Mass-Energy Equivalence

Up to now in this course, mass and energy have been separate, conserved quantities. In mechanics, you conserved kinetic energy (in elastic collisions) and momentum; in thermodynamics, you conserved internal energy. Mass was a fixed property of each object. The great insight of Einstein's special relativity (1905) was that mass and energy are not independent at all — they are two different expressions of the same underlying quantity, related by what is probably the most famous equation in all of physics:

E = mc²

This equation underpins the whole of nuclear and particle physics, from the fission reactor to the PET scanner. In this lesson we unpack what it means, introduce the atomic mass unit and the MeV/c², and apply it to simple nuclear processes. The material forms section 6.4.3 of the OCR A-Level Physics A specification (H556).

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The Meaning of E = mc²

Einstein's equation says that any object with mass m has an associated rest energy:

E₀ = m₀ c²

where m₀ is the rest mass (the mass of the object in its own rest frame) and c = 3.00 × 10⁸ m s⁻¹ is the speed of light in vacuum. The prefactor c² is enormous — 9 × 10¹⁶ m² s⁻² — so even a small amount of mass corresponds to a huge amount of energy. A single gram of matter has a rest energy of:

E = (10⁻³)(3 × 10⁸)² = 9 × 10¹³ J

which is the energy released by burning about two thousand tonnes of petrol. This is why nuclear processes are so energetic: they directly convert small amounts of mass into large amounts of energy.

The equation has a subtler meaning too. If an object absorbs energy ΔE (becomes hotter, compresses a spring, gains kinetic energy), its mass increases by Δm = ΔE/c². Conversely, if it loses energy, its mass decreases. This effect is undetectably small in everyday processes (heating a cup of coffee increases its mass by about 10⁻¹² g), but in nuclear processes the mass changes are a significant fraction of a percent — large enough to be measured directly on a mass spectrometer.

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The Atomic Mass Unit

Working with kilograms is inconvenient at the nuclear scale, where masses are of order 10⁻²⁷ kg. Physicists use instead the atomic mass unit (u), defined so that a carbon-12 atom has a mass of exactly 12 u:

1 u = 1.661 × 10⁻²⁷ kg

On this scale:

• Proton mass: m_p ≈ 1.00728 u
• Neutron mass: m_n ≈ 1.00867 u
• Electron mass: m_e ≈ 0.000549 u
• Alpha particle: m_α ≈ 4.00150 u

The atomic mass unit is also called the dalton (Da) in chemistry, but OCR uses "u". You will see both in the literature.

Note: "atomic mass" and "nuclear mass" are subtly different. The atomic mass of a neutral atom includes its orbiting electrons; the nuclear mass does not. For most A-Level calculations the distinction is small enough to ignore, but you should be alert to it when a question specifies one or the other.

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MeV and MeV/c²

Multiplying the atomic mass unit by c² gives its rest energy. Converting to MeV (since 1 MeV = 1.60 × 10⁻¹³ J):

1 u × c² = (1.661 × 10⁻²⁷)(3.00 × 10⁸)² J
        = 1.495 × 10⁻¹⁰ J
        = 931.5 MeV

So:

1 u = 931.5 MeV/c²

This conversion factor is one of the most useful numbers in nuclear physics. Whenever you see a mass in atomic mass units and need the corresponding energy in MeV, just multiply by 931.5.

Particle physicists commonly use "MeV/c²" (or "GeV/c²") as a unit of mass directly, since it saves the conversion step. The rest energies of common particles are:

Particle	Rest energy (MeV)	Mass (u)
Electron	0.511	0.000549
Proton	938.3	1.00728
Neutron	939.6	1.00867
Alpha	3727	4.00150
Pion (π⁺)	139.6	0.1499
Muon (μ⁻)	105.7	0.1135

Note that the neutron is slightly heavier than the proton — by about 1.3 MeV/c² — which is why a free neutron decays via beta-minus to a proton + electron + antineutrino. The energy difference appears partly as the rest energy of the emitted electron (0.511 MeV) and partly as the kinetic energy of the decay products.

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Energy Released in a Nuclear Process

In any nuclear process — decay, fission, fusion — the total relativistic energy is conserved. If the rest masses of the initial particles differ from those of the final particles, the mass difference is converted to (or from) kinetic energy:

Q = (Σ m_initial - Σ m_final) c²

where Q is called the Q-value of the reaction. A positive Q-value means energy is released (the process is exoergic); a negative Q-value means energy must be supplied (endoergic).

Worked Example 1: Q-value of alpha decay

Polonium-210 alpha-decays to lead-206. The relevant atomic masses are:

• m(²¹⁰Po) = 209.98286 u
• m(²⁰⁶Pb) = 205.97445 u
• m(⁴He) = 4.00260 u

What is the Q-value in MeV?

Solution.

Δm = 209.98286 - (205.97445 + 4.00260) = 0.00581 u
Q = Δm × 931.5 MeV/u ≈ 5.41 MeV

This 5.41 MeV appears as kinetic energy of the emitted alpha particle (most of it, because the heavy daughter nucleus barely moves) and a tiny amount of recoil energy of the Pb-206 nucleus. The observed alpha energy of polonium-210 is indeed about 5.3 MeV — the small discrepancy is the recoil energy of the lead daughter.

Worked Example 2: Q-value of beta-minus decay

Carbon-14 beta-minus-decays to nitrogen-14. Atomic masses:

• m(¹⁴C) = 14.00324 u
• m(¹⁴N) = 14.00307 u

What is the Q-value?

Solution.