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Chapter 21: Problem 79

(a) Calculate the energy released when a \({ }^{238} \mathrm{U}\) isotope decays to \({ }^{234} \mathrm{Th} .\) The atomic masses are given by: \(^{238} \mathrm{U}: 238.0508 \mathrm{amu} ;{ }^{234} \mathrm{Th}: 234.0436 \mathrm{amu} ;{ }^{4} \mathrm{He}\) 4.0026 amu. (b) The energy released in (a) is transformed into the kinetic energy of the recoiling \({ }^{234} \mathrm{Th}\) nucleus and the \(\alpha\) particle. Which of the two will move away faster? Explain.

Short Answer

Expert verified
The energy released during the decay is \(4.28 \, \text{MeV}\). The alpha particle (Helium-4 nucleus) moves away faster post-decay.

Step by step solution

01

Calculate Mass Deficit

The change in mass, which is often referred to as the 'mass deficit', results in energy release. Calculate it by subtracting the mass of the products (Thorium-234 and Helium-4) from the initial mass of Uranium-238. We get: \( \Delta m = m_{^{238}U} - (m_{^{234}Th} + m_{^{4}He}) = 238.0508 \, \text{amu} - (234.0436 \, \text{amu} + 4.0026 \, \text{amu}) = 0.0046 \, \text{amu} \)
02

Convert Mass Deficit to Energy

Using Einstein's relation \(E = \Delta mc^2\), where \(\Delta m\) is the mass difference and \(c\) is the speed of light. However, when working in atomic mass units, we use the conversion factor that \(1 \, \text{amu} = 931.5 \, \text{MeV}/c^2\). Thus, the energy released (E) is:\(E = \Delta m \times 931.5 \, \text{MeV}/c^2 = 0.0046 \, \text{amu} \times 931.5 \, \text{MeV}/c^2 = 4.28 \, \text{MeV}\)
03

Consider Momentum Conservation

The momentum of the system before decay equals the momentum of the system after decay. Before decay, the Uranium-238 atom was at rest, so its momentum was zero. After decay, the momentum of the Thorium-234 and the alpha particle must therefore also add up to zero due to the law of conservation of momentum. This results in an 'equal and opposite' momentum scenario, meaning the larger mass (Thorium-234) will have a smaller speed, while the smaller mass (alpha particle) will have a larger speed.

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