Some Basic Concepts of Chemistry · Lesson 7 of 11

The Mole and Molar Mass

“Connect laboratory masses with the enormous numbers of atoms, molecules, ions, and formula units they contain.”

Learning Objectives

• Explain the mole as a counting unit for specified entities. • Use the exact Avogadro constant and distinguish it from a rounded working value. • Interpret molar mass and distinguish it from particle mass. • Convert between mass, amount, and entity count. • Count constituent atoms from the amount of a molecular compound.

Counting far more than we can handle individually

We count eggs by the dozen because grouping makes counting convenient. Atoms require a much larger grouping. Even a small sample contains so many entities that direct counting is impractical, but its mass can be measured. The mole provides a connection between a measurable sample and a specified number of microscopic entities.

Definition

Mole

The SI unit of amount of substance. One mole contains exactly $6.02214076 \times 1 0^{23}$ specified elementary entities.

Avogadro constantLaTeX

N_{A} = 6.02214076 \times 1 0^{23} mo l^{- 1}

The defining value is exact. For many calculations here, use

6.022 \times 1 0^{23} mo l^{- 1}

as a rounded working value.

The entities must be named. One mole of hydrogen atoms and one mole of hydrogen molecules contain the same number of specified entities, but each hydrogen molecule has two atoms. Mole can also count ions, electrons, formula units, or specified groups. A mole is an amount unit, not a special type of particle.

One mole of water molecules and one mole of carbon dioxide molecules have equal molecule counts, but different masses because their molecules have different masses. A photograph showing one-mole samples in different containers illustrates this distinction: equal amounts do not imply equal masses or equal volumes.

Molar mass gives mass per mole

Molar mass tells us how much mass corresponds to one mole of a substance. It depends on what substance and entities are being counted. At ordinary school-calculation precision, the numerical molecular or formula mass in u gives the numerical molar mass in grams per mole.

Definition

Molar mass

Mass divided by amount of substance, usually expressed here in grams per mole. Its symbol is M.

A water molecule has mass about $18.02 u$ and one mole of water has mass about $18.02 g$ . The units tell you which description is intended. Similarly, sodium chloride has formula mass about $58.5 u$ and molar mass about $58.5 g mo l^{- 1}$ . These numerical correspondences are used at the stated precision; they do not make the mass of one molecule equal to the mass of a mole.

Historically, carbon-12 linked the mole to a 12 g sample. The modern mole is defined directly by the exact entity count. Measuring the mass of a carbon-12 atom and dividing a bulk carbon mass by it illustrates the counting idea, but that measurement is not needed to redefine the already fixed Avogadro constant.

Example — Why the constant has this enormous scale

Problem
Using the source’s approximate carbon-12 atom mass $1.992648 \times 1 0^{- 23} g$ , estimate the number of atoms in $12 g$ .

1.Divide sample mass by mass per atom: $12/ (1.992648 \times 1 0^{- 23}) \approx 6.02214 \times 1 0^{23}$ atoms.
2.The gram units cancel, leaving the number of atoms. The result is enormous because each atom is extremely light.
3.This is an approximate historical counting illustration. The modern value of $N_{A}$ is fixed exactly by definition.

A conversion bridge that can be used in either direction

If every mole has mass M, a sample of n moles has mass nM. If every mole contains the Avogadro number of entities, it contains n times that count. These two relationships form a reliable conversion route; keep the entities and units visible at every step.

Mass and amountLaTeX

n = \frac{m}{M} m = n M

n

is amount in mol,

m

is sample mass in g, and

M

is molar mass in

g mo l^{- 1}

Amount and entity countLaTeX

N = n N_{A} n = \frac{N}{N _{A}}

N

is the number of the specified entities, while

N_{A}

is the number per mole. A particle count is a count, not a mass.

The mass–amount–count bridge— Use molar mass only for the mass–mole step and the Avogadro constant only for the mole–count step.

Example — From water mass to molecule count

Problem
How many moles and molecules are in $9.01 g$ water? Use $M = 18.02 g mo l^{- 1}$ .

1.First convert mass to amount: $n = 9.01/18.02 = 0.500 mol$ .
2.Then convert amount to count: $N = (0.500) (6.022 \times 1 0^{23}) = 3.011 \times 1 0^{23}$ molecules before final rounding.
3.The input mass has three significant figures, so report $3.01 \times 1 0^{23}$ water molecules. The sample is half a mole, so half the Avogadro number is sensible.

Example — From count back to mass

Problem
A carbon dioxide sample contains $3.011 \times 1 0^{23}$ molecules. Find its mass using $M = 44.01 g mo l^{- 1}$ .

1.Convert count to amount: $n = (3.011 \times 1 0^{23}) / (6.022 \times 1 0^{23}) = 0.5000 mol$ .
2.Convert amount to mass: $m = (0.5000) (44.01) = 22.005 g$ .
3.Report $22.00 g$ to four significant figures under the stated round-to-even convention: the halfway digit leaves the retained even zero unchanged. The units show why multiplying the count directly by 44.01 grams would be wrong.

Counting atoms inside molecules

A molecular subscript describes each molecule, so it also supplies a ratio of amounts. One mole of water molecules contains two moles of hydrogen atoms and one mole of oxygen atoms. This counts atoms present within the molecules; it does not claim that separate elemental hydrogen and oxygen samples are physically present.

Example — Three moles of ethane

Problem
For $3.00 mol$ of $C_{2} H_{6}$ , find the amounts of carbon and hydrogen atoms and the number of ethane molecules.

1.Each ethane molecule contains 2 C atoms and 6 H atoms. Thus carbon atom amount is $2 (3.00) = 6.00 mol$ and hydrogen atom amount is $6 (3.00) = 18.0 mol$ .
2.The ethane molecule count is $(3.00) (6.022 \times 1 0^{23}) = 1.81 \times 1 0^{24}$ molecules to three significant figures.
3.The total atom amount is $8 (3.00) = 24.0 mol$ of atoms. Keep this separate from $3.00 mol$ of molecules.

Example — Comparing equal masses

Problem
Which contains more atoms: $1.00 g$ sodium or $1.00 g$ gold? Use molar masses 23.0 and 197 respectively.

1.Sodium amount is $1.00/23.0 = 0.0435 mol$ of atoms; gold amount is $1.00/197 = 0.00508 mol$ of atoms.
2.Both amounts would be multiplied by the same $N_{A}$ to find their counts. Therefore sodium contains more atoms.
3.For equal masses of elements counted as atoms, a lower atomic molar mass gives more atoms. For molecular elements, account for the number of atoms per molecule too.

Do not mix mass, amount, and concentration

$0.50 mol$ names an amount. It does not state a volume and is not a concentration. Also distinguish a sample mass of $52 g$ from a particle-scale mass of $52 u$ : the numerical equality 52 does not make the samples equal.

Quick check

One mole of any specified entity contains:

Quick check

How many moles of hydrogen atoms are in $2.00 mol$ water?

Quick check

What is the amount in $11.0 g$ of a substance with $M = 44.0 g mo l^{- 1}$ ?

Quick check

Equal amounts of two different molecular substances must have:

Quick check

Which conversion gives a particle count from sample mass?

Practice Problems

Find moles and molecules in $4.40 g$ carbon dioxide using $M = 44.0 g mo l^{- 1}$ .
Find the mass of $0.250 mol$ NaCl using $M = 58.5 g mo l^{- 1}$ .
Find the amounts of carbon and hydrogen atoms in $0.500 mol$ methane.
Find the atom count in $52.0 mol$ argon and in $52.0 g$ helium, using He = 4.00.
Explain how $52.0 u$ helium can correspond to only about 13 atoms when He = 4.00 u.
Compare the numbers of atoms in $1.00 g$ each of Au, Na, Li, and $C l_{2}$ using Au = 197, Na = 23.0, Li = 6.94, Cl = 35.45.

Key Takeaways

The mole counts a specified kind of entity and has an exact defining entity number. Molar mass is mass per mole; particle mass is expressed on a microscopic scale. Use molar mass to convert mass and amount, then the Avogadro constant to convert amount and count. Formula subscripts convert molecular amounts into constituent atom amounts. Equal moles need not mean equal masses, volumes, or total atom counts.

Previous · Lesson 6

Atomic, Molecular, and Formula Masses

Next · Lesson 8

Percentage Composition and Chemical Formulas

Counting far more than we can handle individually

Molar mass gives mass per mole

A conversion bridge that can be used in either direction

Counting atoms inside molecules

Quiz

Practice Problems

Key Takeaways