Question
A meteorite is determined to be 4.5 billion years old. It now has 78 atoms of lead-206. Assuming no loss of daughter isotopes, how many atoms of uranium-238 did the meteorite have to begin with? The half-life of uranium-238 is 4.5 billion years.
78
238
156
312
78
238
156
312
Answers
GPT-4o mini
To determine how many atoms of uranium-238 the meteorite initially had, we can use the concept of radioactive decay. Uranium-238 decays to lead-206, and since we know the half-life of uranium-238 is 4.5 billion years, the meteorite has gone through one half-life to reach its current state.
1. After one half-life (4.5 billion years), half of the original uranium-238 would have decayed. If the initial number of uranium-238 atoms is \(N_0\), after one half-life (4.5 billion years), the number of uranium-238 atoms remaining would be \(N_0 / 2\), and the number of lead-206 atoms produced would be equal to the number of uranium-238 atoms that have decayed.
Thus, if there are now 78 atoms of lead-206, that means 78 atoms of uranium-238 have decayed.
2. Therefore, the number of remaining uranium-238 atoms is:
\[
N_0 / 2 = N_0 - 78
\]
3. Rearranging this gives us:
\[
N_0 = 2 \times 78 = 156 \text{ atoms}
\]
So the initial number of uranium-238 atoms in the meteorite was 156.
The answer is **156**.
1. After one half-life (4.5 billion years), half of the original uranium-238 would have decayed. If the initial number of uranium-238 atoms is \(N_0\), after one half-life (4.5 billion years), the number of uranium-238 atoms remaining would be \(N_0 / 2\), and the number of lead-206 atoms produced would be equal to the number of uranium-238 atoms that have decayed.
Thus, if there are now 78 atoms of lead-206, that means 78 atoms of uranium-238 have decayed.
2. Therefore, the number of remaining uranium-238 atoms is:
\[
N_0 / 2 = N_0 - 78
\]
3. Rearranging this gives us:
\[
N_0 = 2 \times 78 = 156 \text{ atoms}
\]
So the initial number of uranium-238 atoms in the meteorite was 156.
The answer is **156**.