Question

7.3 The decay rate of the isotope carbon-14 (14C) is often used to establish the date on which carbon-containing matter died. In the upper atmosphere, cosmic radi- ation synthesizes 14C. This process balances the loss of 14C through radioactive decay. Living matter, which exchanges carbon with atmospheric carbon diox- ide and maintains its 14C level, produces 15.3 disintegrations per minute per gram of carbon it contains. Dead organisms no longer exchange carbon with the atmosphere, and the 14C content decreases with time because of radioactive decay. A 0.5 g sample of a plant from an excavation shows 3.5 disintegrations per minute from its 14C. How long ago did the plant die?

Answer 1

Half life of ¹⁴C = 5730 years = 5730 * 365* 24*60 min = 3011688000 min

then decay constant ( $\small \lambda$​​​​​​) = (ln 2/5730) = ( 0.693 /3011688000) = 2.30 *10^-10 min^-1

now,

initial activity; N₀ = 15.3 disintegration per minute

activity after time t  ; N_t = 3.5 disintegration per minute

now radioactive decay equation is

N_t = N₀ * e^{-$\small \lambda$t}

or , 3.5 = 15.3 * e^{-$\small \lambda$t}

or, e^{-$\small \lambda$t} = (3.5/15.3)

or, e^{-$\small \lambda$t} = 0.228

or, - $\small \lambda$t = ln (0.228)

or, - $\small \lambda$t = -1.475

or, 2.30 *10^-10 * t = 1.81

or, t = (1.475/2.30*10^-10) min

or, t = 6.41*10⁹ min

or, t = 12195 year.

hence the plant died 12195 years ago.