sample questions from Nuclei

1. Q1. What fraction of a radioactive sample remains undecayed after a time equal to half of one half-life?

   • A.0.75
   • B.0.50
   • C.0.25
   • D.About 0.71✓
   Solution

   N/N0 = (1/2)^(t/T) with t/T = 1/2, so N/N0 = 2^(-1/2) = 1/sqrt2 = 0.707. Decay is exponential, not linear, so it is NOT 0.75.

2. Q2. For a radioactive sample, the ratio of the number of undecayed nuclei at t = T(1/2) to that at t = 3T(1/2) is:

   • A.4✓
   • B.3
   • C.8
   • D.2
   Solution

   N(T) = N0/2 and N(3T) = N0/8. Ratio = (1/2)/(1/8) = 4, corresponding to the two half-lives elapsed between the two instants.

3. Q3. In a nuclear reactor, a moderator of low mass number is chosen because in an elastic head-on collision a neutron of mass m with a nucleus of mass M transfers a fraction 4mM/(m + M)^2 of its energy. For ordinary hydrogen (M = m) this fraction is:

   • A.0.5
   • B.1, so a single head-on collision can stop the neutron✓
   • C.0.28
   • D.0.02
   Solution

   Putting M = m gives 4m^2/(2m)^2 = 1: a head-on collision with an equal mass transfers all the kinetic energy. Ordinary water is still bypassed for natural uranium reactors because hydrogen absorbs neutrons by forming deuterium, whereas deuterium (fraction 8/9) scatters without absorbing.

4. Q4. The number of half-lives needed for the activity of a sample to fall to 1% of its initial value is closest to:

   • A.5.0
   • B.10.0
   • C.6.6✓
   • D.3.3
   Solution

   (1/2)^n = 0.01 gives n = log2(100) = ln100/ln2 = 4.605/0.693 = 6.64 half-lives. Note 2^6 = 64 and 2^7 = 128, so the answer lies between 6 and 7.

5. Q5. In the alpha decay of a parent of mass number 226, the fraction of the disintegration energy carried away by the recoiling daughter is closest to:

   • A.1.8%✓
   • B.50%
   • C.98%
   • D.4%
   Solution

   Fraction to the daughter = 4/226 = 0.0177, i.e. about 1.8 per cent; the alpha keeps 222/226 = 98.2 per cent. Because momenta are equal, kinetic energy divides in inverse proportion to mass.

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