9.5 Nuclear Reactions and Energy Release

Nuclear vs Chemical Reactions

Nuclear reactions involve changes to atomic nuclei, releasing enormous energy through mass-to-energy conversion. Unlike chemical reactions, they can transmute one element into another and emit subatomic particles. From fusion powering stars to fission in nuclear reactors, these processes shape both our universe and our technology.

Fundamental Differences

Nuclear and chemical reactions differ in almost every way that matters:

• What changes: Nuclear reactions alter the nucleus itself, while chemical reactions only rearrange electrons around unchanged nuclei.
• Element identity: Nuclear reactions transmute elements (turning one element into another). Chemical reactions never change elemental identity.
• Energy scale: Nuclear reactions release millions of times more energy than chemical reactions, because they convert mass directly into energy via $E = mc^2$.
• Particles emitted: Nuclear reactions can emit protons, neutrons, electrons, positrons, neutrinos, and gamma rays. Chemical reactions exchange or share electrons but don't produce subatomic particles.
• Timescale: Individual nuclear reactions happen in fractions of a second. A fission chain reaction can proceed extraordinarily fast once initiated.

Governing Forces and Scale

• Nuclear reactions are governed by the strong nuclear force (which binds nucleons together) and the weak nuclear force (responsible for beta decay). Chemical reactions are governed by the electromagnetic force between charged particles.
• Nuclear reactions occur at the scale of the nucleus, roughly $10^{-15}$ m. Chemical reactions involve electron clouds at atomic scales, roughly $10^{-10}$ m.
• Energies in nuclear reactions are measured in MeV (mega-electron volts), while chemical reaction energies are typically just a few eV. That's a factor of about $10^6$ difference.
• Nuclear reactions can change which isotope you have. Chemical reactions cannot alter isotopic composition.

Q-value in Nuclear Reactions

Definition and Significance

The Q-value tells you how much energy a nuclear reaction releases or absorbs. It's calculated from the mass difference between reactants and products:

$Q = (m_{\text{reactants}} - m_{\text{products}}) \cdot c^2$

• A positive Q-value means the reaction is exothermic: it releases energy. The products have less total mass than the reactants, and that "missing" mass has become kinetic energy and radiation.
• A negative Q-value means the reaction is endothermic: you need to supply energy to make it happen. The products are more massive than the reactants.
• The Q-value is directly tied to the binding energy per nucleon of the nuclei involved. Reactions that produce more tightly bound nuclei (higher binding energy per nucleon) release energy.
• The Q-value determines how the released energy is distributed among the kinetic energies of the products and any emitted radiation.

Q-value in Fusion and Fission

The binding energy curve explains why both fusion and fission can release energy:

• Fusion of light nuclei produces nuclei closer to the peak of the binding energy curve (near iron-56, $A \approx 56$), so the products are more tightly bound and energy is released. The fusion of deuterium and tritium, for example, releases 17.6 MeV per reaction.
• Fission of heavy nuclei (like uranium-235) splits them into medium-mass fragments that are also more tightly bound than the original heavy nucleus. Fission of $^{235}_{92}U$ releases approximately 200 MeV per event.

Both processes move nuclei toward the iron peak from opposite directions on the binding energy curve. That's why iron and nickel are the most stable nuclei: you can't extract energy by either fusing or fissioning them.

Balancing Nuclear Reactions

Fundamental Rules

Every nuclear equation must obey two conservation laws:

1. Mass number (A) is conserved. The total number of nucleons (protons + neutrons) on the left must equal the total on the right.
2. Atomic number (Z) is conserved. The total charge (number of protons) on the left must equal the total on the right.

Common particles you'll see in nuclear equations:

• Proton: $^{1}_{1}H$
• Neutron: $^{1}_{0}n$
• Alpha particle: $^{4}_{2}He$
• Beta particle (electron): $^{0}_{-1}e$
• Positron: $^{0}_{+1}e$
• Gamma ray: $\gamma$ (no mass number or charge)

To balance a nuclear equation, add up all the A values on each side and confirm they match, then do the same for Z values. If a particle is unknown, solve for its A and Z using these two constraints.

Types of Nuclear Reactions

Decay processes are spontaneous nuclear reactions:

• Alpha decay reduces A by 4 and Z by 2:

$^{238}_{92}U \rightarrow ^{234}_{90}Th + ^{4}_{2}He$

• Beta decay (electron emission) converts a neutron into a proton, increasing Z by 1 while A stays the same. An antineutrino is also emitted:

$^{14}_{6}C \rightarrow ^{14}_{7}N + ^{0}_{-1}e + \bar{\nu}_e$

• Gamma decay releases energy as a high-energy photon without changing A or Z. It typically follows alpha or beta decay when the daughter nucleus is in an excited state.

Induced reactions are triggered by bombarding a nucleus with a particle:

• Neutron capture occurs when a nucleus absorbs a neutron:

$^{235}_{92}U + ^{1}_{0}n \rightarrow ^{236}_{92}U^*$

The asterisk indicates the resulting nucleus is in an excited state. Induced reactions follow the same balancing rules as spontaneous decay.

Energy Release in Nuclear Reactions

Calculation Methods

Here's how to calculate the energy released in a nuclear reaction, step by step:

1. Look up the atomic masses of all reactants and products. Use atomic mass units (u). Using atomic masses (rather than nuclear masses) automatically accounts for electron masses in most reactions.

2. Calculate the mass defect $\Delta m$: $\Delta m = \sum m_{\text{reactants}} - \sum m_{\text{products}}$

3. Convert to energy using $E = \Delta m \cdot c^2$. The conversion factor is $1 \text{ u} = 931.5 \text{ MeV}/c^2$, so: $E = \Delta m \times 931.5 \text{ MeV/u}$

A positive $\Delta m$ means mass was lost and converted to energy (exothermic). The released energy appears as kinetic energy of the products and as gamma radiation.

Energy Release in Fusion and Fission

Fusion example: Deuterium-tritium fusion

$^{2}_{1}H + ^{3}_{1}H \rightarrow ^{4}_{2}He + ^{1}_{0}n + 17.6 \text{ MeV}$

The 17.6 MeV comes from the mass difference between the reactants and products. Most of this energy goes into kinetic energy of the neutron (about 14.1 MeV) and the alpha particle (about 3.5 MeV). Per nucleon, fusion releases more energy than fission, which is why it powers stars.

Fission example: Uranium-235 fission

$^{235}_{92}U + ^{1}_{0}n \rightarrow ^{141}_{56}Ba + ^{92}_{36}Kr + 3 \, ^{1}_{0}n + \sim 200 \text{ MeV}$

The ~200 MeV is distributed among kinetic energy of the fission fragments (~165 MeV), kinetic energy of the neutrons (~5 MeV), gamma rays, and energy carried by neutrinos from subsequent beta decays of the fragments. The three released neutrons are what makes a chain reaction possible: each fission event can trigger additional fissions.

On the binding energy curve, both reactions move their products toward the iron-56 peak, which is why both release energy despite going in opposite directions (light nuclei fusing vs. heavy nuclei splitting).