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 $\mathrm{U}-238$ is $4.5 \times 10^9$ years; $\log _e 2=0.693$ ]
An organic compound $\mathbf{P}$ having molecular formula $\mathrm{C}_6 \mathrm{H}_6 \mathrm{O}_3$ gives ferric chloride test and does not have intramolecular hydrogen bond. The compound $\mathbf{P}$ reacts with 3 equivalents of $\mathrm{NH}_2 \mathrm{OH}$ to produce oxime $\mathbf{Q}$. Treatment of $\mathbf{P}$ with excess methyl iodide in the presence of $\mathrm{KOH}$ produces compound $\mathbf{R}$ as the major product. Reaction of $\mathbf{R}$ with excess iso-butylmagnesium bromide followed by treatment with $\mathrm{H}_3 \mathrm{O}^{+}$gives compound $\mathbf{S}$ as the major product.
The total number of methyl $\left(-\mathrm{CH}_3\right)$ group(s) in compound $\mathbf{S}$ is ___.
An organic compound $\mathbf{P}$ with molecular formula $\mathrm{C}_9 \mathrm{H}_{18} \mathrm{O}_2$ decolorizes bromine water and also shows positive iodoform test. $\mathbf{P}$ on ozonolysis followed by treatment with $\mathrm{H}_2 \mathrm{O}_2$ gives $\mathbf{Q}$ and $\mathbf{R}$. While compound $\mathbf{Q}$ shows positive iodoform test, compound $\mathbf{R}$ does not give positive iodoform test. $\mathbf{Q}$ and $\mathbf{R}$ on oxidation with pyridinium chlorochromate (PCC) followed by heating give $\mathbf{S}$ and $\mathbf{T}$, respectively. Both $\mathbf{S}$ and $\mathbf{T}$ show positive iodoform test.
Complete copolymerization of 500 moles of $\mathbf{Q}$ and 500 moles of $\mathbf{R}$ gives one mole of a single acyclic copolymer $\mathbf{U}$.
[Given, atomic mass: $\mathrm{H}=1, \mathrm{C}=12, \mathrm{O}=16$ ]