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Reading a formula as a count of atoms, multiplying by the relative atomic masses, and adding.
Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.
By the end of this lesson you will be able to work out the molar mass of any compound whose formula you can read: count the atoms of each element, multiply each count by the element's relative atomic mass, and add the results. You will be able to do it for a formula with brackets and for a salt that crystallizes with water in it, and to say why adding the atomic masses once each — the commonest wrong answer there is — leaves every subscript out.
You can read a formula as a count of atoms, brackets included. A molar mass is that count with a multiplication and an addition on the end, so if the reading is right the mass is right — and if the reading is wrong, nothing later can rescue it.
| Term | What it means |
|---|---|
| Relative atomic mass | How heavy one atom of an element is on a standard scale; it has no unit. |
| Relative formula mass | The sum of the relative atomic masses of every atom in a formula. |
| Molar mass | The mass of one mole of a substance, in grams per mole. |
| Mole | A fixed, very large number of particles. |
| Water of crystallization | Water locked into a crystal, written after a dot in the formula. |
The molar mass of a substance is worked out in three moves, and none of them is new.
That is all. For $\mathrm{CaCO_3}$: one calcium at 40, one carbon at 12, three oxygens at 16 each, so $40 + 12 + 48 = 100$ g/mol.
The unit is grams per mole, and the per mole is the whole idea. A molar mass is not the mass of a molecule — a molecule of calcium carbonate weighs an unimaginably small amount. It is the mass of a mole of them, which is a number of particles chosen precisely so that the answer comes out in grams you can weigh on a balance.
That is why the mole exists at all. The periodic table gives you a number per atom; the mole turns it into a number per gram, and the bridge is the same number in both directions.
Another way: picture
Think of a grocery receipt with quantities. Three cans at 400 g and one bag at 1000 g is not four items at 1400 g each; it is $3 \times 400 + 1000$. A formula is the quantities and the periodic table is the price list.
Another way: steps
For $\mathrm{Mg(OH)_2}$:
Read the formula. List each element once, with the number of its atoms in one formula unit. A subscript belongs to the symbol just before it; a subscript after a bracket multiplies everything inside the bracket; a number after a dot multiplies the whole water molecule.
Look up the masses. Use the course's table below, so every answer is built from the same numbers.
Multiply along. For each element, atoms times relative atomic mass. Writing this as a small table — element, count, mass, contribution — is the surest way to keep a long formula straight.
Add down. The molar mass is the sum of the contributions, in grams per mole.
Check the work. Three quick tests catch almost every error. First, is the answer larger than the heaviest single atom in the formula? It must be. Second, add up your atom counts and compare them with the formula: did you count every atom, including those inside a bracket? Third, recompute the total by adding the atomic masses once each; if that gives your answer and the formula has any subscript bigger than 1, you have dropped the subscripts.
Reading the formula as a count is allowed because that is what a formula is: a statement of how many atoms of each element are in one unit of the substance.
Multiplying by the atomic mass is allowed because every atom of an element has, on average, the same mass. Three oxygen atoms weigh three times what one does.
Adding the contributions is allowed because mass is additive. A formula unit weighs exactly what its atoms weigh together; bonding does not measurably change the total on this scale.
Attaching grams per mole is allowed because the mole was defined so that this works: a mole of carbon-12 atoms has a mass of exactly 12 grams, so a mole of anything has a mass in grams equal to its relative formula mass. The unitless number from the periodic table and the weighable number on a balance are the same number by design.
Every molar mass in this course is built from this list, rounded as shown. Using a different rounding gives a slightly different answer and is not wrong chemistry — but one course has to pick one set, or two items disagree about the mass of chlorine and a correct answer gets marked wrong.
| Element | Symbol | Relative atomic mass |
|---|---|---|
| hydrogen | $\mathrm{H}$ | 1 |
| carbon | $\mathrm{C}$ | 12 |
| nitrogen | $\mathrm{N}$ | 14 |
| oxygen | $\mathrm{O}$ | 16 |
| sodium | $\mathrm{Na}$ | 23 |
| magnesium | $\mathrm{Mg}$ | 24 |
| aluminum | $\mathrm{Al}$ | 27 |
| sulfur | $\mathrm{S}$ | 32 |
| chlorine | $\mathrm{Cl}$ | 35.5 |
| potassium | $\mathrm{K}$ | 39 |
| calcium | $\mathrm{Ca}$ | 40 |
| iron | $\mathrm{Fe}$ | 56 |
| copper | $\mathrm{Cu}$ | 63.5 |
| zinc | $\mathrm{Zn}$ | 65 |
Significant figures. You will see them in every worked example in this course, and they are never what an answer is graded on. Chlorine's $35.5$ carries three, and a molar mass built from it carries three; a mass worked out from it is quoted to the same precision. The answers here are all exact, so this is a habit being shown rather than a rule being enforced.
Most of the masses in the table are close to whole numbers, and chlorine's $35.5$ stands out. The reason is isotopes, from the mass spectrum lesson. Natural chlorine is about three quarters chlorine-35 and one quarter chlorine-37, and the relative atomic mass is the weighted average of the two: $0.75 \times 35 + 0.25 \times 37 = 35.5$.
No single chlorine atom weighs 35.5. But a sample big enough to weigh holds so many atoms that the mixture is always the same, so the average is the number that predicts what a balance will read. That is the only thing a molar mass has to do.
Copper's $63.5$ comes from the same arithmetic, with copper-63 and copper-65. Elements with one dominant isotope — carbon, oxygen, sodium — come out close to whole numbers, which is why most molar masses do too.
Some salts crystallize with water molecules locked into the structure, and the formula says so with a dot: $\mathrm{CuSO_4 \cdot 5H_2O}$ means five water molecules for every copper sulfate.
The dot is a multiplier like any other, and the water is part of the substance. So the molar mass is the salt plus the water:
$$159.5 + 5 \times 18 = 249.5 \text{ g/mol}.$$
This matters more than it looks. Weighing out $159.5$ g of the blue crystals in the belief that it is a mole gives you about two thirds of a mole of copper sulfate and a quantity of water you did not intend, and every calculation afterwards is out by that factor with nothing to show for it.
Every winter, highway departments across the northern United States spread millions of tons of salt. Rock salt, sodium chloride, stops working below about 15 °F, so for the coldest nights many states switch to calcium chloride, $\mathrm{CaCl_2}$, which keeps melting ice down to around −20 °F.
What melts ice is the number of dissolved particles, not their mass, and that is where the molar mass comes in. Calcium chloride's molar mass is $40 + 2 \times 35.5 = 111$ g/mol, and each formula unit dissolves into three ions. Sodium chloride's is $23 + 35.5 = 58.5$ g/mol, giving two ions. Per ton, sodium chloride gives about $2 \times 1{,}000{,}000 \div 58.5 \approx 34{,}000$ moles of ions, and calcium chloride about $3 \times 1{,}000{,}000 \div 111 \approx 27{,}000$.
So pound for pound, plain salt actually puts more particles into the slush. Calcium chloride earns its higher price — often three or four times as much per ton — because it works at temperatures where sodium chloride does not, and because it releases heat as it dissolves. A road engineer comparing the two is doing exactly this lesson's arithmetic, and a supplier quoting a molar mass of 75.5 for calcium chloride — the subscript dropped — would throw every comparison off by nearly a third.
A calcium supplement listing 600 mg of calcium per tablet contains $600 \times 100 \div 40 = 1500$ mg of calcium carbonate, because calcium is only 40 of the 100 g in each mole. The molar mass is how the label's two numbers are connected, and it is why a calcium citrate tablet, with a much heavier molar mass per calcium, has to be larger to deliver the same dose.
Adding the atomic masses once each. $\mathrm{H_2SO_4}$ read as $1 + 32 + 16 = 49$ rather than $2 + 32 + 64 = 98$. Every subscript has been dropped. This is by far the commonest error, and it is not an arithmetic slip — it is the formula being read as a list of ingredients rather than as a count.
Adding the atom counts. $\mathrm{H_2SO_4}$ has 7 atoms in a formula unit, and 7 is not a mass. Counting and weighing are different questions, and the two numbers are only equal by coincidence.
Missing the bracket. $\mathrm{Ca(NO_3)_2}$ with three oxygens instead of six. Lesson 1's error, arriving with a bigger consequence attached.
Leaving out the water of crystallization. The dot is not a footnote. If it is in the formula, it is in the mass.
Using the atomic number. The periodic table shows two numbers for each element. The smaller, whole one counts protons; the mass is the other. Oxygen is 8 by number and 16 by mass, and using 8 halves its contribution.
Read the formula as counts.
$\mathrm{H_2SO_4}: 2 \text{ H}, 1 \text{ S}, 4 \text{ O}$
Read it as a count first.
Weigh the hydrogen atoms.
$2 \times 1 = 2$
Two at one each.
Weigh the sulfur atom.
$1 \times 32 = 32$
A single atom.
Weigh the oxygen atoms.
$4 \times 16 = 64$
Most of the mass.
Add the three contributions.
$2 + 32 + 64 = 98 \text{ g/mol}$
Dropping subscripts gives 49, exactly half — a coincidence that hides the error.
Read the bracket first.
$\mathrm{Al_2(SO_4)_3}$
Three sulfates, each one S and four O.
Count every element.
$2 \text{ Al}, 3 \text{ S}, 12 \text{ O}$
Twelve oxygens, not four.
Weigh the aluminum atoms.
$2 \times 27 = 54$
Two at twenty-seven.
Weigh the sulfur atoms.
$3 \times 32 = 96$
One per sulfate.
Weigh the oxygen atoms.
$12 \times 16 = 192$
Four per sulfate.
Add the contributions.
$54 + 96 + 192 = 342 \text{ g/mol}$
Missing the bracket gives 150, which looks plausible.
Split the formula at the dot.
$\mathrm{CuSO_4} \text{ and } \mathrm{5H_2O}$
Salt and water separately.
Weigh the copper atom.
$1 \times 63.5 = 63.5$
From the course table.
Weigh the sulfate group.
$32 + 4 \times 16 = 96$
One sulfur, four oxygens.
Add for the salt.
$63.5 + 96 = 159.5$
The anhydrous molar mass.
Weigh one water molecule.
$2 \times 1 + 16 = 18$
Two hydrogens, one oxygen.
Weigh all five waters.
$5 \times 18 = 90$
The dot's number multiplies the whole molecule.
Add salt and water.
$159.5 + 90 = 249.5 \text{ g/mol}$
The mass you weigh for a mole of crystals.
Read the bracket first.
$1 \text{ Ca}, 2 \text{ N}, 6 \text{ O}$
The subscript 2 reaches nitrogen and oxygen.
Weigh each element.
$40; \ 2 \times 14 = 28; \ 6 \times 16 = 96$
Count times atomic mass.
Add the contributions.
Three students work out a number for calcium carbonate, $\mathrm{CaCO_3}$. Which of them has the molar mass?
Complete the worked solution: baking soda is $\mathrm{NaHCO_3}$. Use sodium at twenty-three, hydrogen at one, carbon at twelve and oxygen at sixteen. Find the oxygen's contribution, the three single atoms together, and the molar mass.
Weigh the oxygen atoms.
$\text{three} \times \text{sixteen} =$ o
The subscript reaches only the oxygen.
Add the single atoms.
$\text{sodium} + \text{hydrogen} + \text{carbon} =$ s
One atom each, so no multiplying.
Add the two parts.
$(\text{oxygen}) + (\text{single atoms}) =$ m
That is grams per mole.
What is the molar mass of copper(II) sulfate, $\mathrm{CuSO_4}$? Give it in grams per mole.
Answer: g/mol
Complete the two numbers for potassium carbonate.
In $\mathrm{K_2CO_3}$, the $\mathrm{O}$ contributes a g to every mole, and the molar mass is b g/mol.
$\mathrm{CaCl_2}$, calcium chloride, is the road salt Michigan trucks spread below about 0 °F. A supplier's spreadsheet needs its molar mass. Using the course's atomic masses, what is it in grams per mole?
The answer: a g/mol.
A bottle of gypsum is labeled $\mathrm{CaSO_4 \cdot 2H_2O}$. The water in the crystal is part of the substance and has to be weighed with it. Work out the molar mass in two parts.
| mass in one mole, in g | |
|---|---|
| $\mathrm{CaSO_4}$ | |
| the water | |
| the whole crystal |
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
Work out the molar mass of aluminum sulfate, $\mathrm{Al_2(SO_4)_3}$, one element at a time. The relative atomic masses are already filled in.
| atoms in the formula | relative atomic mass | mass in one mole, in g | |
|---|---|---|---|
| $\mathrm{Al}$ | 27 | ||
| $\mathrm{S}$ | 32 | ||
| $\mathrm{O}$ | 16 | ||
| Molar mass | — | — |
You can turn a formula into a molar mass in grams per mole. Say what the molar mass of magnesium hydroxide is, and what number you would get by forgetting the bracket. Next: what that number is for — turning a mass you can weigh into a number of moles you can put into an equation.
16. Your turn: the molar mass of calcium nitrate, $\mathrm{Ca(NO_3)_2}$., step 3
$40 + 28 + 96 = 164 \text{ g/mol}$
The mass of one mole.