Answer :
Final answer:
Using the half-life of Uranium-235, nearly all of the original 64 kg remains today, as its half-life is 700 million years compared to the 78 years since the bomb was dropped. Therefore, approximately 1.64 x 10²⁵ atoms of U-235 are still present. This demonstrates the stability and longevity of radioactive isotopes.
Explanation:
Calculating Remaining Atoms of Uranium-235
To determine how many atoms of Uranium-235 (U-235) remain today from the original 64 kg used in the Little Boy bomb dropped on Hiroshima, we can use the concept of half-life and the following formula:
Formula
N = N₀ (1/2)^(t/T_half)
- N = number of atoms remaining
- N₀ = initial number of atoms
- t = time elapsed
- T_half = half-life of U-235
First, we need to calculate the initial number of U-235 atoms:
1. Convert mass to moles:
Using the molar mass of U-235, which is approximately 235 g/mol:
64 kg = 64000 g
Number of moles (n) = mass/molar mass = 64000 g / 235 g/mol ≈ 272.34 moles.
2. Calculate number of atoms:
Using Avogadro's number (approximately 6.022 x 10²³ atoms/mol):
Number of atoms (N₀) = n × Avogadro's number = 272.34 moles × 6.022 x 10²³ atoms/mol ≈ 1.64 x 10²⁵ atoms.
3. Time elapsed since August 6, 1945:
As of today (let's assume it's October 2023), the time elapsed is:
2023 - 1945 = 78 years.
4. Convert years to years (since the half-life is already in years):
T_half = 700 million years.
5. Calculate remaining atoms:
N = N₀ (1/2)^(t/T_half)
N = 1.64 x 10²⁵ * (1/2)^(78/700000000) ≈ 1.64 x 10²⁵.
This indicates that nearly all of the original U-235 atoms would still be present and undergoing radioactive decay today due to the extremely long half-life relative to the time elapsed since the bomb's detonation.
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