Atoms that have too many neutrons in their nucleus can become unstable. Large unstable atoms can become more stable by emitting radiation. Radioactive decay is the process in which an unstable atomic nucleus loses energy by spontaneous emission of radiation in the form of particles or electromagnetic waves. This radiation can be emitted in the form of a positively charged alpha particle (α), a negatively or positively charged beta particle (β- or β+), or gamma (γ) rays. Often it is a combination of more than one form of radiation. Many isotopes have to go trough several steps, forming a decay-chain, to transform each parent nuclide into a stable daughter nuclide.
Radioactive decay is a random process at the atomic level. In that respect, it is impossible to predict when a particular atom will decay, but the average decay rate for a large number of the same isotope is predictable. Based on experimental observations, a radioactive decay law has been established:
Rate of decay= A(activity)= -dN/dt= λN
Where N is the number of radioactive atoms in a sample, A is the rate of radioactive decay (or activity), expressed as the number of atoms per time unit, and λ is the decay constant expressed in units of 1/time. Using equation 1, we can derive an integrated radioactive decay law:
Where, N0 is the number of atoms at the initial time (t=0) and Nt is the number of atoms at some later time, t. Using equation 2, we can write the relationship between the decay constant (λ) and the half-life of the process (t1/2) - the time required for half of a radioactive sample to disintegrate:
Table 1: Some Representative Half-Lives
1. The radioactive isotope is a β- emitter used as a tracer for radio immunoassays in biological systems.
If the initial sample weighs 1.0 g, approximately what mass of 131I will disintegrate after 24 days?
A) 125.0 mg
B) 250.0 mg
C) 500.0 mg
D) 875.0 mg
One way to solve this question is to calculate the mass of 131I remaining after 24 days and subtract this amount from the initial 1.0 g sample. This is done by first calculating the decay constant (λ) of the process using Equation 3 and the t1/2 of 131I from Table 1, which is 8.0 days. We could then apply Equation 2, using the calculated λ, N0 (converted from 1.0 gr into number of atoms using iodine's molecular weight as specified in the periodic table and Avogadro's number (6.02 x 1023 1/mol)) and 24 days for t. However, this calculation is quite complicated without a calculator. An easier and shorter way to solve this question is to notice that 24 days is exactly 3 times the half-time of
131I: 24/t1/2 = 24/8 = 3 half lives.
The quantity of radioactive material decreases by one half for every half life. So the quantity of 131I remaining after 24 days is (1/2)3 of the original quantity:
1.0 g x (1/2)3 = 0.125 g
But this is not the final answer. The question asks for the mass of 131I that will disintegrate after 24 days so this amount needs to be subtracted from the initial mass:
1.0 g - 0.125 g = 0.875 g = 875 mg
Therefore, D is the correct answer. The principle illustrated in this solution can be applied in general to phenomena involving exponential decay. If the quotient of total decay time divided by half-life can be easily computed, it is advisable to use this computation for determining the amount of substance remaining after the decay.
2. Which of the following curves represents a radioactive source with the longest half-life?
D) Half-life time can not be inferred from this graph
This is a stand alone question that requires a basic understanding of the meaning of half-life and of graph analysis. The half-life (t1/2) of a radioactive sample is the time it takes for half of the sample to decay. In the graph above, all of the radioactive sources have the same initial mass, proportional to N0. Thus, in order to solve this problem, one has to estimate the location of N0/2 on the y axis and draw an imaginary line parallel to the x axis that intersects the four curves representing the four radioactive sources. A perpendicular line from each intersection point on the curves will intersect with the x axis at a point which is the half-life of each radioactive source. Using this method, it can be easily seen that the radioactive source represented by curve A has the longest half-life. Therefore, A is the correct answer.
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