Question 9.9: Calculate the weight in grams, W, of 1 mCi (3.700×10^7 dps) ......
Calculate the weight in grams, W, of 1 mCi (3.700\times 10^{7} dps) of ^{14}C from its half-life of 5720 years.
Step-by-Step
Half-life = t_{1/2} = \frac{0.693}{k} ; where k is a characteristic decay rate constant for the radioisotope.
The number of disintegrations per second = activity = \mathscr{A}=-\frac{d\mathscr{N}}{dt} = k\mathscr{N} , \text{ where } \mathscr{N} is the number of ^{14}C atoms present.
\mathscr{N} = \frac{grams of ^{14}C}{\text{ mass number of } ^{14}C}\times \text{ Avogadro’s number } = \frac{W}{14}\times 6.02\times 10^{23}k = \frac{0.693}{t_{1/2}} = \frac{0.693}{5720 years} = 1.211\times 10^{-4}/years
\mathscr{A} = \frac{1.211\times 10^{-4} y^{-1}}{3.156\times 10^{7} s y^{-1}}\times \frac{W}{14}\times 6.02\times 10^{23} = 1.65\times 10^{12} s^{-1}\times W
Let \mathscr{A} = \text{1 mCi} = 3.700\times 10^{7} s^{-1}
W = \text{grams of } ^{14}C \text{emitting 1 mCi }= \frac{3.700\times 10^{7} s^{-1}}{1.65\times 10^{12} s^{-1} g^{-1}} = 2.24\times 10^{-5} g
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