Discussion
The half-life of \(^{238}\)U is 4468×10\(^{6}\) years and \(^{235}\)U is 704×10\(^{6}\) years. The ratio \(^{235}\)U/\(^{238}\)U in a metamorphosed sample is 0.007257. Given that the ratio was 0.4 at the time of formation, calculate the metamorphosed sample age.
Answer: B
The age is 4.8 × 10\(^{9}\) years.
Both uranium isotopes decay, so their ratio R changes as \(R = R_0 e^{-(\lambda_{235} - \lambda_{238})t}\), with \(\lambda = \ln 2 / t_{1/2}\).
\[ \begin{aligned} \lambda_{235} &= 0.693/7.04\times10^{8} = 9.85\times10^{-10} \\ \lambda_{238} &= 0.693/4.468\times10^{9} = 1.55\times10^{-10} \\ \Delta\lambda &= 8.29\times10^{-10}\ \text{yr}^{-1} \end{aligned} \]
\[ \begin{aligned} t &= \frac{\ln(0.4/0.007257)}{\Delta\lambda} \\ &= \frac{4.01}{8.29\times10^{-10}} \\ &\approx 4.8\times10^{9}\ \text{years} \end{aligned} \]
The other options have the same mantissa with the wrong power of ten. An age of 10\(^{6}\) or 10\(^{7}\) years would barely change the ratio, since both half-lives are hundreds of millions of years or more.
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