Answer: The age of the fossilized leaf is 13829 years.
Step-by-step explanation:
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
t = age of sample
a = let initial amount of the reactant = 100
a - x = amount left after decay process = [tex]\frac{18}{100}\times 100=18[/tex]
a) for completion of half life:
Half life is the amount of time taken by a radioactive material to decay to half of its original value.
[tex]t_{\frac{1}{2}}=\frac{0.693}{k}[/tex]
[tex]k=\frac{0.693}{5600years}=0.00012years^{-1}[/tex]
b) for finding the age
[tex]t=\frac{2.303}{0.00012}\log\frac{100}{18}[/tex]
[tex]t=13829years[/tex]
The age of the fossilized leaf is 13829 years.
The fossilized leaf is 13855 years old.
Half life is given by:
[tex]N(t)=N_o(\frac{1}{2} )^\frac{t}{t_\frac{1}{2} } \\\\Where\ N(t)\ is \ the \ value\ of\ the\ substance\ after\ t\ years,N_o\ is\ the\\original\ value\ and\ t_\frac{1}{2} \ is\ the\ half\ life[/tex]
Given that half life = 5600 years, N(t) = 18% of normal amount = 0.18N₀. Hence:
[tex]0.18N_o=N_o(\frac{1}{2} )^\frac{t}{5600} \\\\0.18=(\frac{1}{2} )^\frac{t}{5600}\\\\ln(0.18)=\frac{t}{5600}ln(0.5)\\\\t=13855\ years[/tex]
The fossilized leaf is 13855 years old.
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