Question

Archaeologists can determine the age of artifacts made of wood or bone by measuring the amount of the radioactive isotope 14C present in the object. The amount of isotope decreases in a first-order process. If 25.7% of the original amount of 14C is present in a wooden tool at the time of analysis, what is the age of the tool? The half-life of 14C is 5,730 yr. Give your answer in scientific notation.

Answer 1

Answer: The age of the tool is $1.12\times 10^4years$

Explanation:

Half-life of carbon-14 = 5730 years

First we have to calculate the rate constant, we use the formula :

$k=\frac{0.693}{5730\text{years}}$

$k=1.21\times 10^{-4}\text{years}^{-1}$

Now we have to calculate the age of the tool:

Expression for rate law for first order kinetics is given by:

$t=\frac{2.303}{k}\log\frac{a}{a-x}$

where,

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

t = age of sample  = ?

a = let initial amount of the reactant  = 100 g

a - x = amount left after decay process  = $\frac{25.7}{100}\times 100=25.7$

Now put all the given values in above equation, we get

$t==\frac{2.303}{1.21\times 10^{-4}}\log\frac{100}{25.7}$

$t=1.12\times 10^4years$

Thus the age of the tool is $1.12\times 10^4years$