Chemistry Labs

Problem 1

131I^{131}\ce{I} is a radioactive isotope of iodine (a β−\beta^{-} emitter) used in nuclear medicine for scintigraphy of thyroid disorders. Its decay rate constant is k=9.93×10−7 s−1k = 9.93\times10^{-7}\ \mathrm{s^{-1}}. (1.1) Write the decay reaction of 131I^{131}\ce{I}. (1.2) Calculate the half-life of 131I^{131}\ce{I} in days. (1.3) Calculate the time (in days) needed for a sample of 131I^{131}\ce{I} to fall to 30 % of its initial activity. (1.4) A Geiger counter detects activities of order 10−4 μ10^{-4}\ \muCi; calculate the minimum mass (in grams) of 131I^{131}\ce{I} detectable by this counter. (1 curie is the amount of a radioisotope producing 3.7×10103.7\times10^{10} disintegrations per second.)
Step 4 of 4: Minimum detectable mass
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Intuition

Radioactive counting is extraordinarily sensitive: a femtogram-scale sample is still millions of atoms.

N=Ak=10−4×3.7×1049.93×10−7=3.73×106 atoms;m=3.73×1066.02×1023×131=8.1×10−16 gN = \frac{A}{k} = \frac{10^{-4}\times3.7\times10^{4}}{9.93\times10^{-7}} = 3.73\times10^{6}\ \text{atoms};\quad m = \frac{3.73\times10^{6}}{6.02\times10^{23}}\times131 = 8.1\times10^{-16}\ \mathrm{g}
Analysis

The detection limit 10−4 μ10^{-4}\ \muCi equals 10−4×3.7×104=3.710^{-4}\times3.7\times10^{4} = 3.7 disintegrations s⁻¹. From A=kNA = kN, N=3.7/9.93×10−7=3.7×106N = 3.7/9.93\times10^{-7} = 3.7\times10^{6} atoms, which at 131 g mol⁻¹ is only 8.1×10−168.1\times10^{-16} g.