Radioactive dating: Today, the uranium found on Earth contains 0.720% ²³⁵U (with a half-life of 0.700 billion years) and 99.28% ²³⁸U (with a half-life of 4.50 billion years). At a time 2.20 billion years ago, what percent of the uranium on Earth was ²³⁸U (assuming that no other uranium isotopes were present)?

Question

Quizplus · Accepted Answer

To find the percentage of Uranium-238 (U-238) 2.20 billion years ago, we use the decay formula for half-life. The half-life of U-238 is 4.50 billion years. The formula to calculate the remaining amount of a substance after a certain time is: $$N = N_0 \times (1/2)^{t/T}$$, where $N$ is the final amount, $N_0$ is the initial amount, $t$ is the time elapsed, and $T$ is the half-life of the substance. Since we are interested in the percentage of U-238, we can simplify our calculations by focusing on the ratio of U-238 to the total uranium, which initially is 99.28%. The time elapsed is 2.20 billion years, and the half-life of U-238 is 4.50 billion years. However, since the question asks for the percentage of U-238 2.20 billion years ago, and given that U-238 decays much slower than U-235, the percentage of U-238 would have been slightly higher than it is today, but the calculation provided does not directly apply to finding the initial percentage of U-238. Instead, we should consider the decay of U-235 and its effect on the overall uranium composition. The rapid decay of U-235 compared to U-238 would mean that U-238's relative abundance would increase over time. Given the options and the nature of radioactive decay, where U-238 becomes more dominant over long periods due to its longer half-life, the correct answer reflects the need to understand the dynamics of both isotopes' decay rates and their impact on the uranium composition over time. The calculation mistake here is in the explanation approach, which should focus on the comparative decay of both isotopes and the fact that the question implies a calculation error by not directly applying the decay formula to find the initial percentage of U-238. The correct approach would involve understanding that U-238, with its longer half-life, would have constituted a larger percentage of the uranium on Earth 2.20 billion years ago due to the faster decay of U-235, leading to an increase in the relative abundance of U-238 over time.

Radioactive Dating: Today, the Uranium Found on Earth Contains 0