Answer: It will take 5 billion years for the parent isotope to get reduced to one-sixteenth of its original amount.
Explanation:
Initial amount of an isotope =[tex]N_o[/tex]
Amount left after the radio decay = N=[tex]\frac{N_o}{16}=0.0625 N_o[/tex]
Decay constant= [tex]\lambda [/tex]
Half life of the sample: [tex]t_{\frac{1}{2}}[/tex]=1.25 billion years
Years pass before a sample of potassium-40 contains one-sixteenth the original amount of parent isotope be ,t= T
[tex]\lambda =\frac{0.693}{1.25 \text{billion years}}=0.5544 \text{billion years}^{-1}[/tex]
[tex]N=N_o\time e^{-\lambda t}[/tex]
[tex]\ln[\frac{N_o}{16}]=\ln[N_o]-0.5544 \text{billion years}^{-1}\times T[/tex]
[tex]\ln[\frac{1}{16}]=-0.5544 \text{billion years}^{-1}\times T[/tex]
[tex]-2.7725=-0.5544\text{billion years}^{-1}\times t[/tex]
[tex]T=\frac{-2.7725}{-0.5544\text{billion years}^{-1}}=5.00 \text{billion years}[/tex]
It will take 5 billion years for the parent isotope to get reduced to one-sixteenth of its original amount.
