Answer: 35,000 years
Explanation: Radioactive process follow first order kinetics.
To calculate the rate constant, we use the formula:
\(k=\frac{0.693}{t_{1/2}}\)
\(k=\frac{0.693}{5370years}}\)
\(k=1.2\times 10^{-4} years^{-1}\)
Also : \(a=\frac{a_o}{2^n}\)
where,
a = amount of reactant left after n-half lives
\(a_o\) = Initial amount of the reactant
n = number of half lives
= 6
Putting values in above equation, we get:
\(2^6=\frac{a_0}{a}\)
\(64=\frac{a_0}{a}\)
The rate expression for first order kinetics is given by:
\(t=\frac{2.303}{k}\log\frac{a_0}{a}\)
where,
k = rate constant
t = time taken for decay process
\(a_0\) = initial amount of the reactant
a = amount left after decay process
Putting values in above equation, we get:
\(t=\frac{2.303}{1.2\times 10^{-4}}\log{64}\)
\(t=35000years\)
