D. Energy and Society · Work, Energy and PowerMass–Energy Equivalence: E = mc²
D2.8 D3.1 D2.1 A2.2About 65 minutes
Learning goals- I can explain how Einstein's mass–energy equivalence links the law of conservation of mass and the law of conservation of energy.
- I can use E = mc² to calculate the energy equivalent of a mass, and the mass equivalent of an energy.
- I can explain why mass changes are measurable in nuclear reactions but not in chemical reactions.
- I can calculate the energy released in a nuclear reaction from the mass difference.
Words to know
law of conservation of mass
In a chemical reaction, the total mass of the products equals the total mass of the reactants (to within any measurable amount).
mass–energy equivalence
Mass is a form of energy. A mass m is equivalent to an energy E = mc².
speed of light (c)
c = 3.00 × 10⁸ m/s in a vacuum. In E = mc², c² = 9.00 × 10¹⁶ m²/s².
mass defect (Δm)
The difference between the total mass before and after a nuclear reaction. It is released as energy: E = Δmc².
law of conservation of mass–energy
The total of mass and energy (with mass counted as mc²) in an isolated system stays constant.
Two laws, or one?
In chemistry you learned the law of conservation of mass: burn a log in a sealed container and the total mass inside does not change. In this unit you have used the law of conservation of energy. Through the 1800s these were treated as two separate laws. In 1905, Albert Einstein showed, as part of his special theory of relativity, that they are really one law: mass is itself a form of energy. His equation is probably the most famous in physics:
E = mc²
What E = mc² says
- E is energy in joules (J), m is mass in kilograms (kg), and c = 3.00 × 10⁸ m/s is the speed of light.
- Because c² = 9.00 × 10¹⁶ m²/s², a tiny mass is equivalent to an enormous energy. One kilogram of mass is equivalent to 9.0 × 10¹⁶ J.
- It works both ways: any energy change also changes mass, by Δm = ΔE ÷ c².
When a system gives out energy (light, thermal energy, kinetic energy of particles), its mass goes down by ΔE ÷ c². When a system gains energy, its mass goes up. A hot cup of tea has very slightly more mass than the same tea when cold; a stretched spring has very slightly more mass than an unstretched one.
So the two old laws combine into the law of conservation of mass–energy: the total of all energy, counting mass as mc², is constant. In Grade 9 you used E = mc² to see how the Sun shines; here you will use it quantitatively.
Worked exampleExample 1: The energy in one gram
What is the energy equivalent of 1.0 g of mass? How many years could it supply a home that uses 750 kWh of electrical energy a month?
- Convert: m = 1.0 g = 1.0 × 10⁻³ kg.
- E = mc² = (1.0 × 10⁻³ kg)(3.00 × 10⁸ m/s)² = (1.0 × 10⁻³)(9.00 × 10¹⁶) = 9.0 × 10¹³ J.
- Home's yearly use: 750 kWh × 12 = 9000 kWh = 9000 × 3.6 × 10⁶ J = 3.24 × 10¹⁰ J.
- Years = 9.0 × 10¹³ ÷ 3.24 × 10¹⁰ ≈ 2.8 × 10³ years.
9.0 × 10¹³ J, enough for that home for about 2800 years, if all of it could be captured as electrical energy.
Worked exampleExample 2: Why chemists don't notice
Burning 1.0 kg of gasoline releases about 4.6 × 10⁷ J. By how much does the total mass of the reactants and products decrease?
- Δm = ΔE ÷ c² = 4.6 × 10⁷ J ÷ 9.00 × 10¹⁶ m²/s²
- Δm ≈ 5.1 × 10⁻¹⁰ kg (about half a microgram, out of more than 4 kg of gasoline and oxygen).
About 5 × 10⁻¹⁰ kg, roughly one part in ten billion. No balance used in chemistry can detect this, which is why the law of conservation of mass works perfectly in practice for chemical reactions.
Nuclear reactions: where the mass change shows
Chemical reactions rearrange electrons, and the energies involved are small. Nuclear reactions rearrange the protons and neutrons inside nuclei, and the energies are about a million times larger per particle. Now the mass difference is big enough to measure.
Carefully measured masses show that the products of a nuclear reaction have less mass than the reactants. This mass defect Δm has become energy (mostly kinetic energy of the products, which becomes thermal energy):
E = Δmc²
In fission of uranium-235, roughly 0.1% of the mass that reacts becomes energy. In fusion of hydrogen isotopes, it is several times more (about 0.4% for deuterium–tritium fusion). The Thermal and Nuclear Energy unit explains fission, fusion and isotopes in detail; here we use the masses to find the energy.
Worked exampleExample 3: Energy from one fusion reaction
Use the masses in the diagram to find the energy released when one deuterium nucleus fuses with one tritium nucleus to form helium-4 and a neutron.
- Mass before: 3.3436 × 10⁻²⁷ + 5.0074 × 10⁻²⁷ = 8.3510 × 10⁻²⁷ kg.
- Mass after: 6.6447 × 10⁻²⁷ + 1.6749 × 10⁻²⁷ = 8.3196 × 10⁻²⁷ kg.
- Mass defect: Δm = 8.3510 × 10⁻²⁷ − 8.3196 × 10⁻²⁷ = 0.0314 × 10⁻²⁷ = 3.14 × 10⁻²⁹ kg.
- E = Δmc² = (3.14 × 10⁻²⁹)(9.00 × 10¹⁶) ≈ 2.8 × 10⁻¹² J.
About 2.8 × 10⁻¹² J per reaction. That seems tiny, but 1 kg of fuel contains about 1.2 × 10²⁶ such pairs, so it would release roughly 3.4 × 10¹⁴ J.
Worked exampleExample 4: The Sun loses mass
The Sun gives out energy at a rate of about 3.8 × 10²⁶ W. How much mass does it convert into energy each second?
- In 1 s the Sun gives out 3.8 × 10²⁶ J.
- Δm = E ÷ c² = 3.8 × 10²⁶ ÷ 9.00 × 10¹⁶
- Δm ≈ 4.2 × 10⁹ kg each second
About 4.2 × 10⁹ kg (over four million tonnes) every second. The Sun is so massive that it can keep this up for billions of years.
Energy released per kilogram of fuel, on a scale where each step is 100 times bigger (a logarithmic scale). Nuclear fuels release millions of times more energy per kilogram than chemical fuels. Complete conversion of mass, E = mc², is the upper limit.Calculator and unit tips
- Always use kilograms (g ÷ 1000) and joules.
- Square c before multiplying: c² = 9.00 × 10¹⁶.
- Use scientific notation keys (EXP or ×10ˣ). (3.00 × 10⁸)² = 9.00 × 10¹⁶, not 9.00 × 10⁶⁴.
- To find a mass from an energy, divide: m = E ÷ c².
- In a nuclear problem, subtract total mass after from total mass before to get Δm. Keep all the digits until the subtraction; the difference is a tiny part of the total.
Just for funBy E = mc², a 60 kg student is equivalent to about 5.4 × 10¹⁸ J of energy. So "I don't have the energy to do my chores" is, technically, not true.
Hands-on activityCan a kitchen scale see E = mc²?
You need: a sealed plastic bottle of water, a kitchen scale (preferably reading to 1 g), a sink or bowl of warm tap water and a towel.
- Weigh the sealed, dry bottle. Record the mass.
- Stand it in warm tap water for 10 minutes so the water inside warms up. Dry the outside completely and weigh it again.
- Did the reading change?
- Now calculate. Warming 0.50 kg of water by 20 °C takes about 4.2 × 10⁴ J. Find the mass increase Δm = E ÷ c².
- Compare Δm with the smallest mass your scale can show. Explain why we can say the mass increased, even though the scale cannot show it.
What to look for: the scale reading should not change (if it does, the bottle was probably still wet). Δm = 4.2 × 10⁴ ÷ 9.00 × 10¹⁶ ≈ 4.7 × 10⁻¹³ kg, which is about 2 billion times smaller than 1 g. The mass really did increase, but by far too little to measure, just like in chemical reactions. (The energy needed to warm water, Q = mcΔT, is taught in the Thermal and Nuclear Energy unit.)
Your turn
1Find the energy equivalent of 2.0 kg of mass.
E = (2.0)(9.00 × 10¹⁶) = 1.8 × 10¹⁷ J.
2What mass is equivalent to 4.5 × 10¹⁵ J?
m = 4.5 × 10¹⁵ ÷ 9.00 × 10¹⁶ = 0.050 kg (50 g).
3Find the energy equivalent of 1.0 mg (1.0 × 10⁻⁶ kg).
E = (1.0 × 10⁻⁶)(9.00 × 10¹⁶) = 9.0 × 10¹⁰ J.
4A nuclear reaction has reactants of total mass 6.6950 × 10⁻²⁷ kg and products of total mass 6.6900 × 10⁻²⁷ kg. Find the energy released.
Δm = 0.0050 × 10⁻²⁷ = 5.0 × 10⁻³⁰ kg. E = (5.0 × 10⁻³⁰)(9.00 × 10¹⁶) = 4.5 × 10⁻¹³ J.
5Why don't we notice mass changes in chemical reactions, like burning wood?
The energy released is small, so Δm = E ÷ c² is extremely small (about one part in ten billion), far too small for any balance to detect.
6Explain how E = mc² joins the laws of conservation of mass and conservation of energy.
Mass is a form of energy (mc²). Neither mass nor energy is conserved on its own when they convert, but their total, mass–energy, is always conserved. Chemistry's conservation of mass is an excellent approximation because the mass changes are so tiny.
More practice: warm-up
1What does c stand for in E = mc², and what is its value?
The speed of light, 3.00 × 10⁸ m/s.
3Who first proposed E = mc², and in what year?
Albert Einstein, in 1905.
5When a system releases energy, what happens to its mass?
Its mass decreases (by ΔE ÷ c²).
More practice: on your own
1A spring is compressed, storing 50 J of elastic potential energy. By how much does its mass increase?
Δm = 50 ÷ 9.00 × 10¹⁶ ≈ 5.6 × 10⁻¹⁶ kg.
2A nuclear generating unit delivers 880 MW of electrical power at 33% efficiency. How much mass is converted into energy each second in the reactor?
Reactor thermal power = 880 ÷ 0.33 ≈ 2.67 × 10⁹ W. Δm per second = 2.67 × 10⁹ ÷ 9.00 × 10¹⁶ ≈ 3.0 × 10⁻⁸ kg (30 micrograms) each second.
3How many kilograms of mass would have to be converted completely to supply 3.24 × 10¹⁰ J (a home's yearly electrical energy)?
m = 3.24 × 10¹⁰ ÷ 9.00 × 10¹⁶ = 3.6 × 10⁻⁷ kg (0.36 mg).
4A 1500 kg car speeds up from rest to 30 m/s. By how much does its mass increase because of its kinetic energy? Is this noticeable?
Ek = ½(1500)(30)² = 6.75 × 10⁵ J. Δm = 6.75 × 10⁵ ÷ 9.00 × 10¹⁶ = 7.5 × 10⁻¹² kg. Not noticeable at all.
5Fission releases about 8 × 10¹³ J per kilogram of uranium-235. What fraction of the uranium's mass becomes energy?
Mass converted per kg = 8 × 10¹³ ÷ 9.00 × 10¹⁶ ≈ 9 × 10⁻⁴ kg. Fraction ≈ 0.0009 ≈ 0.09% (roughly 0.1%).
Challenge
1A nuclear reactor produces thermal energy at 2700 MW for a full year (3.15 × 10⁷ s). How much mass is converted into energy in that year?
E = (2.7 × 10⁹ W)(3.15 × 10⁷ s) ≈ 8.5 × 10¹⁶ J. Δm = 8.5 × 10¹⁶ ÷ 9.00 × 10¹⁶ ≈ 0.95 kg: less than a kilogram of mass for a year of output.
2In one fusion reaction in the Sun's core, four hydrogen nuclei (total mass 6.6905 × 10⁻²⁷ kg) eventually become one helium-4 nucleus (6.6447 × 10⁻²⁷ kg), with other tiny particles carrying away very little mass. Estimate the energy released and the percentage of mass converted.
Δm = 0.0458 × 10⁻²⁷ = 4.58 × 10⁻²⁹ kg. E = (4.58 × 10⁻²⁹)(9.00 × 10¹⁶) ≈ 4.1 × 10⁻¹² J. Percentage: 4.58 × 10⁻²⁹ ÷ 6.6905 × 10⁻²⁷ ≈ 0.68%.
3A student says, "E = mc² means that in a nuclear power station, matter is destroyed and energy is created, so conservation of energy is broken." Correct the statement.
No energy is created. Mass is itself a form of energy. In the reactor, energy stored as mass (rest energy) is transformed into kinetic and then thermal energy. The total mass–energy is conserved; the total number of nucleons (protons plus neutrons) is unchanged too, only their arrangement and binding change.
Quiz
2. The energy equivalent of 0.010 kg of mass is:
3. Why is the law of conservation of mass reliable in chemistry?
4. In a nuclear reaction, the mass of the products is:
5. What mass is equivalent to 1.8 × 10¹⁷ J?
6. A cup of tea cools down. Its mass:
Big idea: D2.8 asks students to connect conservation of mass and conservation of energy through E = mc². The key insight is that mass is a form of energy, and that the conversion matters measurably only in nuclear reactions. Einstein's contribution (A2.2) is a good example of how a new theory can unify older laws.
How to help: Practise scientific notation on the calculator together: (3.00 × 10⁸)² should give 9 × 10¹⁶. Ask your teen to explain why a kitchen scale can't detect the mass change of warm water.
Common difficulties:
- Using grams instead of kilograms.
- Forgetting to square c, or squaring incorrectly on the calculator.
- Rounding masses before subtracting in mass-defect problems (the difference disappears).
- Thinking E = mc² means energy is created. It is a transformation; mass–energy is conserved.
Fission, fusion, isotopes and radiation are taught in the Thermal and Nuclear Energy unit.