Explanation:
Rate constant \({\rm{(k)}}\)
\(\,\,\,\,\,\,\,\,\,\,\,\,{\rm{ = }}\frac{{{\rm{0}}{\rm{.693}}}}{{{{\rm{t}}_{{\rm{1/2}}}}}}{\rm{ = }}\frac{{{\rm{0}}{\rm{.693}}}}{{{\rm{1}}{\rm{.28 \times 1}}{{\rm{0}}^{\rm{9}}}}}\)
\(\,\,\,\,\,\,\,\,\,\,\,\, = 5.414 \times {10^{ - 10}}{\rm{y}}{{\rm{r}}^{ - 1}}\)
\({\rm{Number}}\,{\rm{o}}{{\rm{f}}^{{\rm{40}}}}{\rm{\;K}}\,{\rm{atoms}}\,{\rm{in}}\,{\rm{lg}}\,{\rm{KCl}}\)
\(\,\,\,\,\,\,\,\,\,\,{\rm{ = 1/74}}.{\rm{5}}\;{\rm{mol}}\)
\(\,\,\,\,\,\,\,\,\, = \frac{1}{{74.5}} \times 6.02 \times {10^{23}}{\rm{ molecules }}\)
\({\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\rm{ = }}\frac{{\rm{1}}}{{{\rm{74}}.{\rm{5}}}} \times {\rm{6}}.{\rm{02}} \times {\rm{1}}{{\rm{0}}^{{\rm{23}}}} \times \frac{{{\rm{0}}.{\rm{01}}}}{{{\rm{100}}}}{\;^{{\rm{40}}}}{\rm{K}}\,{\rm{atoms}}{\mkern 1mu} \)
\({\mkern 1mu} \,\,\,\,\, = 8.08 \times {10^{17}}{\rm{atoms}}\)
\({\rm{Activity = }}\,{\rm{k\lambda }}\,{\rm{ = 5}}{\rm{.414 \times 1}}{{\rm{0}}^{ - {\rm{10}}}}{\rm{ = 8}}{\rm{.08 \times 1}}{{\rm{0}}^{{\rm{17}}}}\)
\(\,\,\,\,\,\,\,\, = 4.37 \times {10^8}{\rm{ year}}{{\rm{ }}^{ - 1}}\)