A sample initially contains only U-238 isotope of uranium. With time, some of the U-238 radioactively decays into \(\mathrm{Pb}-206\) while the rest of it remains undisintegrated.
When the age of the sample is \(\mathbf{P} \times 10^8\) years, the ratio of mass of \(\mathrm{Pb}-206\) to that of \(\mathrm{U}-238\) in the sample is found to be 7 . The value of \(\mathbf{P}\) is ______
[Given: Half-life of U-238 is \(4.5 \times 10^9\) years; \(\log _c 2=0.693\) ]

\({ }_{92} \mathrm{U}^{238} \quad \longrightarrow \quad{ }_{82} \mathrm{~Pb}^{206}+8{ }_2 \mathrm{He}^4+6_{-1} \beta^0\)
\(\mathrm{t}=0 \quad \mathrm{~N}_0=\left(\frac{1}{238}+\frac{7}{206}\right)\) moles \(\mathrm{t}=\mathrm{tN}_{\mathrm{t}}=\frac{1}{238}\) moles \(\mathrm{x}=\frac{7}{206}\) moles
Life of sample \(\rightarrow t\) years
\([A]_0 \propto\) Initial mole of \(\mathrm{U}-238\)
\([A]]_t \propto\) Final mole of U-238
\(\begin{aligned}
& \frac{|A|_0}{|A|_t}=\frac{\frac{1}{228}+\frac{7}{206}}{\frac{1}{238}} \\
& =\frac{0.0042+0.0340}{0.0042} \\
& =9.1 \\
& =\frac{2.303 \log 2 \times t}{4.5 \times 10^9}=2.303 \log 9.1 \\
& t=14.27 \times 10^9 \text { years } \\
& =142.7 \times 10^8 \text { years } \\
& P=142.7 \\
& P=143
\end{aligned}\)