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  4. Half-life of Ra is 1600 years if initial mass of radium is 400 g. The time after which it decays 100 gm, will be:

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Q. Half-life of Ra is 1600 years if initial mass of radium is 400 g. The time after which it decays 100 gm, will be:

JIPMERJIPMER 2002

Solution:

For the relation $ N={{N}_{0}}{{e}^{-\lambda t}} $ where $ \lambda =\frac{0.693}{{{T}_{1/2}}} $ Hence $ {{T}_{1/2}}=1600\, $ years, $ {{N}_{0}}=400g, $ $ N=400-100=300g $ so, $ -\frac{0.693t}{{{T}_{1/2}}}=\frac{300}{400} $ or $ \frac{0.693}{{{T}_{1/2}}}t=\ln \frac{3}{4} $ so $ t=\frac{{{T}_{1/2}}\ln \frac{3}{4}}{0.693}=\frac{1600\,\ln \frac{3}{4}}{0.693} $ $ =4800\text{ }years $