Answer : The age of the rock is, 2.60 billion years
Explanation :
Half-life = 1.3 billion 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}{1.3\text{ billion years}}[/tex]
[tex]k=0.533\text{ billion 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]0.533\text{ billion years}^{-1}[/tex]
t = time passed by the sample = ?
a = initial amount of the reactant = 12 mg
a - x = amount left after decay process = 3 mg
Now put all the given values in above equation, we get
[tex]t=\frac{2.303}{0.533}\log\frac{12}{3}[/tex]
[tex]t=2.60\text{ billion years}[/tex]
Therefore, the age of the rock is, 2.60 billion years
