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