Problem 2: A certain radioactive isotope has half-life of 8 hours. A solution containing 500 million atoms of this isotope is prepared. How many atoms of this isotope have not disintegrated (a) 8 hours (b) 24 hours?
Solution
Number of atoms decayed in the first half-life = ½ (N0) = ½ (500 × 106) = 250 × 106 atoms.
Remaining amount = (500 – 250) × 106 = 250 × 106 = 250 million atoms.
Similarly, in the second half-life, the atoms decayed are = ½ (250 × 106) = 125× 106 atoms. The same amount will be left after 2nd half-life, too.
And the number of atoms decayed in the 3rd half-life = ½ (125 × 106) = 62.5 × 106 atoms.
The amount of isotope remained after 3rd half-life, 24 hours = (125 – 62.5) × 106 = 62.5 × 106 atoms = 62 million atoms.
Note: An alternate method is, amount of the substance left after 3 half-lifes = (½)1/3 of the total initial amount. Therefore, amount left after three half-lifes = (½)1/3 × 500 × 106 = 0.125 × 500 × 106 = 62.5 × 106 = 62.5 million atoms.
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