5cMass–Energy Equivalence
Uncover the meaning of E = mc², explore how mass converts to energy in nuclear reactions, and discover why fission and fusion release such extraordinary amounts of energy.
E = mc² is arguably the most famous equation in science. It explains the energy of stars, the power of nuclear reactors, and the annihilation of matter — and it fundamentally changed our understanding of the universe.
Lesson Overview
Einstein's most famous equation, E = mc², reveals that mass and energy are equivalent and interconvertible. The rest energy of an object is E₀ = mc², where c ≈ 3×10⁸ m/s. The total relativistic energy is E = γmc², and the relativistic kinetic energy is KE = (γ − 1)mc². In nuclear reactions, a small mass defect Δm is converted into a large amount of energy ΔE = Δmc². This principle underlies nuclear fission (splitting heavy nuclei) and nuclear fusion (combining light nuclei), both of which release enormous amounts of energy.
Key Concepts
Mass–Energy Equivalence
E = mc²; mass and energy are interconvertible
Rest Energy
E₀ = mc²; energy an object has due to its mass alone
Relativistic KE
KE = (γ − 1)mc²; reduces to ½mv² at low speeds
Mass Defect (Δm)
Difference between reactant and product masses in a nuclear reaction
Binding Energy
Energy released when nucleons bind together; equals Δmc²
Fission vs Fusion
Fission: heavy nucleus splits; Fusion: light nuclei combine; both release energy
Calculate the rest energy of a 1 kg object. (c = 3×10⁸ m/s)
A nuclear reaction has a mass defect of Δm = 3.2×10⁻²⁸ kg. How much energy is released?
A proton (m = 1.67×10⁻²⁷ kg) moves at 0.6c (γ = 1.25). Calculate its relativistic kinetic energy.
The Sun converts about 4×10⁹ kg of mass to energy every second via fusion. How much power does this represent?
In a fission reaction, a uranium-235 nucleus absorbs a neutron and splits. The mass defect is 3.1×10⁻²⁸ kg. How many such reactions are needed to release 1 J of energy?
Calculate the rest energy of an electron (m = 9.11×10⁻³¹ kg). Express your answer in joules.
Hint: Use E₀ = mc². Square the speed of light: c² = 9×10¹⁶ m²/s².
A nuclear reaction releases 4.5×10⁻¹² J of energy. What mass was converted to energy?
Hint: Rearrange E = mc² to find m = E/c².
A particle with mass m = 2×10⁻²⁷ kg moves at 0.8c (γ = 5/3). Calculate its relativistic kinetic energy.
Hint: Use KE = (γ − 1)mc². First find (γ − 1), then multiply by mc².
Explain why nuclear reactions release far more energy per kilogram than chemical reactions.
Hint: Think about what is being converted in each case — chemical bonds vs. nuclear mass defect.
In nuclear fusion, two deuterium nuclei fuse to form helium-3 plus a neutron. The mass defect is 5.5×10⁻³⁰ kg. Calculate the energy released per reaction.
Hint: Use ΔE = Δmc². This is the binding energy released.
Key Vocabulary
Mass–Energy Equivalence
Einstein's principle that mass and energy are interconvertible, expressed as E = mc². A small mass corresponds to an enormous amount of energy.
Example: The rest energy of 1 gram of matter is 9×10¹³ J — equivalent to about 21 kilotons of TNT.
Mass Defect
The difference between the total mass of separate nucleons and the mass of the assembled nucleus; this 'missing' mass is converted to binding energy.
Example: A helium-4 nucleus has less mass than two separate protons and two neutrons; the mass defect equals the binding energy divided by c².
Nuclear Fission
A nuclear reaction in which a heavy nucleus (such as uranium-235) splits into two smaller nuclei, releasing a large amount of energy and several neutrons.
Example: Nuclear power plants use controlled fission of uranium-235 to generate electricity.
Nuclear Fusion
A nuclear reaction in which two light nuclei (such as hydrogen isotopes) combine to form a heavier nucleus, releasing even more energy per kilogram than fission.
Example: The Sun generates energy through fusion of hydrogen nuclei into helium in its core.
Interactive Practice — 5 Questions
What does E = mc² tell us about mass and energy?
A nuclear reaction has a mass defect of 1×10⁻²⁸ kg. How much energy is released? (c = 3×10⁸ m/s)
The relativistic kinetic energy formula KE = (γ − 1)mc² reduces to which classical formula at low speeds?
Which process releases more energy per kilogram of fuel?
What is the "binding energy" of a nucleus?
Independent Practice
State E = mc² in words and explain what each symbol represents. Why is the energy so large even for small masses?
Calculate the rest energy of a proton (m = 1.67×10⁻²⁷ kg). Express your answer in joules and in MeV (1 MeV = 1.6×10⁻¹³ J).
A nuclear fission reaction releases 3.2×10⁻¹¹ J of energy. What mass was converted? How does this compare to the mass of a proton (1.67×10⁻²⁷ kg)?
Compare nuclear fission and nuclear fusion: what nuclei are involved, what products are formed, and which releases more energy per kilogram?
★ A hydrogen bomb uses fusion of deuterium (²H, m = 3.344×10⁻²⁷ kg) and tritium (³H, m = 5.008×10⁻²⁷ kg) to form helium-4 (m = 6.646×10⁻²⁷ kg) and a neutron (m = 1.675×10⁻²⁷ kg). Calculate the mass defect and energy released per reaction. How many reactions are needed to release 1 MJ of energy?
ChallengeCommon Mistakes
Thinking E = mc² only applies to nuclear bombs or extreme situations.
E = mc² applies to all energy changes, including chemical reactions — but the mass changes in chemical reactions are too tiny to measure with ordinary scales.
Using the classical KE = ½mv² formula for particles moving near the speed of light.
At relativistic speeds, use KE = (γ − 1)mc². The classical formula significantly underestimates KE at high speeds.
Confusing mass defect with the total mass of the nucleus.
Mass defect Δm is the small difference between the sum of nucleon masses and the actual nuclear mass. It is always a tiny fraction of the total mass.
Math Tips
Key formulas: Rest energy E₀ = mc². Energy from mass defect: ΔE = Δmc². Relativistic KE = (γ−1)mc². Always use c = 3×10⁸ m/s, so c² = 9×10¹⁶ m²/s².
To find mass from energy: m = E/c² = E / (9×10¹⁶). For nuclear problems, mass defect Δm = m_reactants − m_products (always positive for exothermic reactions).