Nuclear energy (fission, fusion)：練習問題

In a nuclear fusion study, the following reaction is considered for energy production:
\( ^2_1\text{H} + ^3_1\text{H} \rightarrow ^4_2\text{He} + ^1_0\text{n} \)
The rest masses of the involved particles are:
\( m(^2_1\text{H}) = 2.0141 \, \text{u} \)
\( m(^3_1\text{H}) = 3.0160 \, \text{u} \)
\( m(^4_2\text{He}) = 4.0026 \, \text{u} \)
\( m(^1_0\text{n}) = 1.0087 \, \text{u} \)
If a \( 1.00 \, \text{kg} \) fuel mixture contains deuterium (\( ^2_1\text{H} \)) and tritium (\( ^3_1\text{H} \)) in equal numbers of nuclei, what is the total energy released when all nuclei in the mixture undergo fusion? (Given: \( 1 \, \text{u} = 1.66 \times 10^{-27} \, \text{kg} \), \( c = 3.00 \times 10^8 \, \text{m/s} \))

Consider two nuclear reactions:
Reaction 1 (Fusion): A deuteron (\(_{1}^{2}\text{H}\)) and a triton (\(_{1}^{3}\text{H}\)) fuse to form a helium nucleus (\(_{2}^{4}\text{He}\)) and a neutron (\(_{0}^{1}\text{n}\)). This reaction releases approximately \(17.6 \text{ MeV}\) of energy.
Reaction 2 (Fission): A neutron induces fission in a Uranium-235 nucleus (\(_{92}^{235}\text{U}\)) to produce Barium-141 (\(_{56}^{141}\text{Ba}\)), Krypton-92 (\(_{36}^{92}\text{Kr}\)), and three neutrons. This reaction releases approximately \(200 \text{ MeV}\) of energy.

Using the approximate atomic masses for the primary fuel nuclei: mass of deuteron (\(_{1}^{2}\text{H}\)) \(\approx 2 \text{ u}\), mass of triton (\(_{1}^{3}\text{H}\)) \(\approx 3 \text{ u}\), and mass of Uranium-235 (\(_{92}^{235}\text{U}\)) \(\approx 235 \text{ u}\). Which of the following statements is a correct comparison between these two nuclear energy processes?

A nuclear fission reactor utilizes control rods and a moderator to maintain a stable and safe power output. Describe the distinct physical mechanisms by which these two components regulate the population of neutrons within the reactor core.

The following structured data compares the binding energy per nucleon (\(BE/A\)) for light and heavy nuclei:

1. Deuterium (\(_{1}^{2}\text{H}\)): \(1.11 \text{ MeV}\)
2. Helium-4 (\(_{2}^{4}\text{He}\)): \(7.07 \text{ MeV}\)
3. Uranium-235 (\(_{92}^{235}\text{U}\)): \(7.59 \text{ MeV}\)
4. Fission fragments (average): \(8.40 \text{ MeV}\)

Based on this data, show why nuclear fusion of light nuclei is more energetic per unit mass than the fission of heavy nuclei.

The binding energy per nucleon of \(_{92}^{235}\text{U}\) is \(7.59 \text{ MeV}\), and the average binding energy per nucleon of its resulting fragments after fission is \(8.40 \text{ MeV}\). Calculate the total energy released in \(\text{MeV}\) when a single Uranium-235 nucleus undergoes fission, assuming the number of nucleons is conserved and no energy is carried away by neutrons.

The binding energy per nucleon is a key factor in determining the energy released in nuclear reactions.
(a) Consider a fusion reaction where two deuterium nuclei (\(_{1}^{2}\text{H}\)) fuse to form a helium nucleus (\(_{2}^{4}\text{He}\)). Given the binding energy per nucleon of \(_{1}^{2}\text{H}\) is 1.11 MeV and for \(_{2}^{4}\text{He}\) is 7.07 MeV, calculate the energy released in this single fusion event.
(b) Explain, with reference to the binding energy per nucleon curve, why energy is released during the fission of very heavy nuclei but absorbed during the fusion of the same heavy nuclei.

To achieve nuclear fusion on Earth, deuterium nuclei must overcome their mutual electrostatic repulsion.
(a) Calculate the electrostatic potential energy between two deuterium nuclei when they are separated by a distance of \(2.5 \times 10^{-15}\) m.
(b) Assuming the nuclei must have an average kinetic energy equal to this potential energy to fuse, estimate the temperature required for a plasma of deuterium.
(c) Explain why, in practice, fusion can occur at temperatures lower than the value calculated in (b).
(Given: \(\epsilon_0 = 8.85 \times 10^{-12} \text{ F m}^{-1}\), \(e = 1.6 \times 10^{-19}\) C, \(k = 1.38 \times 10^{-23} \text{ J K}^{-1}\), and \(\text{K.E.}_{avg} = \frac{3}{2}kT\).)