A rock sample contains $X_{238} X_{2}^{2238} U$ and its final decay product $X_{206} X_{2}^{2206} Pb$ in the mole ratio $Pb : U = 0.25 : 1$ . Assuming no lead was present initially and no loss of either, and taking the half-life of $X_{238} X_{2}^{2238} U$ as $4.5 \times 1 0^{9}$ years, the age of the rock in years is closest to

Answer: C

If U left is

N

, original U

= N (1 + 0.25) = 1.25 N

. So

N / N_{0} = 1/1.25

and

t = t_{1/2} lo g_{2} (1.25) = 4.5 \times 1 0^{9} \times 0.3219 = 1.45 \times 1 0^{9}

years. The values

1.13 \times 1 0^{9}

(

0.25 t_{1/2}

) and

1.87 \times 1 0^{9}

(

lo g_{2} (1/0.75)

) come from wrongly treating

0.25

as a fraction of the total or of the original.