An isotope of potassium K⁴⁰undergoes two parallel types of decay, one by electron capture and other by $β$ decay with half lives as 1.3 x 10⁹ years and $\frac{1.3}{9} \times 10^{9}$ . If in a sample of a mineral, mass ratio of K : Ar : Ca is 5,1.5 and 16 then calculate age of the mineral if it is known that all Ca⁴⁰ in the mineral is not from the radioactive decay of potassium.
[Express your answer in terms of 10⁷ years for e.g. if you answer is 2 x 10⁹ years fill 0200 in OMR sheet]

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answer is 26.

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Detailed Solution

$k_{1} = \frac{\ln 2}{\frac{1.3}{9} \times 10^{9}}; k_{2} = \frac{\ln 2}{1.3 \times 10^{9}}$

$\frac{m_{{CB}_{B}}}{m_{Ar}} = \frac{k_{1}}{k_{2}} = 9$

$\begin{array}{l} \Rightarrow m_{K} \to 5; m_{Ca} \to 13.5; m_{Ar} \to 1.5 \\ k = k_{1} + k_{2} = \frac{\ln 2}{1.3 \times 10^{9}} [9 + 1] = \frac{\ln 2}{1.3 \times 10^{8}} \end{array}$

$\begin{array}{l} (\frac{\ln 2}{1.3 \times 10^{8}}) \times t = \ln (\frac{5 + 13.5 + 1.5}{5}) \\ \Rightarrow t = 2.6 \times 10^{8} years = 0026 \end{array}$