Lesson 2 of 7Revolutions in Modern Physics

Unit 5 · Lesson 2 of 7

Mass-Energy Equivalence and Relativistic Energy

What E = mc² really says: every mass carries a rest energy, a moving object's total energy is γmc², and the same bookkeeping explains why nothing with mass reaches the speed of light, where the energy of fission and fusion comes from, and how light can turn into matter.

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Key things to remember

1Rest energy: $E_{0} = m c^{2}$ ; one gram of mass is equivalent to $9.0 \times 1 0^{13} J$ .
2Total energy $E = γm c^{2}$ and kinetic energy $E_{k} = (γ - 1) m c^{2}$ ; at low speeds $E_{k} \approx \frac{1}{2} m v^{2}$ .
3No object with mass can reach $c$ , because $γ$ and the kinetic energy grow without limit as $v \to c$ .
4Mass defect: a nucleus is lighter than its separate nucleons; the binding energy is $Δ m c^{2}$ . Fission and fusion release energy because the products are more tightly bound.
5Pair production needs a photon of at least $2 m_{e} c^{2} = 1.02 MeV$ ; electron-positron annihilation at rest gives two $0.511 MeV$ photons.
6Conversions: $1 MeV = 1.60 \times 1 0^{- 13} J$ and $1 u = 1.66 \times 1 0^{- 27} kg$ .

Special Relativity showed that time and length depend on the observer. Einstein followed it in 1905 with a short paper that drew an even stranger conclusion: mass itself is a form of energy. A brick sitting perfectly still holds an enormous store of energy simply because it has mass. Usually that energy stays locked up. But in a nuclear reactor, in the core of the Sun, and when matter meets antimatter, measurable amounts of mass disappear and the energy shows up as heat, light and fast-moving particles. This lesson does the energy accounting, using the Lorentz factor $γ = 1/ 1 - v^{2} / c^{2}$ you have already met.

Rest energy: mass is stored energy

E_{0} = m c^{2} c = 3.00 \times 1 0^{8} m/s 1 MeV = 1.60 \times 1 0^{- 13} J

E0 is the rest energy in joules when m is in kilograms. Particle physicists often quote masses in MeV per c squared.

Because $c^{2} = 9.00 \times 1 0^{16} m^{2} / s^{2}$ is so large, a tiny mass corresponds to a huge energy. One gram has a rest energy of $9.0 \times 1 0^{13} J$ , about what a large power station delivers in a day. For particles, the joule is an awkward size, so rest energies are given in mega-electron-volts. An electron's rest energy is $8.20 \times 1 0^{- 14} J$ , which is $0.512 MeV$ with this course's rounded constants (the precise value is $0.511 MeV$ ), and a proton's is about $939 MeV$ . The equation also runs the other way: any change in an object's energy changes its mass. Burning a kilogram of gasoline releases about $46 MJ$ , so the products are lighter than the reactants by about $5 \times 1 0^{- 10} kg$ , far too little for any balance to detect. That is why chemists can treat mass as conserved, and why nuclear reactions, which release millions of times more energy per atom, are where mass changes become obvious.

Total energy and kinetic energy

E_{t o t a l} = γm c^{2} E_{k} = E_{t o t a l} - m c^{2} = (γ - 1) m c^{2} γ = \frac{1}{1 - v ^{2} / c ^{2}}

Total energy is rest energy plus kinetic energy. At low speed (gamma minus 1) times mc squared becomes one half m v squared.

Speed $v / c$	Lorentz factor $γ$	Relativistic $E_{k}$ divided by $\frac{1}{2} m v^{2}$
$0.10$	$1.005$	$1.008$
$0.50$	$1.155$	$1.24$
$0.90$	$2.29$	$3.20$
$0.99$	$7.09$	$12.4$
$0.999$	$22.4$	$42.8$

Below about a tenth of light speed the classical formula is accurate to better than 1 percent. Near c it badly underestimates the energy.

Why c is a speed limit

v

approaches

c

, the factor

1 - v^{2} / c^{2}

approaches zero, so

γ

and therefore

(γ - 1) m c^{2}

grow without limit. Reaching

v = c

would take infinite energy, so no object with mass can ever get there. Each extra burst of energy buys a smaller gain in speed: an electron with

3.1 MeV

of kinetic energy moves at

0.99 c

, but it needs about

10.9 MeV

to reach

0.999 c

. Accelerators confirm this every day: their particles gain energy almost without limit while their speed creeps toward, and never past,

c

. Only massless particles, such as photons, travel at

c

, and they can never travel at any other speed in a vacuum.

Mass defect: fission and fusion

Fig 1Fission splits a heavy nucleus into two middle-sized ones; fusion joins two light nuclei. In both, the products have less mass than the reactants, and the missing mass appears as energy.

A nucleus has less mass than the protons and neutrons that make it up, measured separately. The difference is the mass defect, and $Δ m c^{2}$ is the binding energy: the energy that would have to be supplied to pull the nucleus apart. Nuclei in the middle of the periodic table, around iron, are the most tightly bound per nucleon. So splitting a very heavy nucleus (fission) or joining very light ones (fusion) both produce more tightly bound products, and the mass that disappears is released as energy. Each fission of uranium-235 releases about $200 MeV$ . Ontario's CANDU reactors at Pickering, Darlington and Bruce, which supply about half of the province's electricity, use natural uranium fuel with heavy water to slow the neutrons. The Sun runs on fusion, turning hydrogen into helium; its power output of $3.83 \times 1 0^{26} W$ means it loses about $4.3 \times 1 0^{9} kg$ of mass every second.

E = Δ m c^{2} Δ m = (mass of separate particles) - (mass of nucleus) 1 u = 1.66 \times 1 0^{- 27} kg

Nuclear masses are usually given in atomic mass units, u. Convert to kilograms before multiplying by c squared.

Pair production and annihilation: energy budgets

The Standard Model of Particle Physics describes how a gamma-ray photon passing near a nucleus can turn into an electron and a positron, and how an electron and a positron destroy each other to make photons. With $E = m c^{2}$ you can now balance the energy books. In pair production, the photon must supply the rest energy of both particles, $2 m_{e} c^{2} = 1.02 MeV$ ; any energy beyond that becomes the pair's kinetic energy. In annihilation of an electron and positron at rest, the total energy $1.02 MeV$ leaves as two photons of $0.511 MeV$ each, travelling in opposite directions so that momentum stays zero. A proton meeting an antiproton releases far more, about $1.88 GeV$ , because the proton is about 1800 times as massive.

Process	Energy going in	Energy coming out
Pair production ( $γ \to e^{-} + e^{+}$ )	Photon energy $h f$	$2 m_{e} c^{2} = 1.02 MeV$ of rest energy, plus kinetic energy of the pair
Electron-positron annihilation at rest	$2 m_{e} c^{2} = 1.02 MeV$	Two $0.511 MeV$ gamma photons, back to back
Fission of one uranium-235 nucleus	Rest energy of the nucleus and a neutron	About $200 MeV$ , mostly kinetic energy of the fragments
Deuterium-tritium fusion	Rest energy of $^{2} H$ and $^{3} H$	About $17.6 MeV$ , carried by a helium-4 nucleus and a neutron

Mass is not destroyed; it is counted as energy

It is tempting to say mass is 'converted' into energy, as though the mass vanishes and something new appears. A better picture is that mass is energy. The total energy, rest energy included, is always conserved. In fission the rest energy of the fuel becomes kinetic energy of the fragments, which becomes heat in the reactor. A hot cup of coffee is, in principle, very slightly more massive than the same cup cold.

Worked examples

Reveal one step at a time and try to beat it to the next line.

Example 1

Find the rest energy of a

1.0 g

paper clip. If it could all be turned into electricity, how long would it supply an Ontario home that uses about

9000 kWh

per year?

Example 2

A proton (

m = 1.67 \times 1 0^{- 27} kg

) moves at

0.80 c

. Find its rest energy, total energy and kinetic energy, and compare the kinetic energy with the classical

\frac{1}{2} m v^{2}

Example 3

A helium-4 nucleus has a mass of

4.001506 u

. A proton has a mass of

1.007276 u

and a neutron

1.008665 u

. Find the mass defect and the binding energy of the nucleus in MeV.

Example 4

A reactor produces thermal power of

2.0 \times 1 0^{9} W

. Each fission of uranium-235 releases about

200 MeV

. How many fissions occur each second, and how much mass is converted to energy each day?

Where marks get lost

Using $E = γm c^{2}$ as the kinetic energy. That is the total energy; subtract the rest energy to get $E_{k} = (γ - 1) m c^{2}$ .
Using $\frac{1}{2} m v^{2}$ for particles moving faster than about $0.1 c$ . Above that speed the classical formula underestimates the kinetic energy, badly so near $c$ .
Forgetting to convert atomic mass units or grams to kilograms before multiplying by $c^{2}$ , or forgetting to square $c$ .
Thinking the mass defect means a nucleus has more mass than its parts. A bound nucleus is lighter; energy must be added to separate it.
Saying a photon of exactly $0.511 MeV$ can create an electron-positron pair. Pair production needs at least $1.02 MeV$ , enough for the rest energy of both particles.

Flashcards 9

Rest energy

Total relativistic energy

Relativistic kinetic energy

Why can't a massive object reach c?

+ 5 more

Practice

Using

m_{e} = 9.11 \times 1 0^{- 31} kg

, what is the rest energy of an electron, in MeV? Give three significant figures.

MeV

At what speed, as a fraction of

c

, does a particle's kinetic energy equal its rest energy? Give three significant figures.

An electron (rest energy

0.511 MeV

) moves at

0.60 c

. What is its kinetic energy, in MeV? Give three significant figures.

MeV

The Sun radiates

3.83 \times 1 0^{26} W

. How much mass does it lose each second as a result, in units of

1 0^{9} kg

? Give three significant figures.

× 10⁹ kg

3.00 MeV

gamma photon creates an electron-positron pair near a nucleus. What is the total kinetic energy of the pair, in MeV? Use

1.02 MeV

for the rest energy of the pair, and give three significant figures.

MeV

Why can a proton in an accelerator never be pushed to the speed of light?

In a fission reaction, how does the total mass of the products compare with the mass of the reactants?

What is the name for the difference between the mass of a nucleus's separate protons and neutrons and the mass of the nucleus itself?

Order these particles by rest energy, from smallest to largest.

1Helium-4 nucleus
2Deuteron (hydrogen-2 nucleus)
3Proton
4Electron
5Muon

Match each process to its energy budget.

Electron-positron annihilation at rest

Pair production

Fission of uranium-235

Deuterium-tritium fusion

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