Can you make something from nothing? Conservation, no fear

♣ Newbie

Think of a recipe. If you put three eggs into an omelette, there are three eggs in the omelette. Beaten, cooked and a different shape, but all three. No fourth egg appears and none vanishes.

Atoms work the same way. In a chemical reaction, such as cooking, burning or rusting, atoms come apart and join up again in a different way. But they are the same atoms: none is created and none is lost. That’s why what goes in weighs the same as what comes out.

And the log? As it burns, it takes oxygen from the air, and almost all its atoms go up the chimney as gases you can’t see: the smoke. Only the ash stays behind. If you could put the fire inside a sealed bag and weigh the bag before and after, it would weigh the same.

♦ Intermediate

In 1789, Antoine Lavoisier wrote it down in his Elementary Treatise of Chemistry, while studying how grape juice turns into wine:

“Nothing is created, either in the operations of art or in those of nature, and it can be taken as a principle that in every operation there is an equal quantity of matter before and after” (our translation).

This is the law of conservation of mass. Lavoisier even wrote a kind of equation: grape juice = carbonic gas + alcohol. Today we write it with formulas. Kitchen gas, methane, burns like this:

CH₄ + 2 O₂ → CO₂ + 2 H₂O

The number in front of each substance says how many molecules go in or come out. It’s chosen so that there are the same atoms of each element on each side: that’s called balancing the reaction. Here there are 1 carbon, 4 hydrogens and 4 oxygens before and after.

If each number is a mole, the mass comes from adding up the mass of one mole of each substance (atoms lesson): 16 g of methane and 64 g of oxygen give 44 g of CO₂ and 36 g of water. That’s 80 g before and 80 g after.

The catch is where you measure. The law holds in a closed system, where nothing gets in or out. In an open fireplace, oxygen comes in from the air and the gases leave, which is why the ash weighs so little.

♠ Technical

A reaction is written as Σⱼ νⱼ Xⱼ = 0, with νⱼ negative for reactants and positive for products. Conserving each element k means Σⱼ A_kj νⱼ = 0 for every k: A·ν = 0. Balancing a reaction is finding the null space of A.

For methane, A is 3 × 4 with rank 3, so the null space has dimension 1 and the reaction is fixed up to a factor: ν = (−1, −2, 1, 2). In Lavoisier’s fermentation, C₆H₁₂O₆ → 2 C₂H₅OH + 2 CO₂, there are three elements and three species, but the rank is 2 and again there is a single solution. When the null space has dimension greater than 1 (more species than rank(A) + 1), several reactions are possible, and chemistry, not algebra, decides which one happens.

Conservation of mass is a corollary. With Mⱼ = Σₖ mₖ A_kj:

Σⱼ νⱼ Mⱼ = Σₖ mₖ (A·ν)ₖ = 0

This assumes each atom weighs the same inside and outside a molecule. It isn’t exact: bond energy has mass too (block 4), but the error is one part in 10¹⁰ and no scale can see it. OpenStax puts it this way: there is no detectable change in the total quantity of matter (Chemistry 2e, 1.2).

♣ Newbie

Climb to the top of a slide. Up there you have energy just from being high, even if you don’t move. As you slide down, that height energy turns into speed: the lower you go, the faster you move. And since you rub against the slide, part of it turns into a little heat: you feel it in your hands if you brake with them.

If you add everything up (the height you have left, the speed you’re going and the heat from rubbing), the total is the same at every moment. Energy doesn’t appear or disappear: it changes form.

That’s why there’s no machine that gives out energy without taking it from somewhere. And when we say we “use up” energy, what we really do is turn it into heat that spreads everywhere and can no longer move anything. The total hasn’t changed.

♦ Intermediate

The law of conservation of energy says the total energy is constant in any process: it can change form or pass from one system to another, but the total doesn’t change (OpenStax, College Physics 2e, 7.6). Some of its forms:

• Height energy, or gravitational potential energy: m·g·h, mass times gravity (g = 9.8 m/s²) times height.
• Motion energy, or kinetic energy: ½·m·v², with v the speed.
• Heat: energy that flows from a hot body to a cold one (energy lesson).
• And chemical, electrical, light…

Example: drop a kilogram from 10 m. At the top it has 1 × 9.8 × 10 = 98 J of height and no motion. Halfway down, 49 J and 49 J. Just before the ground, 98 J of motion: it’s going at 14 m/s. When it hits, it stops, and those 98 J are shared out as heat and sound.

There’s another way to change something’s energy: pushing or pulling it, which is called work. The energy a body stores inside (the jiggling of its atoms, its bonds) is its internal energy. Heating it or doing work on it raises it. That’s why no machine gives out energy without taking it from somewhere.

♠ Technical

In thermodynamics, conservation is written as the first law, ΔU = Q − W, with Q the heat that enters and W the work the system does. OpenStax calls it “the law of conservation of energy stated in a form most useful in thermodynamics” (College Physics 2e, 15.1). A machine that, over each cycle (ΔU = 0), gave W > 0 with Q = 0 would break it: that’s a perpetual motion machine of the first kind.

In relativity, energy is still conserved if you count it properly: a body of mass m has, besides its kinetic energy K, a rest energy mc² (OpenStax, 28.6). So E = mc² + K. Block 4 draws the consequences.

Why is it conserved? Because of a symmetry. Emmy Noether proved in 1918 that every continuous symmetry of the laws comes with a conserved quantity. If the laws don’t change from t to t + τ (an experiment gives the same result today as tomorrow), energy is conserved. Feynman tells the story in his Lectures (I, 52-3), together with the other two classic symmetries: space and rotation.

♣ Newbie

Rub a balloon on a sweater and it will stick to the wall. As you rub, a few electrons, which have a negative charge, move from the sweater to the balloon. The balloon ends up with extra negative charge, and the sweater is missing exactly the same amount.

It’s like passing cards between two players: one has more and the other fewer, but the table holds the same cards. Electric charge isn’t created: it’s moved from one place to another. Add the balloon and the sweater together and the total charge is still zero.

In the atoms lesson you saw that the number of protons decides the element. In the kitchen, as with the omelette, the atoms of each element don’t change. To change element you have to touch the nucleus, and that’s no longer chemistry. But even there some accounts are kept: the total charge, and the number of protons and neutrons together.

♦ Intermediate

The law of conservation of charge says the total charge is constant in any process (OpenStax, College Physics 2e, 18.1). When two bodies are rubbed together, the charge one gains is the charge the other loses.

In chemistry, conservation of charge means reactions with ions must balance in charge too: if a sodium atom gives up an electron, Na → Na⁺ + e⁻, the charge is 0 before and +1 − 1 = 0 after.

In a nuclear reaction, the element can change. The carbon-14 from the atoms lesson decays by giving off an electron (OpenStax, Chemistry 2e, 21.3):

¹⁴C → ¹⁴N + e⁻ + antineutrino

A neutron in the nucleus has turned into a proton: there were 6 protons and now there are 7, so it’s nitrogen. But two counts still add up, and they are the rules for balancing any nuclear reaction (OpenStax, 21.2):

• Protons + neutrons: 14 before and 14 after. The electron and the antineutrino, a particle with no charge and almost no mass, don’t count here.
• Charge: +6 before; +7 − 1 = +6 after.

And what if a proton or an electron just vanished? People have looked for decades and it has never been seen: if it happens, an electron takes on average more than 10²⁸ years, and the universe is about 10¹⁰ years old.

♠ Technical

In particle physics, three numbers are checked separately (OpenStax, University Physics vol. 3, 11.2). In n → p + e⁻ + ν̄ₑ:

The reaction n → e⁺ + e⁻ conserves charge and energy, but not B (1 versus 0), and it doesn’t happen.

The experimental limits (PDG 2024, review by Pich and Ramsey-Musolf):

• Charge: τ(e → νₑγ) > 6.6×10²⁸ years, from the Borexino experiment (PRL 115, 231802, 2015). That’s about 4.8×10¹⁸ ages of the universe. The PDG links charge conservation to the gauge symmetry of quantum electrodynamics.
• Baryon number: τₚ > 9×10²⁹ years in any channel, and τ(p → e⁺π⁰) > 2.4×10³⁴ years. Grand unified theories predict that B is violated, but no signal has been seen.

So chemistry’s conservation “of elements” is a special case: without nuclear reactions, each nucleus keeps its Z. With them, what’s conserved is charge and B.

♣ Newbie

So far it looked like there were two separate accounts: mass and energy. Einstein discovered they’re the same one. Mass is energy packed very, very tightly.

How much? A huge amount. A single gram of mass, if it all turned into energy, would feed one person for about 29,000 years, or keep a kettle on non-stop for about 1,400 years.

So why don’t we notice? Because in everyday life almost none of it converts. When you burn kitchen gas, what comes out weighs one part in 8 billion less than what went in: that’s the mass that left as heat. No scale notices, and that’s why Lavoisier was right in practice.

And making something from nothing? You can make matter, but not from nothing: you have to pay for it in full with energy, and it always comes with its “twin” of opposite charge, so the charge account stays at zero.

♦ Intermediate

The formula is E = mc²: the energy carried by a mass m, with c the speed of light, exactly 299,792,458 m/s. Since c² is enormous, about 9×10¹⁶ m²/s², one gram is worth 9×10¹³ J (OpenStax, College Physics 2e, 28.6). That energy of a body at rest is called its rest energy.

It works both ways. If a body gains energy, it gains mass; if it gives energy out, it loses mass. The mass that disappears is Δm = E / c²:

• Boiling 1 L of water puts 334 kJ into it (energy lesson): it weighs 3.7 nanograms more.
• Burning a mole of methane gives out 890 kJ: the 80 g of gas and oxygen lose 9.9 nanograms. One part in 8 billion.

In nuclear reactions the share is big enough to measure. How much of the reacting mass becomes released energy:

And “making something from nothing”: a ray of light with enough energy, passing close to an atomic nucleus, can turn into an electron and a positron, its antiparticle, with charge +. That’s an electron-positron pair: the total charge is still zero (OpenStax, 18.1). It needs at least the rest energy of the two, 1.022 MeV (the electronvolt from the energy lesson, times a million).

♠ Technical

The general relation is E² = (pc)² + (mc²)²; at rest (p = 0) it becomes E₀ = mc², and for a photon (m = 0), E = pc (OpenStax, 28.6). Einstein proposed it in 1905, in a three-page paper whose title asks whether the inertia (the mass) of a body depends on the energy it contains (Ann. Phys. 323, 639). He wrote it with other letters: a body that gives off an energy L as radiation loses a mass L/V², with V the speed of light.

Mass-defect calculations, with Q = Δm·c²:

• Combustion of methane: ΔcH° = −890.7 ± 0.4 kJ/mol (NIST WebBook; OpenStax gives −890.8). Δm = 8.907×10⁵ J / c² = 9.9×10⁻¹² kg per mole, out of 80.04 g of reactants: 1.24×10⁻¹⁰.
• Induced fission of ²³⁵U: Q ≈ 200 MeV (OpenStax UP3, 10.5). Δm/m = 200 / [(235.0439 + 1.0087) × 931.494] = 9.1×10⁻⁴.
• ²H + ³H → ⁴He + n: Q = 17.6 MeV (OpenStax UP3, 10.6). With NIST masses, Δm/m = 17.6 / (5.0302 × 931.494) = 3.8×10⁻³.

Pair creation conserves charge (0 = −1 + 1) and Lₑ (0 = 1 − 1). A photon on its own can’t do it in empty space: in the pair’s reference frame the total momentum is zero, and a photon has nonzero momentum in every frame (E = pc ≠ 0). A third body has to take the leftover momentum, usually a nucleus: γ + N → e⁻ + e⁺ + N. NIST’s XCOM tables compute pair production “in the field of the atomic nucleus” and in that of the electrons. It’s the same reason why, the other way round, an electron and a positron can’t annihilate into a single photon (OpenStax UP3, 11.2). Since the nucleus is heavy, it takes almost no energy, and the threshold stays at 2mₑc² = 2 × 0.510 998 950 69 MeV = 1.022 MeV (CODATA 2022; HyperPhysics gives the same threshold). That’s why “making something from nothing” is impossible: the energy is paid in full, and the charge, B and L accounts stay at zero.