Question

A sample initially contains only U-238 isotope of uranium. With time, some of the U-238 radioactively decays into Pb-206 while the rest of it remains undisintegrated.

When the age of the sample is \(P\times10^8\) years, the ratio of mass of Pb-206 to that of U-238 in the sample is found to be 7. The value of P is ______.

[Given : Half-life of U-238 is \(4.5 \times 10^9\) years; \(\log_e2= 0.693\)]

Answer 1

Correct answer: 143

Life of sample \(\rightarrow t\) years

\([A] _0 \propto\) Initial mole of \(\mathrm{U}-238\)

\([A]]_t \propto\) Final mole of U-238

\(\frac{[A]_0}{[A]_t} =\frac{\frac{1}{238}+\frac{7}{206}}{\frac{1}{238}}\)

\(=\frac{0.0042+0.0340}{0.0042}\)

\(=9.1\)

\(=\frac{2.303 \log 2 \times t}{4.5 \times 10^9}=2.303 \log 9.1 \)

\(t =14.27 \times 10^9 \text { years }\)

\(=142.7 \times 10^9 \text { years } \)

\(P =142.7\)

\(P =143\)