calculate the energy released in nuclear reactions using E = c2∆m

Mass Defect & Nuclear Binding Energy

What is Mass Defect?

When a nucleus is formed from protons and neutrons, the total mass of the nucleus is

slightly less than the sum of the masses of its individual nucleons.

The missing mass, called the mass defect (Δm), is converted into binding energy.

Think of it like cutting a cake: the cake you bake (the nucleus) weighs a little less than

the sum of all the ingredients (protons + neutrons). The “missing” weight is the cake’s

binding energy that keeps it together.

Binding Energy per Nucleon

The binding energy per nucleon tells us how tightly each nucleon is held in the nucleus.

It peaks around iron (Fe) and nickel (Ni), meaning these nuclei are the most stable.

🔬 Formula: \$E_{\text{bind}} = \frac{E}{A}\$, where \$E\$ is total binding energy and \$A\$ is mass number.

Calculating Energy Released

The energy released in a nuclear reaction is given by Einstein’s famous equation:


\$E = c^2 \Delta m\$

  1. Find the mass of reactants (sum of individual nucleon masses).

  2. Find the mass of products (mass of the resulting nucleus).

  3. Compute the mass defect:


    \$\Delta m = \text{mass of reactants} - \text{mass of products}\$

  4. Convert Δm to kilograms (if given in atomic mass units, 1 u = 1.6605×10⁻²⁷ kg).

  5. Use \$c = 3.00 \times 10^8 \text{ m/s}\$ and calculate:


    \$E = (3.00 \times 10^8)^2 \times \Delta m\$

  6. Result is in joules (J). To get MeV, divide by \$1.602 \times 10^{-13}\$ J/MeV.

💥 Example: Fusion of two deuterium nuclei (\$^2\$H + \$^2\$H → \$^3\$He + n)

StepValue (u)
Mass of reactants1.007825 + 1.007825 = 2.015650
Mass of products3.016029 (He) + 1.008665 (n) = 4.024694
Δm (u)-2.009044
Δm (kg)-3.34 × 10⁻²⁷
E (J)≈ 3.0 × 10⁻¹⁵ J
E (MeV)≈ 18.6 MeV

Exam Tips 📚

  • Always convert masses to kg before using \$E = c^2 \Delta m\$.

  • Remember \$c = 3.00 \times 10^8 \text{ m/s}\$.

  • Check units: Δm in kg → E in joules; to get MeV, divide by \$1.602 \times 10^{-13}\$ J/MeV.

  • When given mass defect directly, skip steps 1–3.

  • Use the binding energy per nucleon to compare stability of different nuclei.

  • Practice with both fusion (mass decreases) and fission (mass decreases) reactions.

📝 Remember: The larger the mass defect, the more energy is released.