Law of conservation of mass (Lavoisier) — proof, examples: Foundations

This page unpacks every symbol and idea the parent note on the Law of Conservation of Mass leans on — from what "mass" even means, up to that intimidating sum $\sum_{i} N_{i} m_{i}$ . We build each piece from nothing, and never use a symbol before it has both a plain-words meaning and a picture.

1. Matter, mass, and "how much stuff"

Why the topic needs it. The whole law is a sentence about mass: "mass before = mass after." If we don't nail down what mass is, the law is just noise. Picture a kitchen balance — a pan on a spring or lever that tips to show how much stuff sits on it. That balance is the hero of Lavoisier's experiment.

Figure — Law of conservation of mass (Lavoisier) — proof, examples

2. Atoms — the indestructible bricks

Picture a bin of colored bricks: red = hydrogen, blue = oxygen, black = carbon. You can build a "water" shape or a "carbon-dioxide" shape, but you are always using the same bricks.

3. Elements and the index $i$ — labelling the brick colours

Read $i$ out loud as "element number $i$ ". If we sort our bins as bin 1 = hydrogen, bin 2 = oxygen, bin 3 = carbon, then $i = 2$ means "the oxygen bin."

4. Counting atoms — the symbol $N$

We attach the element tag as a little subscript: $N_{i} = number of atoms of element i .$ So $N_{2}$ = how many oxygen atoms.

We also need to say when we counted — before or after the reaction. We write that as a superscript label (a word, not an exponent): $N_{i}^{before}, N_{i}^{after} .$

Why the topic needs it. The law's whole proof is the claim that this count doesn't change: $N_{i}^{before} = N_{i}^{after}$ for every colour. You cannot state that without a symbol that means "count of one colour at one time."

5. Atomic mass — the symbol $m$

Small subscript = which colour: $m_{1}$ = mass of one hydrogen atom, $m_{2}$ = mass of one oxygen atom.

6. Multiplication — turning count × weight into mass

Why multiply and not add? Because "5 bricks, each 2 g" is five lots of 2 g $= 5 \times 2 = 10$ g. Multiplication is the tool for "so-many copies of the same thing." That is precisely our situation: many identical atoms of one colour.

7. The summation sign $\sum$ — adding every colour at once

Now we have the mass of one colour. A real sample has many colours. We must add them all: $(H mass) + (O mass) + (C mass) + \dots$

Writing " $+ \dots$ " is sloppy. The summation sign $\sum$ (a big Greek "S", for Sum) is the tidy tool for "add this pattern over every value of the tag."

Read it as a loop: land on bin 1, work out $N_{1} m_{1}$ ; land on bin 2, add $N_{2} m_{2}$ ; keep going; the running total is the answer.

So the parent note's two central lines simply mean: $M_{R} = \sum_{i} N_{i}^{before} m_{i} = (total mass of all reactant bricks),$ $M_{P} = \sum_{i} N_{i}^{after} m_{i} = (total mass of all product bricks),$ where $M_{R}$ ("mass of reactants") and $M_{P}$ ("mass of products") are just names for those two grand totals — the two balance readings.

Recall Decode

$M_{R} = \sum_{i} N_{i}^{before} m_{i}$ in words Total starting mass equals: for every element, (its atom count before) times (its per-atom mass), all added up. ::: Correct — count each colour, weigh each colour, sum over all colours.

8. Reactants, products, and the arrow $\to$

Picture the before-photo and after-photo of the LEGO bin. The arrow is the act of rebuilding.

Small numbers explained. The big number in front (the coefficient, e.g. the $2$ in $2 Hg$ ) says "two of these." The little subscript (e.g. the $2$ in $O_{2}$ ) says "this piece is made of two atoms stuck together." Making the atom-counts match on both sides is exactly Balancing Chemical Equations, which — as the parent shows — is conservation of mass written at the atom level.

9. Closed vs open system — where the law lives or dies

Picture two jars: one lidded (closed), one lidless (open). Burn a candle in each.

Closed: the smoke and vapour stay trapped → the balance reads the same.
Open: the gas escapes → the balance reads less, even though the universe lost nothing.

10. Where the classical law bows out: $E = m c^{2}$

The parent note warns that in nuclear reactions a tiny bit of mass seems to vanish. That's because there, unlike our chemistry, the bricks (atomic nuclei) do change, and a sliver of mass converts to energy — the realm of Mass–Energy Equivalence (E=mc²). For all the chemistry on the parent page, nuclei stay whole, so this effect is far too small to ever see on a balance. You only need to know the boundary exists.

Prerequisite map

Equipment checklist

Test yourself — cover the right side and answer before revealing.

What does "mass" mean in one plain sentence

How much stuff is present — the reading on a balance.

What does an atom do during an ordinary chemical reaction

It gets rearranged, never created or destroyed.

What does the label

i

stand for

A name-tag for "element number

i

" — one brick colour.

What does

N_{i}

count

The number of atoms of element

i

(a head-count, not a weight).

What does the superscript in

N_{i}^{before}

mean

A time-stamp word ("counted before"), NOT a power.

What does

m_{i}

measure

The mass of a single atom of element

i

Why is the mass of one colour written

N_{i} m_{i}

(multiply)

It's "so many copies of the same weight" = count times weight-each.

What does

\sum_{i}

tell you to do

Loop over every element

i

, form

N_{i} m_{i}

, and add them all up.

What are

M_{R}

and

M_{P}

Total mass of all reactants, and total mass of all products.

What does the arrow

\to

mean in an equation

"Turns into" — reactants on the left become products on the right.

Why must the system be closed for the law to be observed

So no gas leaves or enters; otherwise the balance loses sight of some bricks and reads wrong.

Where does the classical law stop applying

In nuclear reactions, where mass converts to energy via

E = m c^{2}

Connections

Dalton's Atomic Theory — supplies the indestructible-atom idea behind Sections 2–4.
Balancing Chemical Equations — the practical enforcement of equal atom counts (Section 8).
Law of Definite Proportions — the sister mass-law that also builds on atoms and fixed masses.
Stoichiometry and the Mole — where these counts and masses get scaled up to lab amounts.
Mass–Energy Equivalence (E=mc²) — the boundary in Section 10.