At radioactive equilibrium, the ratio of two atoms A and B are 3.1 × 10⁹ = 1. If half life of A is 2 × 10¹⁰ years, then what is the half life of B?

A$\to$B

At equilibrium, rate of formation of B = rate of decay of B

$\therefore K_{\mathrm{A}}{N}_{\mathrm{A}}=K_{\mathrm{B}}{N}_{\mathrm{B}}$

or $\frac{K_{\mathrm{A}}}{K_{\mathrm{B}}}=\frac{N_{\mathrm{B}}}{N_{\mathrm{A}}}$

$\frac{t_{1/2\left(\mathrm{B}\right)}}{t_{1/2\left(\mathrm{A}\right)}}=\frac{N_{\mathrm{B}}}{N_{\mathrm{A}}}$ $\left[K=\frac{0.693}{t_{1/2}}\right]$

$\therefore t_{1/2\left(\mathrm{B}\right)}=\frac{1\times 2\times {10}^{10}}{3.1\times {10}^9};\left[\frac{N_{\mathrm{B}}}{N_{\mathrm{A}}}=1:3.1\times {10}^9\left(\text{ given }\right)\right]$

$\text{ or }{\mathrm{t}}_{1/2\left(\mathrm{B}\right)}=6.45\text{ years }$