**Question 6: Ra**^{226} has half life of 1600 years. (a) What fraction remains after 4800 years. (b) How many half-lives does it have in 9600 years?

**ANSWER**

- Now 4800 ÷ 1600 = 3, therefore, Radium has 3 half-lives in 4800 years.

Now let us suppose that there was N_{0} number of Rn 4800 years (3 half-lives) ago. Then in one half-life, the decayed or the number left is = T_{1/2} = ½ N_{0}.

Therefore, after 1600 years, ½ N_{0} number of atoms will be there. In the next half-life, ½(N_{0}/2) of the atoms will decay and the same number of atoms, N_{0}/4 will left.

Therefore, after 2 half-lives, i-e, 3200 years, the number of atoms present will be N_{0}/4.

After another 1600 years, i-e, after 4800 years (at this moment as initially supposed), half of this amount of atoms would have decayed and half left. Therefore, during 3 half-lives, the number of atoms left is ½(N_{0}/4) = N_{0}/8.

Therefore, number of atoms decayed in 4800 years

Therefore, the fraction of atoms left un-decayed in 4800 years

- Given that 1600 years is one half-life, therefore, 9600 years would have 9600 ÷ 1600 = 6 half-lives.

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