Calculate the number of atoms in each of the following (i) 52 moles of Ar (ii) 52 u of He (iii) 52 g of He.

Exercises · 1.33

Q. Calculate the number of atoms in each of the following

(i) 52 moles of Ar

(ii) 52 u of He

(iii) 52 g of He.

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Concept understanding — Molecular Mass Calculation

What is Molecular Mass? The Intuition

Imagine you're at a market buying apples. You don't weigh each apple individually — you put a dozen on a scale. The total weight tells you something about the apples, but it also depends on how many apples you have.

Atoms and molecules are unimaginably tiny. A single water molecule ( $H_2O$ ) weighs about $3 \times 10^{-23}$ grams. That number is useless for practical chemistry. So instead of working with individual molecules, chemists count them in huge fixed numbers — specifically, $6.022 \times 10^{23}$ molecules, which is called one mole.

Molecular mass is simply the mass of one mole of a substance, expressed in grams per mole (g/mol). It answers the question: "If I have $6.022 \times 10^{23}$ molecules of this compound, how much would they weigh on a lab balance?"

Note

The number $6.022 \times 10^{23}$ is Avogadro's constant. It's chosen so that the mass of one mole of carbon-12 atoms is exactly 12 grams — matching the atomic mass unit scale perfectly.

The Precise Definition

Molecular mass (also called molar mass) is the mass of one mole of a molecular substance. It is numerically equal to the sum of the atomic masses of all atoms in the molecule, expressed in g/mol.

For example:

Water ( $H_2O$ ): 2 hydrogen atoms + 1 oxygen atom
Atomic mass of H = 1.008 g/mol
Atomic mass of O = 16.00 g/mol
Molecular mass of $H_2O$ = $2(1.008) + 16.00 = 18.016$ g/mol

\text{Molecular mass} = \sum (\text{number of atoms of each element} \times \text{atomic mass of that element})

How to Calculate It — Step by Step

Let's take glucose, $C_6H_{12}O_6$ , as a worked example.

Step 1: Identify each element and its count

Carbon (C): 6 atoms
Hydrogen (H): 12 atoms
Oxygen (O): 6 atoms

Step 2: Look up atomic masses (from the periodic table)

C: 12.01 g/mol
H: 1.008 g/mol
O: 16.00 g/mol

Step 3: Multiply and add

\text{Molecular mass} = 6(12.01) + 12(1.008) + 6(16.00)

= 72.06 + 12.096 + 96.00

= 180.156 \text{ g/mol}

Tip

Always keep at least 2 decimal places from the periodic table. For exam problems, they usually give you atomic masses — use exactly what's provided.

Why This Matters

Molecular mass is the bridge between the microscopic world (atoms and molecules) and the macroscopic world (grams you can weigh). Once you know the molecular mass, you can:

Convert grams to moles: $\text{moles} = \frac{\text{mass in grams}}{\text{molecular mass}}$
Convert moles to grams: $\text{mass} = \text{moles} \times \text{molecular mass}$
Determine the number of molecules: $\text{molecules} = \text{moles} \times 6.022 \times 10^{23}$

Watch out

Do not confuse molecular mass with atomic mass. Atomic mass refers to a single element (like oxygen = 16.00 g/mol). Molecular mass refers to a compound (like $CO_2$ = 44.01 g/mol). Also, for ionic compounds like NaCl, we use formula mass (same calculation, but the substance isn't molecular).

Common Exam Pitfalls

Forgetting to multiply by the subscript. In $H_2SO_4$ , there are 2 hydrogens, not 1. …

Why this formula?

Stoichiometry & Mole Calculation: The "Why" Behind the Formula

Let's build this from the ground up — not as a list of formulas to memorise, but as a logical chain of reasoning.

1. The Core Question: What is a Mole?

A mole is simply a counting unit, like a dozen (12) or a gross (144). But instead of 12, a mole contains $6.022 \times 10^{23}$ particles (Avogadro's number, $N_A$ ).

Why this number?

It was chosen so that 1 mole of any substance has a mass in grams equal to its atomic/molecular mass in amu.

Example: 1 atom of carbon-12 has mass 12 amu.
1 mole of carbon-12 has mass 12 grams.

This is the bridge between the microscopic (atoms/molecules) and the macroscopic (grams we can weigh).

2. The Fundamental Relationship

The key formula is:

n = \frac{m}{M}

Where:

$n$ = number of moles
$m$ = mass of substance (in grams)
$M$ = molar mass (in g/mol)

Why does this work?

Think of it as a conversion factor:

If 1 mole of a substance weighs $M$ grams, then $m$ grams contains $\frac{m}{M}$ moles.

Derivation logic:

Molar mass $M$ tells you: "1 mol = $M$ g"
So the conversion factor is $\frac{1 \text{ mol}}{M \text{ g}}$
Multiply mass $m$ by this factor: $m \times \frac{1}{M} = \frac{m}{M}$ moles

3. Connecting to Number of Particles

n = \frac{N}{N_A}

Where:

$N$ = number of particles (atoms, molecules, ions)
$N_A = 6.022 \times 10^{23}$ particles/mol

Why?

1 mole = $N_A$ particles
So $N$ particles = $\frac{N}{N_A}$ moles

Combine both formulas:

\frac{m}{M} = \frac{N}{N_A}

This single equation ties mass, molar mass, number of particles, and Avogadro's number together.

4. The Gas Volume Connection (for gases at STP)

For gases only:

n = \frac{V}{22.4 \text{ L/mol}}

Why 22.4 L?

From the ideal gas law: $PV = nRT$

At STP (Standard Temperature and Pressure: 0°C, 1 atm):

$P = 1$ atm
$T = 273.15$ K
$R = 0.0821$ L·atm/(mol·K)

For $n = 1$ mole:

V = \frac{nRT}{P} = \frac{1 \times 0.0821 \times 273.15}{1} \approx 22.4 \text{ L}

So 1 mole of any ideal gas occupies 22.4 L at STP. This is a consequence of the gas laws, not a definition.

5. The Stoichiometry Chain: From One Substance to Another

In a balanced chemical equation like:

aA + bB \rightarrow cC + dD

The coefficients tell you the mole ratio:

\frac{\text{moles of A}}{\text{moles of B}} = \frac{a}{b}

Why this works: …

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Convert moles, atomic mass units, and grams to atoms using Avogadro's number ( $N_A = 6.022 \times 10^{23}$ atoms/mol). For 52 moles of Ar: $3.13 \times 10^{25}$ atoms; for 52 u of He: 13 atoms; for 52 g of He: $7.83 \times 10^{24}$ atoms.

The bridge between the macroscopic world (grams, moles) and the atomic world (individual atoms) is Avogadro's number, $N_A = 6.022 \times 10^{23}$ particles per mole. The strategy depends on what you're given: moles connect directly to atoms, atomic mass units tell you how many atoms by comparing to one atom's mass, and grams require conversion through molar mass.

(i) 52 moles of Ar

When you already have moles, the path is immediate.

1. Recognize the direct relationship

One mole of any substance contains exactly $N_A$ particles. For argon atoms:

\text{Number of atoms} = \text{moles} \times N_A

2. Calculate

N = 52 \times 6.022 \times 10^{23} = 3.13 \times 10^{25} \text{ atoms}

(ii) 52 u of He

Atomic mass units measure mass on the atomic scale, where 1 u is defined as $\frac{1}{12}$ the mass of a carbon-12 atom.

1. Find the mass of one helium atom

Helium has an atomic mass of approximately 4 u. This means one He atom weighs 4 u.

2. Divide total mass by mass per atom

\text{Number of atoms} = \frac{\text{Total mass}}{\text{Mass per atom}} = \frac{52 \text{ u}}{4 \text{ u/atom}} = 13 \text{ atoms}

Note

This is a tiny sample—just 13 atoms! The atomic mass unit is so small that even 52 u represents only a handful of helium atoms.

(iii) 52 g of He

Grams are macroscopic units, so we convert through moles.

1. Convert grams to moles using molar mass

Helium's molar mass is 4 g/mol (numerically equal to its atomic mass, but in grams per mole instead of u per atom). …

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Here are the common mistakes students make when solving molecular mass and atom-counting problems like this one, along with how to avoid each.

Mistake 1: Confusing Moles, Grams, and Atomic Mass Units (u)

Students often treat 52 u of He the same as 52 g of He or 52 moles of Ar.

Why it happens: The number "52" looks the same, but the units are completely different.
How to avoid: Always underline the unit before solving. Ask yourself:
- Is it moles? → Multiply by $N_A$ (Avogadro’s number).
- Is it grams? → First convert to moles using molar mass.
- Is it u (atomic mass units)? → The number of atoms = the given number (since 1 u = mass of 1 nucleon, and He has mass 4 u per atom).

Key rule:

For a substance in u: number of atoms = given value ÷ atomic mass in u.

For a substance in g: use $n = \frac{\text{mass}}{\text{molar mass}}$ , then multiply by $N_A$ .

For a substance in moles: directly multiply by $N_A$ .

Mistake 2: Using Wrong Molar Mass for Helium

Students sometimes take molar mass of He as 2 g/mol (thinking of He₂ molecule) instead of 4 g/mol.

Why it happens: Confusion between atomic mass (He is monatomic) and molecular mass (like O₂, N₂).
How to avoid: Remember: Noble gases are monatomic.
- Atomic mass of He = 4 u
- Molar mass of He = 4 g/mol

Mistake 3: Forgetting Avogadro’s Number ( $N_A$ ) in Mole-to-Atom Conversion

Students write:

52 moles of Ar = 52 atoms ✗

Why it happens: They forget that 1 mole = $6.022 \times 10^{23}$ particles.
How to avoid: Always write the conversion factor:

1 \text{ mole} = 6.022 \times 10^{23} \text{ atoms}

So for 52 moles:

\text{Number of atoms} = 52 \times 6.022 \times 10^{23}

Mistake 4: Mixing Up “u” and “g” in the Same Problem

In part (ii): 52 u of He

Students incorrectly convert it like grams:

52 u → 52 g → then divide by 4 → then multiply by $N_A$ ✗

Why it happens: They don’t realize that u is a unit of mass per atom, not a bulk mass.
How to avoid:
- 1 atom of He has mass 4 u.
- So number of atoms = $\frac{52 \text{ u}}{4 \text{ u/atom}} = 13$ atoms. …

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Q.Calculate the number of atoms in each of the following

Concept understanding — Molecular Mass Calculation

What is Molecular Mass? The Intuition

The Precise Definition

How to Calculate It — Step by Step

Why This Matters

Common Exam Pitfalls

Why this formula?

Stoichiometry & Mole Calculation: The "Why" Behind the Formula

1. The Core Question: What is a Mole?

2. The Fundamental Relationship

Why does this work?

3. Connecting to Number of Particles

Why?

4. The Gas Volume Connection (for gases at STP)

Why 22.4 L?

5. The Stoichiometry Chain: From One Substance to Another

Why this works: …

(i) 52 moles of Ar

(ii) 52 u of He

(iii) 52 g of He

Method: Mole-Atom Conversion via Avogadro’s Number

Steps for each case

(i) 52 moles of Ar

(ii) 52 u of He

(iii) 52 g of He

Mistake 1: Confusing Moles, Grams, and Atomic Mass Units (u)

Mistake 2: Using Wrong Molar Mass for Helium

Mistake 3: Forgetting Avogadro’s Number ( $N_A$ ) in Mole-to-Atom Conversion

Mistake 4: Mixing Up “u” and “g” in the Same Problem

Q.Calculate the number of atoms in each of the following

(i) 52 moles of Ar

(ii) 52 u of He

(iii) 52 g of He

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