Half life is defined as the time when the final concentration reduced to half of the initial value.
Thus, for first half life, the remaining percentage of the reactant is 50 percent that is [tex]\frac{50}{100}=\frac{1}{2}[/tex]
Now, for second half life, the remaining percentage will be 50 percent of the remaining 50 percentage that is [tex]\frac{50}{2}=25[/tex] [tex]\frac{25}{100}=\frac{1}{4}[/tex] or [tex]\frac{1}{2^{2}}[/tex]
Thus, general formula for n half lives =[tex]\frac{1}{2^{n}}[/tex]
Now, 98.4 years is equal to eighth times 12.3 years that is [tex]\frac{1}{2^{8}}[/tex] =[tex]\frac{1}{256}[/tex] remaining material.
Therefore, mass of the nuclide will remain after 98.4 years= [tex]\frac{48.0 mg}{256}[/tex]
= [tex]0.1875 mg[/tex]
Mass of remaining nuclide after 98.4 years = [tex]0.1875 mg[/tex]
