Answer : The time passed in years is [tex]2.74\times 10^2\text{ years}[/tex]
Explanation :
Half-life of carbon-14 = 5730 years
First we have to calculate the rate constant, we use the formula :
[tex]k=\frac{0.693}{t_{1/2}}[/tex]
[tex]k=\frac{0.693}{5730\text{ years}}[/tex]
[tex]k=1.21\times 10^{-4}\text{ years}^{-1}[/tex]
Now we have to calculate the time passed.
Expression for rate law for first order kinetics is given by:
[tex]t=\frac{2.303}{k}\log\frac{a}{a-x}[/tex]
where,
k = rate constant = [tex]1.21\times 10^{-4}\text{ years}^{-1}[/tex]
t = time passed by the sample = ?
a = initial amount of the reactant disintegrate = 15.3
a - x = amount left after decay process = 14.8
Now put all the given values in above equation, we get
[tex]t=\frac{2.303}{1.21\times 10^{-4}}\log\frac{15.3}{14.8}[/tex]
[tex]t=274.64\text{ years}=2.74\times 10^2\text{ years}[/tex]
Therefore, the time passed in years is [tex]2.74\times 10^2\text{ years}[/tex]
