91Ó°ÊÓ

Radioactivity is a phenomenon in which a few substances spontaneously release energy and subatomic particles. The nuclear instability of an atom causes radioactivity.

The process is given by,

\(\begin{array}{c}R = \frac{{0.693N}}{{{t_{1/2}}}}\\N = \frac{{R{t_{1/2}}}}{{0.693}}\end{array}\)

\({\rm{131}}\)iodine has a half-life of\({t_{1/2}} = 8.04\,{\rm{day}} = 6.95 \times {10^5}\,{\rm{s}}\). The iodine’sgiven activity is,

\(\begin{array}{c}R = 1.00\,{\rm{PCi}}\\ = (1.00 \times {10^{ - 12}}\,{\rm{Ci}})\left( {3.70 \times {{10}^{10}}\,{\rm{Bq/Ci}}} \right)\\ = 3.7 \times {10^{ - 2}}\,{\rm{Bq}}\end{array}\)

As a result, the number of\({}^{{\rm{131}}}{\rm{I}}\)is,

\(\begin{array}{c}N = \frac{{\left( {3.7 \times {{10}^{ - 2}}\,{\rm{Bq}}} \right)\left( {6.95 \times {{10}^5}\,{\rm{s}}} \right)}}{{0.693}}\\ = 3.71 \times {10^4}\,{\rm{atoms}}\end{array}\)

\({}^{{\rm{131}}}{\rm{I}}\)has a molar atomic mass of\({m_0} = 131\,{\rm{g}}\). As a result, the mass that generates the activity is,

\(\begin{array}{c}m = \frac{N}{{{N_A}}}{m_0}\\ = \frac{{3.71 \times {{10}^4}}}{{6.02 \times {{10}^{23}}}} \times (131\,{\rm{g}})\\ = 8.07 \times {10^{ - 18}}\,{\rm{g}}\\ = 8.07 \times {10^{ - 21}}\,{\rm{Kg}}\end{array}\)

Therefore, the mass is \(8.07 \times {10^{ - 21}}\,{\rm{Kg}}\).