Respuesta :
Answer : The time passed in years is [tex]2.83\times 10^3\text{ years}[/tex]
Explanation :
Half-life = 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 = let initial amount of the reactant = X g
a - x = amount left after decay process = [tex]71\% \times (x)=\frac{71}{100}\times (X)=0.71Xg[/tex]
Now put all the given values in above equation, we get
[tex]t=\frac{2.303}{1.21\times 10^{-4}}\log\frac{X}{0.71X}[/tex]
[tex]t=2831.00\text{ years}=2.83\times 10^3\text{ years}[/tex]
Therefore, the time passed in years is [tex]2.83\times 10^3\text{ years}[/tex]
