The mole concept — counting by weighing: Question bank

Before we start, three words we lean on constantly:

Entity — the thing we are counting: an atom, a molecule, an ion, or even an electron. You must always name it.
Mole — a fixed count of entities, == $6.022 \times 1 0^{23}$ == of them (see Avogadro's Number).
Molar mass $M$ — grams per one mole of a named entity, in $g mol^{- 1}$ (see Molar Mass Calculations).

True or false — justify

TF1. "One mole of any substance always weighs the same."

False. A mole is a fixed count, not a fixed mass; 1 mol of H weighs

\approx 1

g while 1 mol of Pb weighs

\approx 207

g because their atoms have different masses.

TF2. "One mole of hydrogen atoms and one mole of lead atoms contain the same number of atoms."

True. "Mole" fixes the count at

N_{A}

regardless of what the entity is — that is the whole point of the unit. Their masses differ, their counts do not.

TF3. "Molar mass increases if I scoop out a bigger sample."

False. Molar mass is intensive — grams per mole — so it is fixed by the substance. Only the total mass

m

and moles

n

grow with the sample.

TF4. "1 mole of

O_{2}

contains

1

mole of oxygen atoms."

False. It contains

1

mole of

O_{2}

molecules but

2

moles (

1.204 \times 1 0^{24}

) of oxygen atoms, because each molecule holds two atoms.

TF5. "Avogadro's number is dimensionless — it's just a pure count."

False. It carries the unit

mol^{- 1}

(entities per mole); without it,

N = n N_{A}

would not have units of "entities". See Avogadro's Number.

TF6. "Since molar mass equals atomic mass in u, the two are the same quantity."

False. They are numerically equal by design but different quantities: atomic mass is per-atom in

u

, molar mass is per-mole in

g mol^{- 1}

. The equality is an engineered convenience, not an identity.

TF7. "22 g of

CO_{2}

and 22 g of

Ne

contain the same number of molecules."

False (and a units trap).

M (CO_{2}) = 44

M (Ne) = 20

, so

n

differs (

0.5

1.1

mol); equal mass does not mean equal moles unless molar masses match.

TF8. "Doubling the number of atoms doubles the number of moles."

True.

n = N / N_{A}

is a straight proportionality, so scaling the count scales the moles by the same factor.

TF9. "You can have

0.5

of a mole."

True. The mole is a unit of amount, like a metre; fractional moles (

0.5

mol

= 3.011 \times 1 0^{23}

entities) are perfectly ordinary. The count of entities stays a whole number though.

TF10. "The mole is defined using the mass of a proton."

False. It is fixed by an exact declared count,

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

; historically it was tied to

12

g of carbon-12 (see Atomic Mass & Isotopes), not to a proton's mass.

Spot the error

SE1. "

n = M / m

, so if the sample weighs

18

g and

M = 18

, then

n = 18/18 = 1

— and if

M = 36

n = 2

Error: the formula is inverted. It is

n = m / M

, so a heavier molar mass gives fewer moles:

M = 36

would give

n = 0.5

, not

2

SE2. "To get atoms in

2

mol of

H_{2} O

, compute

N = 2 \times N_{A} = 1.2 \times 1 0^{24}

atoms."

Error: that counts molecules, not atoms. Each

H_{2} O

has

3

atoms, so the atom count is

2 \times 3 \times N_{A} = 3.6 \times 1 0^{24}

. Always ask "moles of what entity?"

SE3. "Mass of one carbon atom

= N_{A} / M = 6.022 \times 1 0^{23} /12

Error: divided the wrong way. One atom's mass is

M / N_{A}

— split the molar mass among all

N_{A}

atoms — giving

\approx 1.99 \times 1 0^{- 23}

g, a tiny number, not a huge one.

SE4. "For

NaCl

, molar mass

= 23 \times 35.5

because there are two elements."

Error: multiplied instead of added. You sum the atomic masses of the atoms in the formula:

M = 23 + 35.5 = 58.5 g mol^{- 1}

. See Molar Mass Calculations.

SE5. "

1

g of any gas contains

1

mole because grams and moles both start at

1

Error: confusing two different units. Grams measure mass, moles measure count;

1

g of

H_{2}

0.5

mol,

1

g of

He

0.25

mol. Only mass numerically equal to

M

gives exactly

1

mol.

SE6. "In

N = n N_{A}

the units are

mol \times mol = mol^{2}

Error: wrong units on $N_{A}$ .

N_{A}

mol^{- 1}

, so

mol \times mol^{- 1}

cancels to a pure count — exactly what "number of entities" should be.

SE7. "Empirical formula

CH_{2} O

means the molar mass is definitely

30 g mol^{- 1}

Error: empirical ≠ molecular. The real molecule could be

(CH_{2} O)_{n}

; glucose is

C_{6} H_{12} O_{6}

with

M = 180

. Empirical formula fixes only the ratio, not the size (see Empirical & Molecular Formulae).

Why questions

WHY1. "Why can we count atoms by weighing at all?"

Because a fixed count of atoms (

N_{A}

) has a fixed, measurable mass (

M

); the balance reads mass, and

N = (m / M) N_{A}

converts that mass into a count.

WHY2. "Why did chemists choose

N_{A}

to be

6.022 \times 1 0^{23}

specifically?"

So that

1 u \times N_{A} = 1 g mol^{- 1}

, making "mass in u" and "molar mass in g/mol" the same number — a deliberate convenience, not an accident.

WHY3. "Why is molar mass an intensive property?"

Because it is a ratio (mass ÷ moles); both numerator and denominator scale together with sample size, so the ratio stays fixed — like density, it doesn't depend on how much you have.

WHY4. "Why must we always state the entity when giving moles?"

Because

1

mol of

O_{2}

1

mol of

O

atoms, and

1

mol of electrons are three different physical amounts; the count is meaningless until "mole of what" is fixed.

WHY5. "Why does a lighter element give more atoms per gram?"

Fewer grams are spent per atom, so a fixed mass buys more of them;

n = m / M

makes

n

inversely proportional to

M

. This is Stoichiometry's reason to work in moles, not grams.

WHY6. "Why is the mole an SI base unit rather than a derived one?"

Because "amount of substance" is treated as a fundamental physical quantity of its own, not reducible to mass, length or time (see Units & Measurement).

Edge cases

EC1. "How many moles are in

0

g of copper?"

Exactly

0

mol, since

n = m / M = 0/ M = 0

; zero mass means zero entities — the formula behaves sensibly at the boundary.

EC2. "What is the molar mass of an isotope mixture like natural chlorine?"

A weighted average of the isotope masses by abundance, giving

35.5 g mol^{- 1}

— no single Cl atom actually weighs

35.5

u (see Atomic Mass & Isotopes).

EC3. "Can the number of entities

N

be a fraction, like

3.5 \times 1 0^{23}

The moles

n

can be fractional, but the true count of physical entities must be a whole number;

N = n N_{A}

rounds to the nearest integer of real atoms.

EC4. "What is the molar mass of a single free electron, and does the mole idea still apply?"

About

5.49 \times 1 0^{- 4} g mol^{- 1}

; the mole applies to any entity, so one mole of electrons is a perfectly valid amount.

EC5. "If I have

N_{A}

atoms of gold, how much do they weigh — and why is that the whole trick?"

They weigh

M = 197

g, one molar mass, because

N_{A}

atoms is one mole by definition; this exact link is what lets the balance "count" for us.

EC6. "Does

n = m / M

still hold for a mixture like air?"

Only if you use an average molar mass for the mixture (

\approx 29 g mol^{- 1}

for air); the formula needs one

M

, so mixtures require a composition-weighted value.

EC7. "Is 'a mole of sand grains' a meaningful quantity?"

Mathematically yes —

6.022 \times 1 0^{23}

grains — but physically absurd, as that mass of sand dwarfs the Earth; the mole is designed for atom-scale entities where such counts are ordinary.

Connections

Avogadro's Number — the constant every trap here pivots on.
Atomic Mass & Isotopes — sources the averaging edge cases.
Molar Mass Calculations — the building of $M$ misfires in SE4/SE7.
Stoichiometry — why moles beat grams as the reaction currency.
Empirical & Molecular Formulae — the empirical-vs-molecular trap.
Units & Measurement — the mole as an SI base unit.