Question

Carbon-14 has a half-life of 5720 years and this is a first-order reaction. If a piece of wood has converted 88.5% of the carbon-14, then how old is it?

Answer 1

Answer: The sample of Carbon-14 isotope is 37056.3 years old

Explanation:

The equation used to calculate rate constant from given half life for first order kinetics:

$t_{1/2}=\frac{0.693}{k}$

where,

$t_{1/2}$ = half life of the reaction = 5720 years

Putting values in above equation, we get:

$k=\frac{0.693}{5720yrs}=1.21\times 10^{-4}yrs^{-1}$

Rate law expression for first order kinetics is given by the equation:

$k=\frac{2.303}{t}\log\frac{[A_o]}{[A]}$

where,

k = rate constant  = $1.21\times 10^{-4}yr^{-1}$

t = time taken for decay process = ? yr

$[A_o]$ = initial amount of the sample = 100 grams

[A] = amount left after decay process =  (100 - 88.5) = 11.5 grams

Putting values in above equation, we get:

$1.21\times 10^{-4}=\frac{2.303}{t}\log\frac{100}{11.5}\\\\t=37056.3yrs$

Hence, the sample of Carbon-14 isotope is 37056.3 years old