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:
[tex]t_{1/2}=\frac{0.693}{k}[/tex]
where,
[tex]t_{1/2}[/tex] = half life of the reaction = 5720 years
Putting values in above equation, we get:
[tex]k=\frac{0.693}{5720yrs}=1.21\times 10^{-4}yrs^{-1}[/tex]
Rate law expression for first order kinetics is given by the equation:
[tex]k=\frac{2.303}{t}\log\frac{[A_o]}{[A]}[/tex]
where,
k = rate constant = [tex]1.21\times 10^{-4}yr^{-1}[/tex]
t = time taken for decay process = ? yr
[tex][A_o][/tex] = 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:
[tex]1.21\times 10^{-4}=\frac{2.303}{t}\log\frac{100}{11.5}\\\\t=37056.3yrs[/tex]
Hence, the sample of Carbon-14 isotope is 37056.3 years old
