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 $\mathbf{P} imes 10^8$ years, the ratio of mass of $\mathrm{Pb}-206$ to that of…

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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 $\mathbf{P} 	imes 10^8$ 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 $4.5 	imes 10^9$ years; $\log _c 2=0.693$ ]

Aicharya · Accepted Answer

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

A sample initially contains only U-238 isotope of uranium. With time, some of the U-238 radioactively decays…

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