Nuclear Reactions, Fission and Fusion: Quick Revision

A brief note on Nuclear Reactions, Fission and Fusion. Read it before a test. Try each question at the end before you open its answer.

Reactions and the Q value

• Conserved: the total mass number, the charge, the energy, the momentum and the angular momentum.
• Q value: $Q = \left(\sum m_\text{before} - \sum m_\text{after}\right) \times 931.494\ \text{MeV/u}$, with atomic masses. $Q > 0$ releases energy, $Q < 0$ needs it.
• Threshold: $K_\text{th} = |Q|\,\dfrac{m_a + m_X}{m_X}$ for a target at rest.
• Sharing from rest: the lighter product takes the larger share, $K_1 = Q\,\dfrac{m_2}{m_1 + m_2}$.

Fission and the reactor

• Fission: a slow neutron splits uranium-235 into two fragments, $2$ or $3$ neutrons and about $200$ MeV. One kilogram gives $8.2 \times 10^{13}$ J.
• Chain reaction: the multiplication factor $k$ is $< 1$ (dies out), $= 1$ (steady, critical) or $> 1$ (grows as $k^g$).
• Reactor: fuel rods, moderator (slows neutrons: water, heavy water, graphite), control rods (absorb neutrons: cadmium, boron), coolant, shield.
• Fissions each second: $\dfrac{P}{E}$. Fuel in a day: the fissions in a day over $N_A$, times $235$ g.

Fusion

• Barrier: $U = \dfrac{1.44\,Z_1Z_2}{d}$ MeV, with $d$ in fm. For two deuterons it is about $0.48$ MeV.
• Heat: the average energy is $\tfrac{3}{2}kT$. Fusion works at about $10^{8}$ K on Earth and $1.5 \times 10^{7}$ K in the Sun, thanks to tunnelling and the fast nuclei.
• The Sun: four protons give helium-4, two positrons, two neutrinos and $26.7$ MeV. On Earth: deuterium and tritium give helium-4, a neutron and $17.59$ MeV.

A worked example

A reactor makes $1500$ MW of heat, and one fission gives $200$ MeV. How many fissions take place in each second?

1. One fission in joule. $200 \times 1.602 \times 10^{-13}\ \text{J} = 3.20 \times 10^{-11}$ J.
2. The rate. $\dfrac{1.5 \times 10^{9}\ \text{W}}{3.20 \times 10^{-11}\ \text{J}} = 4.7 \times 10^{19}\ \text{s}^{-1}$.

Practise with new numbers

Check yourself

1. A reaction has the total mass $12.0000$ u before and $11.9950$ u after. What is $Q$, and is the reaction exothermic?

2. What does a moderator do, and what does a control rod do?

3. Why does fusion in the Sun need a temperature of millions of kelvin?

Sources

• NCERT Physics Part II (Class 12, rationalised edition), Chapter 13 Nuclei.
• OpenStax University Physics Volume 3, Chapter 10 Nuclear Physics.