Carbon ( Z = 6 ) possessing mass number 11 decays into boron ( Z = 5 ). The half-life for this decay scheme is 20.3 minutes . Calculate the time elapsed before a mixture containing 90 % carbon-11 and 10 % boron-11 (by number of atoms) transforms into a mixture of 10 % carbon-11 and 90 % boron-11.

Step-by-step solution

Let the total number of atoms in the mixture be N . Initially, the number of Carbon-11 atoms is N₁ = 0.90 N . Finally, the number of Carbon-11 atoms is N₂ = 0.10 N . According to the law of radioactive decay: N₂ = N₁ e^ - t 0.10 N = 0.90 N e^ - t e^ t = 9 Taking the natural logarithm on both sides: t = 9 = 2 3 Since the decay constant = 2 T_ 1/2 , we get: t = 2 3 2 T_ 1/2 Substituting T_ 1/2 = 20.3 minutes : t = 2 0.4771 0.3010 20.3 t 3.17 20.3 64.35 minutes The closest value is 64.3 minutes .