How Do You Count Atoms by Weighing Them?: Going Further 9–12

Counting by weighing

You learned in Grade 9 that every atom has a mass, measured in atomic mass units (amu): hydrogen 1, carbon 12, oxygen 16, and so on Grade 9: Atomic mass. You also added them up to get a molecule’s molecular mass: water, H₂O, is 2 × 1 + 16 = 18 amu Grade 9: Molecular mass. These numbers tell you how heavy atoms are compared with each other: a carbon atom is 12 times as heavy as a hydrogen atom, and an oxygen atom 16 times.

That is all you need to count by weighing. Think of a hardware shop in Bhaktapur that sells nails and bolts by the kilo. Suppose one bolt weighs exactly 12 times as much as one nail. Then:

1 kg of nails and 12 kg of bolts hold the same number of things;
10 g of nails and 120 g of bolts hold the same number;
in fact, any mass of nails and 12 times that mass of bolts hold the same number, whatever one nail weighs.

Atoms work the same way. A carbon atom is 12 times as heavy as a hydrogen atom, so 1 g of hydrogen atoms and 12 g of carbon atoms hold the same number of atoms. So do 16 g of oxygen atoms, 23 g of sodium atoms and 56 g of iron atoms. Take each element’s atomic mass, in grams, and you always get the same number of atoms. What is that number?

An oxygen atom is 16 times as heavy as a hydrogen atom. Which holds the same number of atoms as 2 g of hydrogen?

Pick an answer to see why.

The mole

People count things in useful batches. You buy bananas by the dozen (12), and paper by the ream (500 sheets). Chemists count particles in a batch of 6.02 × 10²³, called a mole (symbol mol).

One mole of anything is 6.02 × 10²³ of it: 6.02 × 10²³ atoms, or molecules, or ions. This number is the Avogadro constant, written $N_\text{A}$ :

$N_\text{A} = 6.02 \times 10^{23}\ \text{per mole}$

The mole is one of the seven base units of the SI, alongside the meter, the kilogram and the second (you met it in Grade 10’s table of SI units, Fundamental units). It is the chemist’s unit for an amount of substance: “how many”, not “how heavy”.

The number was picked so that the masses come out easy. One mole of atoms of an element has a mass, in grams, equal to its atomic mass: 1 mol of carbon atoms is 12 g, 1 mol of oxygen atoms is 16 g. The same goes for molecules: one mole of water is 18 g, the molecular mass in grams. That is the bridge from atoms, which you can’t see, to grams, which you can weigh.

Molar mass

The mass of one mole of a substance is its molar mass, M, in grams per mole (g/mol). It is the same number as the atomic mass (for an element) or the molecular mass (for a compound), with the unit changed from amu to g/mol:

Substance	Formula	Molecular mass (amu)	Molar mass (g/mol)
hydrogen gas	H₂	2 × 1 = 2	2
oxygen gas	O₂	2 × 16 = 32	32
water	H₂O	2 × 1 + 16 = 18	18
carbon dioxide	CO₂	12 + 2 × 16 = 44	44
methane	CH₄	12 + 4 × 1 = 16	16
common salt	NaCl	23 + 35.5 = 58.5	58.5
calcium carbonate (limestone)	CaCO₃	40 + 12 + 3 × 16 = 100	100

A mole of each of these holds exactly the same number of particles, 6.02 × 10²³, but they weigh different amounts and take up different amounts of room, because their particles are different.

Figure 5.1 One mole of each of seven substances, drawn to the same scale (the cubes as solid blocks: loose crystals would take more room, with air between them). Each holds 6.02 × 10²³ particles: atoms of carbon, aluminium, iron and gold; molecules of water and sugar; and for salt, 6.02 × 10²³ of each of its two ions, sodium and chloride (common salt is made of ions, not molecules).

Grams, moles and particles

Three numbers describe a sample: its mass (what a balance shows), the number of moles, and the number of particles. You move between them with the molar mass and the Avogadro constant:

$n = \frac{m}{M} \qquad\qquad N = n \times N_\text{A}$

where n is the number of moles, m the mass in grams, M the molar mass in g/mol, and N the number of particles.

Figure 5.2 The road between grams and particles always goes through moles. Divide a mass by the molar mass to get moles; multiply moles by the Avogadro constant to get particles. Going back, do the opposite.

Try it

Three to try

a.How many molecules are there in one drop of water, of mass 0.05 g?
b.What is the mass of 3.01 × 10²³ atoms of iron? (Fe = 56)
c.A 14.2 kg cylinder of cooking gas holds butane, C₄H₁₀ (and some propane). If it were all butane, how many moles would that be?

Answer

a. $n = 0.05 \div 18 = 0.00278$ mol; $N = 0.00278 \times 6.02 \times 10^{23} =$ 1.7 × 10²¹ molecules.
b. $n = 3.01 \times 10^{23} \div 6.02 \times 10^{23} = 0.5$ mol; $m = 0.5 \times 56 =$ 28 g.
c.Molar mass of C₄H₁₀ = 4 × 12 + 10 × 1 = 58 g/mol. 14.2 kg = 14,200 g, so $n = 14\,200 \div 58 =$ 245 mol (to three figures).

Which of these holds the most molecules?

Pick an answer to see why.

A story from science

How the number was found

In 1811 the Italian scientist Amedeo Avogadro suggested that equal volumes of different gases, at the same temperature and pressure, hold the same number of molecules. Hardly anyone listened for fifty years. Nobody knew what that number was.

The first rough estimate came in 1865, from the Austrian Josef Loschmidt, who worked out the size of air molecules from how gases flow and spread. In 1909 the French physicist Jean Perrin watched tiny grains of resin jiggling in water under a microscope, knocked about by the water molecules around them (the jiggling is called Brownian motion). From how far they jiggled, he could work out how many molecules there are in a mole, and he named the number after Avogadro. Several quite different methods gave the same answer, which convinced the last doubters that atoms are real. Perrin was given the Nobel Prize in Physics in 1926.

In 2019 the world’s scientists fixed the number exactly, by definition: one mole is now exactly 6.022 140 76 × 10²³ particles. To get the value right before fixing it, teams counted the atoms in two balls of almost pure silicon-28, each weighing 1 kg, polished into some of the roundest objects ever made: from the size of the ball and the spacing of the atoms in its crystal, they knew how many atoms it held.

A man in a white laboratory coat and white gloves holding out a shiny, mirror-like ball the size of a large orange — **Figure 5.3** A 1 kg ball of silicon, made in Australia for the Avogadro project. By measuring its size and the spacing of its atoms, scientists counted how many atoms it holds, and so how many there are in a mole. Photo: The Commonwealth Scientific and Industrial Research Organisation of Australia, CC BY-SA 3.0

Atoms are conserved, so mass is conserved

You learned in Grade 9 that in a chemical reaction the total mass of the products equals the total mass of the reactants: the law of conservation of mass, which is why an equation must be balanced Grade 9: How to balance a chemical equation. Now you can see why it is true, and show it with numbers.

In a reaction, atoms are not made, destroyed or turned into atoms of another element. They only change partners: the bonds between them break and new bonds form. Every atom keeps its own mass. So if the same atoms are there before and after, the total mass before and after must be the same.

A balanced equation lets you check this with arithmetic. Take methane burning in a biogas stove:

$\text{CH}_4 + 2\text{O}_2 \rightarrow \text{CO}_2 + 2\text{H}_2\text{O}$

	Reactants	Products
carbon atoms	1	1
hydrogen atoms	4	2 × 2 = 4
oxygen atoms	2 × 2 = 4	2 + 2 × 1 = 4
mass (amu)	16 + 2 × 32 = 80	44 + 2 × 18 = 80

The same atoms, so the same mass: 80 amu before and 80 amu after. Multiply every particle by 6.02 × 10²³ and it is 80 g before and 80 g after. The numbers are bigger, but the balance holds at every scale, because each atom brings its mass with it.

When the balance seems to say otherwise

If mass is conserved, why does a log get lighter as it burns, while a nail gets heavier as it rusts? Because gases come and go, and you can’t see them. When wood burns, most of its atoms leave as carbon dioxide and water vapor, up the chimney. When iron rusts, it takes oxygen in from the air. Weigh only the solid and the mass seems to change. Weigh everything, gases included, and it doesn’t.

Put a reaction on a balance and run it, first in an open container, then sealed so that no gas can get in or out. Watch the balance. (Old camera flashbulbs worked like the sealed steel wool: a thin metal foil sealed in a glass bulb of oxygen, which burned in one bright flash. A bulb weighs the same before and after.)

Reaction:

Container:

Before: 52.00 g
After: —
Change: —

Press Start, and watch the balance.

A story from science

The man who weighed everything

In 1673 the English scientist Robert Boyle heated metals such as tin and lead in sealed glass vessels, and found that the metal gained weight. He thought particles of fire had passed through the glass into the metal. But Boyle had opened the vessel before weighing it.

A hundred years later, in 1774, the French chemist Antoine Lavoisier did the experiment again, and this time weighed the sealed vessel before he opened it. It weighed exactly the same as before the heating. Only when he opened it did air rush in, and the vessel became heavier, by the same amount the tin had gained. The tin hadn’t taken in fire: it had taken in part of the air inside the vessel. That part he later named oxygen.

Lavoisier weighed everything, gases included, with the best balances in Europe. In his textbook of 1789 he wrote (in Robert Kerr’s English translation of 1790): “We may lay it down as an incontestible axiom, that, in all the operations of art and nature, nothing is created; an equal quantity of matter exists both before and after the experiment.” His wife, Marie-Anne Paulze Lavoisier, worked beside him: she kept the laboratory records, translated English chemistry for him, and drew the plates of his apparatus for the book. Lavoisier had also been a member of the company that collected taxes for the king, and in 1794, during the French Revolution, he was executed by guillotine.

A painting of a man in black seated at a table with a red cloth and glass laboratory apparatus, writing, as a woman in a white dress leans on his shoulder and looks out at the viewer — **Figure 5.4** Antoine and Marie-Anne Lavoisier, painted by Jacques-Louis David in 1788. On the table and floor are some of their instruments. Photo: Jacques-Louis David, Public domain

Activity

At home: a sealed bottle on the kitchen scale

You need: an empty 1.5-liter plastic fizzy-drink bottle with its cap (a fizzy-drink bottle is made to hold pressure; don’t use a thin water bottle or a glass one), about 100 mL of vinegar, a level teaspoon of baking soda (about 5 g; no more, so the pressure stays far below what the bottle is made for), a square of tissue paper, and a kitchen scale. Ask an adult to help.

Pour the vinegar into the bottle. Fold the baking soda into the tissue to make a small packet.
Put the bottle, its cap and the packet on the scale together. Write down the reading.
Drop the packet into the bottle and screw the cap on tightly at once. The tissue slows the reaction for a few seconds.
Put the bottle (and nothing else) back on the scale while it fizzes. Is the reading the same?
When the fizzing stops, point the bottle away from faces and unscrew the cap slowly. Weigh it again.

What you should see

The bottle swells and goes hard as carbon dioxide fills it, but while it is sealed the reading stays the same (to within a gram, if a little gas escaped while you were capping it). When you let the gas out, it hisses, and the reading drops by about 2–3 g: that is the mass of the carbon dioxide that left. The reaction is
$\text{NaHCO}_3 + \text{CH}_3\text{COOH} \rightarrow \text{CH}_3\text{COONa} + \text{H}_2\text{O} + \text{CO}_2$
5 g of baking soda (molar mass 84 g/mol) is 0.06 mol, which makes 0.06 mol of CO₂: 0.06 × 44 = 2.6 g. A little of it stays dissolved in the liquid.

Magnesium ribbon is burned in an open crucible. The white ash weighs more than the ribbon did. Why?

Pick an answer to see why.

From atoms to grams: what a reaction makes

A balanced equation is a recipe in particles. Grade 9 showed you that it tells you the ratio of the masses that react Grade 9: Information from a balanced chemical equation. The mole turns that into a recipe in grams. Take carbon (charcoal) burning:

$\text{C} + \text{O}_2 \rightarrow \text{CO}_2$

Figure 5.5 One equation read at three scales. One carbon atom and one oxygen molecule make one carbon dioxide molecule; a mole of each makes a mole; and since a mole of each has its molar mass in grams, 12 g of carbon and 32 g of oxygen make 44 g of carbon dioxide.

The numbers in front of the formulae, the coefficients, count particles, so they also count moles. 2H₂ + O₂ → 2H₂O says that 2 molecules of hydrogen react with 1 of oxygen, so 2 moles of hydrogen react with 1 mole of oxygen, to make 2 moles of water. The ratio of the coefficients is the mole ratio.

So to work out the mass of a product from the mass of a reactant, you always take the same four steps:

Write the balanced equation.
Change the mass you know into moles ( $n = m/M$ ).
Use the mole ratio from the equation to find the moles of the substance you want.
Change those moles into grams ( $m = n \times M$ ).

Worked example

A cylinder of cooking gas

A full cylinder of cooking gas holds 14.2 kg of LPG. Treat it as butane, which burns like this:
$2\text{C}_4\text{H}_{10} + 13\text{O}_2 \rightarrow 8\text{CO}_2 + 10\text{H}_2\text{O}$

a.How much carbon dioxide does a cylinder make when it is all burned?
b.How much oxygen does it take from the kitchen air? Show that mass is conserved.

Answer

a. Moles of butane: 14,200 g ÷ 58 g/mol = 244.8 mol.
Mole ratio C₄H₁₀ : CO₂ = 2 : 8 = 1 : 4, so CO₂ made = 4 × 244.8 = 979 mol.
Mass of CO₂ = 979 mol × 44 g/mol = 43,100 g = 43.1 kg.
b. Mole ratio C₄H₁₀ : O₂ = 2 : 13, so O₂ used = 244.8 × 13 ÷ 2 = 1591 mol, and 1591 × 32 = 50,900 g = 50.9 kg.
Water made: 244.8 × 10 ÷ 2 = 1224 mol, and 1224 × 18 = 22,000 g = 22.0 kg.
Before: 14.2 + 50.9 = 65.1 kg. After: 43.1 + 22.0 = 65.1 kg. The same.

The 14.2 kg of gas in a cylinder puts about 43 kg of carbon dioxide into the air: three times its own mass. (Real LPG is a mixture of butane and propane; with propane, C₃H₈, the answer comes out almost the same, 42.6 kg.)

In C + O₂ → CO₂, how much carbon dioxide do 24 g of carbon make, with plenty of oxygen?

Pick an answer to see why.

When one reactant runs out

In a real reaction you seldom have the exact amounts the equation asks for. A cook with 12 eggs and 1 kg of flour, and a recipe that needs 2 eggs for every 200 g of flour, can make only five batches, not six: then the flour is gone, and 2 eggs are left over. The flour is what limits the cooking.

Reactions are the same. The reactant that is used up first is the limiting reactant: it decides how much product is made. The other is in excess, and some of it is left over. To find which is which, change both masses to moles and compare them with the equation’s mole ratio.

Pick a reaction and set the masses of the two reactants. The bars show the masses before and after the reaction, to the same scale. Try to find the masses where both reactants are used up exactly, then add more of one.

Reaction:

Carbon: 12 g Oxygen: 32 g

Moles of C: 1.00 mol
Moles of O₂: 1.00 mol
Made: CO₂ 44 g
Left over: nothing

Both are used up exactly: their moles are in the ratio 1 : 1, as the equation says. Every atom ends up in what is made, so the bars are the same height.

Evidence from the laboratory

The mole also works backwards: from masses you have weighed, you can find how many atoms of each kind have joined, and so a formula nobody has seen. This is how chemists first found the formulae of compounds, and it is a test of the whole idea.

In a common school experiment, each group weighs a coiled strip of magnesium ribbon in a crucible, heats it strongly until it has all turned to white ash, and weighs the ash. Here are results of the kind a class gets (made up for this page, but typical of real ones):

Group	Magnesium (g)	Ash (g)	Oxygen gained (g)
1	0.12	0.19	0.07
2	0.18	0.30	0.12
3	0.24	0.39	0.15
4	0.30	0.49	0.19
5	0.36	0.59	0.23
6	0.48	0.79	0.31

If the ash were Mg₂O, MgO or MgO₂, how much oxygen should each gram of magnesium pick up? Each makes a different prediction. Test them against the class’s results.

The dots are the six groups’ results. Pick a formula to draw the line it predicts: how much oxygen each mass of magnesium would gain, worked out from the atomic masses.

If the ash were:

Oxygen per g of Mg, predicted: —
Oxygen per g of Mg, the class’s: 0.64 g

Pick a formula. Its line shows how much oxygen each mass of magnesium would gain if that were the formula.

Why is the class’s oxygen a little low? The white smoke you see when magnesium burns is magnesium oxide escaping, so some is lost; and a little magnesium reacts with the nitrogen of the air instead, making magnesium nitride, Mg₃N₂, which gains less mass than the oxide would. Real data always has errors like these. The point is that the results fall close to one straight line through zero, so the mass of oxygen gained is in proportion to the mass of magnesium (twice the magnesium, twice the oxygen), and the line’s slope matches the one formula whose atoms are in a 1 : 1 ratio.

Questions

How many atoms are there in 0.5 mol of aluminium?

Pick an answer to see why.

A student burns some steel wool on a balance, in an open dish. What happens to the reading?

Pick an answer to see why.

In N₂ + 3H₂ → 2NH₃, 2 mol of nitrogen reacts with 3 mol of hydrogen. Which is the limiting reactant?

Pick an answer to see why.

a.

How many water molecules are there in a 250 mL glass of water (250 g)?

Answer

Moles: $n = 250 \div 18 = 13.9$ mol.
Molecules: $N = 13.9 \times 6.02 \times 10^{23} =$ 8.4 × 10²⁴.

b.

Magnesium burns in oxygen: 2Mg + O₂ → 2MgO. Use the atomic masses (Mg = 24, O = 16) to show that the number of atoms of each element and the total mass are the same before and after.

Answer

	Before	After
magnesium atoms	2	2
oxygen atoms	2	2
mass (amu)	2 × 24 + 32 = 80	2 × (24 + 16) = 80

The atoms are only rearranged, from Mg atoms and O₂ molecules into MgO, and each atom keeps its mass, so the total mass is 80 amu on both sides. For a mole of the equation (6.02 × 10²³ times as much), it is 80 g on both sides: 48 g of magnesium and 32 g of oxygen make 80 g of magnesium oxide.

c.

Lime (calcium oxide) is made by heating limestone in a kiln, and Nepal’s cement factories heat limestone in the same way: CaCO₃ → CaO + CO₂. How much lime, and how much carbon dioxide, does 1 tonne (1,000 kg) of limestone give? (Ca = 40, C = 12, O = 16)

Answer

Molar masses: CaCO₃ 100, CaO 56, CO₂ 44 g/mol.
1,000 kg = 1,000,000 g of CaCO₃ = 10,000 mol.
The mole ratio is 1 : 1 : 1, so 10,000 mol of each product:
CaO: 10,000 × 56 = 560,000 g = 560 kg.
CO₂: 10,000 × 44 = 440,000 g = 440 kg.
Check: 560 + 440 = 1,000 kg. Nearly half of the limestone’s mass goes into the air as carbon dioxide, which is one reason cement making gives out so much of it.

d.

Railway tracks are welded with the thermite reaction, which makes white-hot liquid iron: Fe₂O₃ + 2Al → Al₂O₃ + 2Fe. What mass of aluminium is needed for 320 g of iron(III) oxide, and how much iron is made? (Fe = 56, Al = 27, O = 16)

Answer

Molar mass of Fe₂O₃ = 2 × 56 + 3 × 16 = 160 g/mol, so 320 g is 2 mol.
Aluminium: mole ratio Fe₂O₃ : Al = 1 : 2, so 4 mol, 4 × 27 = 108 g.
Iron: mole ratio Fe₂O₃ : Fe = 1 : 2, so 4 mol, 4 × 56 = 224 g.
(Aluminium oxide made: 2 mol × 102 g/mol = 204 g. Check: 320 + 108 = 428 g = 204 + 224 g.)

e.

A student says: “When paper burns, only a little ash is left. So mass is destroyed in burning.” Explain what is wrong, and describe an experiment that would test the claim properly.

Answer

The paper’s atoms are not destroyed. Paper is made mostly of carbon, hydrogen and oxygen; when it burns, they join oxygen from the air to make carbon dioxide and water vapor, which are gases and drift away. Only the minerals in the paper stay behind as ash. The products, gases included, weigh more than the paper did, because they also contain the oxygen taken from the air.
To test it, burn the paper in a closed container that holds enough air, such as a large sealed glass jar (with an adult; set it alight with a lens and sunlight, or a hot wire, so nothing has to be opened). Weigh the whole sealed jar before and after: the reading stays the same. Opening it afterwards and weighing only the ash would repeat Boyle’s mistake.

f.

a.8 g of methane is burned with 16 g of oxygen: CH₄ + 2O₂ → CO₂ + 2H₂O. Which is the limiting reactant?
b.What masses of carbon dioxide and water are made, and what is left over?
c.Show that the total mass is conserved.

Answer

a.Moles: CH₄ = 8 ÷ 16 = 0.5 mol; O₂ = 16 ÷ 32 = 0.5 mol. 0.5 mol of methane would need 1 mol of oxygen, but there is only 0.5 mol, so oxygen is limiting.
b.0.5 mol of O₂ burns 0.25 mol of CH₄, making 0.25 mol of CO₂ (0.25 × 44 = 11 g) and 0.5 mol of H₂O (0.5 × 18 = 9 g). Left over: 0.25 mol of methane, 0.25 × 16 = 4 g.
c.Before: 8 + 16 = 24 g. After: 11 + 9 + 4 = 24 g.

(In a real flame without enough oxygen, some carbon would be left as soot or carbon monoxide instead; that is why a gas stove needs a good air supply.)

g.

A balanced equation is sometimes described as “a statement that atoms are conserved”. Using 2KClO₃ → 2KCl + 3O₂ and the atomic masses K = 39, Cl = 35.5, O = 16, explain how the numbers support the claim that mass is conserved because atoms are.

Answer

Count each kind of atom: potassium 2 and 2; chlorine 2 and 2; oxygen 2 × 3 = 6 and 3 × 2 = 6. The same atoms are present before and after, only joined differently.
Each atom has a fixed mass, so the total masses must agree: before, 2 × (39 + 35.5 + 48) = 2 × 122.5 = 245 amu; after, 2 × (39 + 35.5) + 3 × 32 = 149 + 96 = 245 amu.
Because a mole of anything is the same number of particles, multiplying every particle by 6.02 × 10²³ turns this into grams: 245 g of potassium chlorate gives 149 g of potassium chloride and 96 g of oxygen. The mass is conserved at every scale, because it is the atoms, each with its own mass, that are conserved.

Think about it

Where does a tree come from?

In the early 1600s the Flemish scientist Jan Baptist van Helmont planted a 5-pound willow shoot in a pot of 200 pounds of dried soil, and for five years gave it nothing but water. The willow grew to 169 pounds, yet the soil, dried and weighed again, had lost only about 2 ounces. He concluded that the tree was made from water (his account was published in 1648, after his death). What was he missing, and where did most of the tree’s mass really come from?

Answer

He was missing a gas he couldn’t see. A tree is made mostly of cellulose and other compounds of carbon, hydrogen and oxygen. The water gives some of the hydrogen and oxygen, but the carbon, and much of the mass, comes from carbon dioxide from the air, taken in through the leaves in photosynthesis: 6CO₂ + 6H₂O → C₆H₁₂O₆ + 6O₂. Only a few grams came from the soil, as minerals. Like the burning paper, it is a case where the mass seemed to come from nowhere because a gas was taking part.

Where to go next

Why Do Some Reactions Get Hot? Now that you can count bonds by the mole, find out why breaking and making them gives out or takes in heat, in kilojoules per mole.
What Makes a Reaction Go Faster? A balanced equation tells you how much a reaction makes, but not how fast. Molecules have to collide hard enough.
Why Do Some Reactions Stop Halfway? Not every reaction makes all the product the masses here predict: some run both ways and settle at a balance.