Physics

Physics · modern-physics

Problem

In the cosmic galaxy, the Sun is a mainsequence star, generating its energy mainly by nuclear fusion of hydrogen nuclei into helium. In its core, the Sun fuses hydrogen to produce deuterium (2H)(2 \mathrm{H}) and tritium (3H)(3 \mathrm{H}), then makes about 600 million metric tons of helium (4He) per second. Of course, there are also some relatively smaller portions of fission reactions in the Sun's core, e.g. a nuclear fission reaction with Uranium-235 (235U). The Fusion reaction: 2H+3H→4He+n+ Released Energy { }^{2} \mathrm{H}+{ }^{3} \mathrm{H} \rightarrow{ }^{4} \mathrm{He}+n+\text { Released Energy } The Fission reaction: 235U+n+( Initial Energy )→144Ba+90Kr+2n+ Released Energy { }^{235} U+n+(\text { Initial Energy }) \rightarrow{ }^{144} \mathrm{Ba}+{ }^{90} \mathrm{Kr}+2 n+\text { Released Energy } Isotope Mass (at rest)
Isotope NamesMass (at rest) (u)(\mathrm{u})
Deuterium (2H)\left({ }^{2} \mathrm{H}\right)2.0141
Tritium (3H)\left({ }^{3} \mathrm{H}\right)3.0160
Helium (4He)\left({ }^{4} \mathrm{He}\right)4.0026
Neutron (n)(\mathrm{n})1.0087
Uranium-235 (235U)\left({ }^{235} \mathrm{U}\right)235.1180
Barium-144 (144Ba)\left({ }^{144} \mathrm{Ba}\right)143.8812
Krypton-90 (90Kr)\left({ }^{90} \mathrm{Kr}\right)89.9471
Calculate the kinetic energy (in MeV\mathrm{MeV} ) released by the products in one fusion reaction. Given constants: - Proton mass, mp=1.67⋅10−27 kgm_{p}=1.67 \cdot 10^{-27} \mathrm{~kg} - Neutron mass, mn=1.67⋅10−27 kgm_{n}=1.67 \cdot 10^{-27} \mathrm{~kg} - Electron mass, me=9.11⋅10−31 kgm_{e}=9.11 \cdot 10^{-31} \mathrm{~kg} - Avogadro's constant, N0=6.02⋅1023 mol−1N_{0}=6.02 \cdot 10^{23} \mathrm{~mol}^{-1} - Universal gas constant, R=8.31 J/(mol⋅K)R=8.31 \mathrm{~J} /(\mathrm{mol} \cdot \mathrm{K}) - Boltzmann's constant, kB=1.38⋅10−23 J/Kk_{B}=1.38 \cdot 10^{-23} \mathrm{~J} / \mathrm{K} - Electron charge magnitude, e=1.60⋅10−19Ce=1.60 \cdot 10^{-19} \mathrm{C} - 1 electron volt, 1eV=1.60⋅10−19 J1 \mathrm{eV}=1.60 \cdot 10^{-19} \mathrm{~J} - Speed of light, c=3.00⋅108 m/sc=3.00 \cdot 10^{8} \mathrm{~m} / \mathrm{s} - Universal Gravitational constant, G=6.67⋅10−11( N⋅m2)/kg2G=6.67 \cdot 10^{-11}\left(\mathrm{~N} \cdot \mathrm{m}^{2}\right) / \mathrm{kg}^{2} - Solar Mass M⊙=1.988⋅1030 kgM_{\odot}=1.988 \cdot 10^{30} \mathrm{~kg} - Acceleration due to gravity, g=9.8 m/s2g=9.8 \mathrm{~m} / \mathrm{s}^{2} - 1 unified atomic mass unit, 1u=1.66⋅10−27 kg=931MeV/c21 \mathrm{u}=1.66 \cdot 10^{-27} \mathrm{~kg}=931 \mathrm{MeV} / \mathrm{c}^{2} - Planck's constant, h=6.63⋅10−34 J⋅s=4.41⋅10−15eV⋅sh=6.63 \cdot 10^{-34} \mathrm{~J} \cdot \mathrm{s}=4.41 \cdot 10^{-15} \mathrm{eV} \cdot \mathrm{s} - Permittivity of free space, ϵ0=8.85⋅10−12C2/(N⋅m2)\epsilon_{0}=8.85 \cdot 10^{-12} \mathrm{C}^{2} /\left(\mathrm{N} \cdot \mathrm{m}^{2}\right) - Coulomb's law constant, k=14πϵ0=8.99⋅109( N⋅m2)/C2k=\frac{1}{4 \pi \epsilon_{0}}=8.99 \cdot 10^{9}\left(\mathrm{~N} \cdot \mathrm{m}^{2}\right) / \mathrm{C}^{2} - Permeability of free space, μ0=4π⋅10−7 T⋅m/A\mu_{0}=4 \pi \cdot 10^{-7} \mathrm{~T} \cdot \mathrm{m} / \mathrm{A} - Magnetic constant, μ04π=1⋅10−7( T⋅m)/A\frac{\mu_{0}}{4 \pi}=1 \cdot 10^{-7}(\mathrm{~T} \cdot \mathrm{m}) / \mathrm{A} - 1 atmospheric pressure, 1 atm=1.01⋅105 N/m2=1.01⋅105 Pa1 \mathrm{~atm}=1.01 \cdot 10^{5} \mathrm{~N} / \mathrm{m}^{2}=1.01 \cdot 10^{5} \mathrm{~Pa} - Wien's displacement constant, b=2.9b=2.9. 10−3 m⋅K10^{-3} \mathrm{~m} \cdot \mathrm{K} - Stefan-Boltzmann constant, σ=5.67⋅10−8 W/m2/K4\sigma=5.67 \cdot 10^{-8} \mathrm{~W} / \mathrm{m}^{2} / \mathrm{K}^{4}
Intuitive explanation

Nuclear reactions release energy because the products weigh less than the reactants—that "missing mass" converts to kinetic energy via Einstein's E=mc2E = mc^2. For this fusion reaction, calculate the mass defect (initial mass minus final mass), convert it to energy using the given conversion factor (1 u = 931 MeV/c2c^2), and that energy becomes the kinetic energy of the helium nucleus and neutron flying apart.

Intuitive explanation

Nuclear reactions release energy because the products weigh less than the reactants—that "missing mass" converts to kinetic energy via Einstein's E=mc2E = mc^2. For this fusion reaction, calculate the mass defect (initial mass minus final mass), convert it to energy using the given conversion factor (1 u = 931 MeV/c2c^2), and that energy becomes the kinetic energy of the helium nucleus and neutron flying apart.

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