The lifetime of uranium 238 is 6.0 × 10⁹ y, and that of uranium 235 is 1.0 × 10⁹ y. The present abundance of uranium 238 is 140 times the abundance of uranium 235. What was the ratio of abundances 1 billion years ago?

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Quizplus · Accepted Answer

The ratio of abundances 1 billion years ago can be calculated using the following formula:$$N_t = N_0e^{-\lambda t}$$where:* $$N_t$$ is the abundance at time t* $$N_0$$ is the initial abundance* $$\lambda$$ is the decay constant* t is the timeFor uranium 238, we have:$$N_t = N_0e^{-(\ln 2/6.0 	imes 10^9) 	imes 10^9}$$$$N_t = N_0e^{-0.115}$$$$N_t = 0.89N_0$$For uranium 235, we have:$$N_t = N_0e^{-(\ln 2/1.0 	imes 10^9) 	imes 10^9}$$$$N_t = N_0e^{-0.693}$$$$N_t = 0.5N_0$$Therefore, the ratio of abundances 1 billion years ago was:$$\frac{N_{238}}{N_{235}} = \frac{0.89N_0}{0.5N_0} = 1.78$$$$\frac{N_{238}}{N_{235}} = 61:1$$

The Lifetime of Uranium 238 Is 6