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A Mole Is a Counting Word Like a Dozen, Only Unimaginably Larger

Understand the mole, Avogadro's number, gram atoms and gram molecules, derive why molecular mass is twice the vapour density, and move confidently between mass, moles, number of particles and gas volume at STP through a set of fully worked problems.

How can chemists count atoms they will never see?

A shopkeeper selling rice never counts the grains. The rice is weighed instead. If one grain has a known mass, then a kilogram of rice contains a known number of grains, and weighing does the counting.

Chemists face the same problem with atoms, only far worse. Atoms are so small that no balance can weigh one, and no one could count even a speck of them. Yet chemical equations are statements about numbers of particles — one carbon atom with two oxygen atoms, two hydrogen molecules with one oxygen molecule. To use them in a laboratory, you need a way of weighing out a known number of particles.

The mole is that way. It is a counting unit, exactly like a dozen:

- A dozen means of anything — eggs, bananas or pencils
- A mole means of anything — atoms, molecules or ions

The number is chosen so that one mole of particles has a mass in grams equal to the particles' relative mass. One mole of carbon atoms weighs ; one mole of water molecules weighs . **So weighing of water is counting out molecules, just as weighing a kilogram of rice counts out its grains.

Everything in this part is a consequence of that one idea.

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Definitions: mole, Avogadro's number, gram atom and gram molecule
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Vapour density, which gives the molecular mass of a gas from a simple weighing
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Conversions between mass, moles, number of particles and volume of a gas
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Problems that combine them

It also joins up with Part 1.** There, litres of any gas at STP weighed its molecular mass in grams. That volume contains exactly one mole of molecules — which is why it behaves the same for every gas.

One habit makes the whole topic easier. Every calculation passes through the number of moles. Whatever you are given — grams, litres or particles — turn it into moles first, and whatever you are asked for, get it from moles last.

This page covers the second part of the ICSE Class 10 Chemistry chapter on mole concept and stoichiometry: the mole and related definitions, vapour density, interconversions, and numerical problems.

What do mole, Avogadro's number, gram atom and gram molecule mean?

A mole is the amount of substance containing Avogadro's number of particles; a gram atom and a gram molecule are one mole of atoms and one mole of molecules, weighing the atomic or molecular mass in grams.

Mole. The amount of a substance that contains as many elementary particles — atoms, molecules or ions — **as there are atoms in exactly of carbon-12. Today that amount is defined directly by fixing the number of particles.

Avogadro's number**, . The number of particles in one mole:



Gram atomic mass. The atomic mass of an element expressed in grams. Oxygen, atomic mass , has a gram atomic mass of .

Gram atom. The quantity of an element whose mass in grams equals its atomic mass. One gram atom is one mole of atoms and contains atoms.

Gram molecular mass. The molecular mass of a substance expressed in grams. Water, molecular mass , has a gram molecular mass of .

Gram molecule. The quantity of a substance whose mass in grams equals its molecular mass. One gram molecule is one mole of molecules and contains molecules.

Worked example 1 — gram atoms. How many gram atoms are in of magnesium, atomic mass ?



Worked example 2 — gram molecules. How many gram molecules are in of carbon dioxide?



Worked example 3 — the mass of a single atom. What is the mass of one carbon atom?



A number so small that no balance could ever detect it — which is why chemists count in moles rather than atoms.

The boundary case that causes most errors — atoms or molecules? Consider oxygen.

- One gram atom of oxygen is and contains atoms
- One gram molecule of oxygen, , is and contains molecules, which is atoms

So the phrase one mole of oxygen is ambiguous until you say whether you mean atoms or molecules — and a question about chlorine, nitrogen or hydrogen needs the same care.

The same counting unit works for ions. One mole of sodium chloride, , contains formula units — and therefore sodium ions and chloride ions.

An everyday comparison. A mole of water is , a little more than one tablespoon; a mole of common salt is ; a mole of cane sugar, , is . Three very different masses, each holding exactly the same number of particles.

What is vapour density, and why is molecular mass twice the vapour density?

Vapour density compares the mass of a volume of gas with the mass of the same volume of hydrogen, and because equal volumes contain equal numbers of molecules, it works out to half the molecular mass.

Definition. The vapour density of a gas or vapour is the ratio of the mass of a certain volume of the gas to the mass of the same volume of hydrogen, both measured at the same temperature and pressure.



Vapour density has no unit, since it is a ratio of two masses.

The derivation, one line at a time. Suppose volume contains molecules. By Avogadro's law, the same volume of hydrogen also contains molecules.





**A hydrogen molecule, , contains two atoms of mass **, so its molecular mass is :





Worked example 1. The vapour density of carbon dioxide is . Find its molecular mass.



Worked example 2. A gas has vapour density . Find its molecular mass and suggest what it could be.



Worked example 3. Find the vapour density of ammonia.



Worked example 4 — from a weighing at STP. One litre of a gas at STP weighs . Find its molecular mass and vapour density.



A useful link with molar volume. Since at STP weighs grams, it also weighs grams. Either route gives the same molecular mass.

The boundary case in the derivation. The factor is two because hydrogen gas is made of molecules of two atoms. **Anyone who takes hydrogen's mass as in the denominator gets , which is wrong by a factor of two — and it is exactly the diatomic nature proved in Part 1 that fixes the correct factor.

A second boundary case.** Worked example 4 gives molecular mass , which fits **both nitrogen, , and carbon monoxide, . Vapour density gives the molecular mass, not the identity — a chemical test is still needed to decide which gas it is.

An everyday sense of the numbers.** Air has an average vapour density of about . Any gas with a vapour density well above that sinks in air, and any gas well below it rises — chlorine, vapour density , collects in low places, while hydrogen, vapour density , escapes upwards.

How do you convert between mass, moles, number of particles and volume at STP?

Divide mass by molar mass to get moles, multiply moles by Avogadro's number to get particles, and multiply moles by 22.4 litres to get the volume of a gas at STP — always passing through moles.

The three relations:







Moles sit at the centre, with mass, particles and volume each one step away. To go from mass to volume, go through moles; to go from particles to mass, go through moles.

Worked example 1 — starting from mass. For of carbon dioxide, find the moles, molecules, total atoms, oxygen atoms and volume at STP.





**Each molecule has atoms**, carbon and oxygen:







Worked example 2 — starting from volume. For of ammonia at STP, find the moles, mass, molecules and hydrogen atoms.







Worked example 3 — starting from particles. Find the mass of molecules of water.



Worked example 4 — the mass of one molecule. Find the mass of a single water molecule.



Atomic masses used on this page: H , C , N , O , Na , Mg , S , Cl .

The boundary case in the volume relation. litres per mole applies only to gases, and only at STP. Nine grams of water is half a mole, but as a liquid it occupies about , not litres. Using the molar volume for a liquid or solid is one of the most frequent errors in this topic.

An everyday way to hold the relations in mind. Think of a mole as a packet in a shop. Mass is the weight printed on the packet, particles are what is inside, and — for a gas at STP — volume is the size of the box. Every question gives you one label and asks for another, and the packet itself, the mole, connects them.

How do you solve mole problems involving mass, volume and Avogadro's number?

Convert what you are given into moles, apply any ratio the question needs, such as atoms per molecule or ions per formula unit, and convert the moles into what is asked.

Problem 1. How many molecules are there in of carbon dioxide?



Problem 2. Which contains more molecules, of oxygen or of nitrogen?



Equal numbers of moles, so equal numbers of molecules — even though the masses differ.

Problem 3. Find the mass of of sulphur dioxide at STP.



Problem 4. How many atoms are there in of oxygen at STP?



**The factor of is there because each oxygen molecule has two atoms.

Problem 5.** mole of a gas weighs . Find its molecular mass and vapour density.



Problem 6. How many sodium ions, and how many ions in total, are there in of sodium chloride?







Problem 7. What volume do molecules of chlorine occupy at STP, and what do they weigh?



Problem 8 — a comparison that surprises. Which contains more atoms, of hydrogen gas or of helium?




Hydrogen has four times as many atoms, because each hydrogen atom is four times lighter than a helium atom.

Problem 9. How many gram atoms of oxygen are present in mole of sulphuric acid, ?



The method, in the order used every time:

- Find the molar mass from the formula
- Convert the given quantity to moles
- Multiply by any count per formula — atoms per molecule or ions per formula unit
- Convert moles to the quantity asked for
- Write the unit, or say molecules, atoms or ions for a number of particles

The boundary case in problems 4, 6 and 9. A question about atoms or ions needs the count per formula unit; a question about molecules or moles does not. Reading the last word of the question — molecules, atoms or ions — decides whether that extra factor belongs in the answer.
Exam tip

What habits prevent errors in mole calculations?

Write the molar mass first, pass every calculation through moles, and state clearly whether each number counts atoms, molecules or ions.

- Calculate the molar mass on its own line, showing the atomic masses used
- Convert to moles first and write the word mol after the value
- **Use and keep powers of ten in standard form throughout
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Multiply by atoms per molecule only when atoms are asked for
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Remember diatomic gases: oxygen, nitrogen, hydrogen and chlorine have two atoms per molecule
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Use **, never
- **Use only for gases at STP
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Label every final answer with grams, litres, or the kind of particle
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Check the size of the answer — a single atom's mass must be tiny, a mole of molecules must be a huge count
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Show the formula for each step so method marks survive an arithmetic slip

The misconception to name. Equal masses do not contain equal numbers of particles. Eight grams of oxygen and eight grams of hydrogen contain very different numbers of molecules — a quarter of a mole against four moles. Equal moles contain equal numbers of particles; equal masses almost never do.

A second trap. Forgetting that one mole of an ionic compound gives more than one mole of ions. One mole of calcium chloride, , contains three moles of ions** — one of calcium and two of chloride — and a question about ions expects that factor.
Did you know

How many water molecules are in a single drop?

A single drop from a dropper is a tiny amount of water — about millilitres. **Taking the density of water as per millilitre, that drop weighs about .** Now apply the method from this lesson.





**About molecules in one drop — a number with twenty-two digits.

The size of Avogadro's number is hard to feel, so try this comparison.** A tablespoon holds about millilitres, and one mole of water is , a little more than a tablespoonful. **So a spoonful of water from your kitchen tap contains about molecules, a number far beyond anything you could ever count one by one.

It also shows why single molecules never matter in everyday measurements.** If you lost a billion molecules from a drop, that is out of :



No balance anywhere could detect such a change. That is exactly why chemistry works with moles: the particles are so numerous that counting them is only possible by weighing.

And one more consequence is worth noticing. One mole of any substance contains the same number of particles, but its mass depends entirely on how heavy each particle is. A mole of hydrogen molecules weighs — about as much as two paper clips — while a mole of cane sugar weighs , enough to fill a small bowl. Same count, very different handfuls, which is the idea of a mole in one sentence.
Exam relevance

Why does the mole concept run through all of JEE and NEET Chemistry?

This is foundation work for Class 11 Some Basic Concepts of Chemistry, examined in both JEE Main and NEET Chemistry — and the mole is then used in almost every numerical in every later chapter.

Where the definitions lead. Class 11 restates the mole, Avogadro's number and molar mass, and adds concentration terms built on them: molarity, the moles of solute per litre of solution; molality, the moles of solute per kilogram of solvent; and mole fraction. Every one of these starts with the mass-to-moles conversion practised here.

Where particle counting leads. A standard question type in both exams asks for the number of electrons, protons or neutrons in a given mass. **For example, of methane is mole of molecules, and each molecule has electrons, so it contains mole of electrons, . The method is exactly problem 4 and problem 6 on this page, with electrons in place of atoms or ions.

Where vapour density leads.** The relation is used in Class 11 and Class 12 questions on the molecular mass of gases and vapours, and it reappears when a gas dissociates or associates and its measured vapour density changes. Recognising that vapour density is half the molar mass is assumed knowledge.

Where Avogadro's number leads beyond Chemistry. In Class 11 Physics, Kinetic Theory relates the gas constant to Avogadro's number, and in Class 12 Chemistry Electrochemistry the Faraday constant is the charge on one mole of electrons. Class 12 Solutions uses moles to find molar masses from colligative properties. All of these appear in both exams.

Question types to expect. At this level: definitions and interconversions. In competitive papers: number of particles or electrons in a mass, molarity and molality numericals, molar mass from vapour density, and multi-step stoichiometry — usually as single numerical-answer questions.

The single trap that costs marks. Confusing moles of atoms with moles of molecules. One mole of oxygen molecules contains two moles of oxygen atoms, and both exams set options that differ by exactly that factor of two.

A second trap. Using litres for a substance that is not a gas under the stated conditions. A question about the volume of water at STP expects the liquid volume, and a candidate who applies the molar volume loses the mark instantly.

Board versus competitive emphasis. The ICSE paper marks definitions, the vapour density derivation and step-by-step conversions with units; a competitive paper marks a final numerical value, often with several conversions compressed into one question. The transferable habit is always going through moles — the same route that works for a Class 10 conversion works for every concentration, electrochemistry and gas-law problem later.
Key takeaways

What must you be able to do from this part?

One counting unit, four definitions, one derived relation and three conversions.

- A mole is the amount containing as many particles as there are atoms in of carbon-12
- Avogadro's number:
- Gram atomic mass is the atomic mass in grams; one gram atom is one mole of atoms
- Gram molecular mass is the molecular mass in grams; one gram molecule is one mole of molecules
- ** of magnesium** is gram atoms; ** of ** is gram molecules
- One carbon atom weighs about ; one water molecule about
- **One mole of ** is and contains atoms
- Vapour density is the mass of a volume of gas divided by the mass of the same volume of hydrogen, with no unit
- **Molecular mass vapour density**, because a hydrogen molecule has mass
- **VD ** means ; ammonia has VD ; ** at STP weighing ** means
- Vapour density gives molecular mass, not identity fits both and
- Moles mass molar mass
- Particles moles
- Volume of a gas at STP moles
- ** of ammonia** is mol, and hydrogen atoms
- ** of and of contain equal numbers of molecules
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of NaCl** contains ions in total
- Multiply by atoms or ions per formula only when atoms or ions are asked for

The sharpest self-test is a single sample described four ways. Pick of oxygen gas and find its moles, molecules, atoms and volume at STP without looking back — then check that all four answers are consistent with one another.

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