U238 and U235 occur in nature in an atomic ratio 140 : 1. Assuming that at the time of earth’s formation the two isotopes were present in equal amounts. If the age of the earth is n × 107 years, then find the value of n. (Round off to the nearest integer)[half-life of U238 = 4.5 × 109 yrs, half-life of U235 = 7.13 × 108 yrs , ln⁡140ln⁡2=7.13]

N238=N0e−λ238TN235=N0e−λ235T⇒1401=e(λ235−λ238)TT=604.1×107 yrs

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$U^{238} a n d U^{235}$ occur in nature in an atomic ratio 140 : 1. Assuming that at the time of earth’s formation the two isotopes were present in equal amounts. If the age of the earth is n × 10⁷ years, then find the value of n. (Round off to the nearest integer)
[half-life of U²³⁸ = 4.5 × 10⁹ yrs, half-life of U²³⁵= 7.13 × 10⁸ yrs , $\frac{\ln 140}{\ln 2} = 7.13$ ]

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answer is 604.

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Detailed Solution

$\frac{N_{238} = N_{0} e^{- λ_{238} T}}{N_{235} = N_{0} e^{- λ_{235} T}} \Rightarrow \frac{140}{1} = e^{(λ_{235} - λ_{238}) T}$

$T = 604.1 \times 10^{7} y r s$

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