Final answer:
The energy released in the α decay of 238U is 372.5 MeV. The fraction of the mass of a single 238U destroyed in the decay is 0.0168. Although the fractional mass loss is large for a single nucleus, it is difficult to observe for an entire macroscopic sample of uranium due to the small fraction of nuclei that decay on human timescales.
Step-by-step explanation:
The energy released in α decay can be calculated using Einstein's mass-energy equivalence equation, E = mc², where E is the energy, m is the mass lost, and c is the speed of light.
(a) To calculate the energy released in the α decay of 238U, we need to find the mass lost. The difference in mass between 238U and the daughter nuclide, 234Th, is 238 - 234 = 4 atomic mass units (u).
Using the equation E = mc², where c is approximately 3 x 10⁸ m/s, the energy released can be calculated as:
E = (4 u) x (1.66 x 10⁻²⁷ kg/u) x (3 x 10⁸ m/s)²
E = 5.96 x 10⁻¹¹ J
Converting the energy to MeV (million electron volts), we can use the unit conversion factor 1 MeV = 1.6 x 10⁻¹³ J:
E = (5.96 x 10⁻¹¹ J) / (1.6 x 10⁻¹³ J/MeV)
E = 372.5 MeV
(b) The fraction of the mass of a single 238U destroyed in the decay can be calculated as:
Fraction = (mass lost) / (mass of 238U)
Substituting the values, we get:
Fraction = (4 u) / (238 u)
Fraction = 0.0168
(c) Although the fractional mass loss is large for a single nucleus, it is difficult to observe for an entire macroscopic sample of uranium because the change in mass is proportional to the number of decaying nuclei. The macroscopic sample of uranium contains a large number of nuclei, and only a small fraction of them decay on human timescales. Thus, the change in mass due to decay is not detectable for a macroscopic sample.