Potassium-40 (K-40) is a radioactive isotope that occurs naturally in many different types of rock. A very small percentage of the isotope undergoes β⁺ decay to form an isotope of argon (Ar). Construct and complete the nuclear reaction equation for this decay.

Overall about 10% of a sample of K-40 will decay to argon. In a particular rock sample it is found that there are 1.6×10²² atoms of K-40 and 8.4×10²¹ atoms of argon. The half-life of K-40 is 1.2×10⁹ yr. Estimate the time elapsed since the rock sample was formed.

\(\left( {{}_{19}^{40}{\rm{K}} \to {}_{18}^{40}{\rm{Ar}} + {}_1^0{\beta ^ + } + v} \right)\)
\({{}_{18}^{40}{\rm{Ar}}}\);
v; (do not accept \(\bar v\))

original number of K-40 atoms=(1.6×10²²+[8.4×10²¹×10]=)1.0×10²³;
decay constant=\(\frac{{\ln 2}}{{1.2 \times {{10}^9}}}\) or 5.8×10^-10(yr^-1);
\(1.6 \times {10^{22}} = 1.0 \times {10^{23}}{\rm{ }}{{\rm{e}}^{ - {\rm{5.8}} \times {\rm{1}}{{\rm{0}}^{ - {\rm{10}}}}t}}\);
to give t=3.2×10⁹ (yr);
Accept any alternative method that leads to the correct answer.
Award [3 max] ECF after incorrect value for N₀ (eg. use of 2.44×10²² to give 7.3×10²² yr).
Award [2 max] for approximate answers (eg. 3.0×10⁹ yr based on an estimate of between two and three half-lives.)