Answer : The correct option is, (C) 3.75 billion years ago.
Explanation :
Half-life = 1.25 billion years
First we have to calculate the rate constant, we use the formula :
\(k=\frac{0.693}{t_{1/2}}\)
\(k=\frac{0.693}{1.25\text{ years}}\)
\(k=0.554\text{ billion years}^{-1}\)
Now we have to calculate the time passed.
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 = \(0.554\text{ billion years}^{-1}\)
t = time passed by the sample = ?
a = let initial amount of the reactant = X g
a - x = amount left after decay process = \(12.5\% \times (x)=\frac{12.5}{100}\times (X)=0.125Xg\)
Now put all the given values in above equation, we get
\(t=\frac{2.303}{0.554}\log\frac{X}{0.125X}\)
\(t=3.75\text{ billion years}\)
Therefore, the time ago the rock formed was \(3.75\text{ billion years}\)
