In three moles of ethane (C₂H₆), calculate the following: (i) Number of moles of carbon atoms. (ii) Number of

Exercises · 1.10

C_2H_6

(i) Number of moles of carbon atoms.

(ii) Number of moles of hydrogen atoms.

(iii) Number of molecules of ethane.

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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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Each molecule of ethane ( $C_2H_6$ ) contains 2 carbon atoms and 6 hydrogen atoms. So, 3 moles of ethane contain 6 moles of carbon atoms, 18 moles of hydrogen atoms, and $1.8066 \times 10^{24}$ molecules of ethane.

The Core Idea: Molecular Composition and the Mole

The mole is a counting unit — like a dozen (12) but much larger ( $6.022 \times 10^{23}$ , Avogadro’s number). When we say “one mole of ethane,” we mean $6.022 \times 10^{23}$ molecules of $C_2H_6$ .

The key is that the chemical formula $C_2H_6$ tells us the ratio of atoms inside each molecule. One molecule of ethane has 2 carbon atoms and 6 hydrogen atoms. Therefore, one mole of ethane molecules contains 2 moles of carbon atoms and 6 moles of hydrogen atoms. This is a direct scaling: the subscript in the formula becomes the mole ratio.

X_aY_b

Now, let’s apply this to the given 3 moles of ethane.

Step-by-Step Calculation

1. Number of moles of carbon atoms.

Each molecule of ethane has 2 carbon atoms. So, each mole of ethane contains 2 moles of carbon atoms.

For 3 moles of ethane:

\text{Moles of C atoms} = 3 \text{ moles } C_2H_6 \times \frac{2 \text{ moles C}}{1 \text{ mole } C_2H_6} = 6 \text{ moles of C atoms}

Tip

Think of it like this: if 1 car has 4 wheels, then 3 cars have $3 \times 4 = 12$ wheels. Here, the “car” is the ethane molecule, and the “wheels” are the carbon atoms.

2. Number of moles of hydrogen atoms.

Each molecule of ethane has 6 hydrogen atoms. So, each mole of ethane contains 6 moles of hydrogen atoms.

For 3 moles of ethane:

\text{Moles of H atoms} = 3 \text{ moles } C_2H_6 \times \frac{6 \text{ moles H}}{1 \text{ mole } C_2H_6} = 18 \text{ moles of H atoms}

Watch out

A common mistake is to forget that the subscript applies to each molecule. Do not add the subscripts (2+6=8) and then multiply — that would give the total number of atoms per molecule, not the separate counts per element. Always use the individual subscripts. …

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Here are the common mistakes students make when solving this problem, along with clear strategies to avoid each.

Mistake 1: Confusing “moles of atoms” with “moles of molecules”

The error:

Students often think that 3 moles of ethane contain 3 moles of carbon atoms total. They forget that each molecule of ethane ( $C_2H_6$ ) has 2 carbon atoms.

How to avoid:

Always write the molecular formula first: $C_2H_6$
For carbon: 1 molecule → 2 C atoms → 1 mole of ethane contains 2 moles of C atoms.
So for 3 moles of ethane:

\text{Moles of C atoms} = 3 \times 2 = 6 \text{ moles}

Key rule:

Moles of atoms = (moles of compound) × (number of that atom per molecule)

Mistake 2: Forgetting to multiply for hydrogen atoms

The error:

Students sometimes use the same multiplier for H as for C, or they only count one H atom per molecule.

How to avoid:

Count carefully: $C_2H_6$ has 6 hydrogen atoms per molecule.
So:

\text{Moles of H atoms} = 3 \times 6 = 18 \text{ moles}

Tip: Write the subscript numbers clearly — $C_2H_6$ means 2 C and 6 H, not 2 and 2.

Mistake 3: Using Avogadro’s number incorrectly for “number of molecules”

The error:

Some students multiply by Avogadro’s number ( $6.022 \times 10^{23}$ ) even when the question asks for moles of molecules, not the actual count.

How to avoid:

Read the question carefully: part (iii) asks for number of molecules of ethane, not moles.
So you do use Avogadro’s number:

\text{Number of molecules} = 3 \times 6.022 \times 10^{23} = 1.8066 \times 10^{24}

If the question had asked for “moles of ethane molecules”, the answer would simply be 3 moles — no multiplication by $N_A$ .

Mistake 4: Mixing up “moles of atoms” and “number of atoms”

The error:

Students sometimes calculate the number of atoms (using $N_A$ ) when the question asks for moles of atoms.

How to avoid:

Moles of atoms = just a number (like 6 or 18) — no $N_A$ needed.
Number of atoms = moles of atoms × $N_A$ — only if the question explicitly asks for the count.

Quick check:

Part (i) and (ii) ask for “number of moles” → answer is a plain number.
Part (iii) asks for “number of molecules” → answer includes $N_A$ .

--- …

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Same / Similar Concept — real previous-year questions on the same or a closely similar concept, not this exact question.

KCET 2024Set B-21 markMCQ
$_2$
›Reveal solutionSolution

Every carbon atom burns to $CO_2$ , so convert the $CO_2$ mass to carbon mass with the factor $12/44$ , then express it as a percentage of the sample.

Step 1 — The principle (Liebig's combustion method)

In quantitative combustion analysis the organic compound is burnt completely in excess oxygen. All its carbon is converted to $CO_2$ , which is absorbed and weighed. So the carbon in the $CO_2$ is the carbon that was in the compound — a simple mass-conservation argument.

Step 2 — The carbon fraction of $CO_2$
$M(CO_2) = 12 + 2(16) = 44\ \text{g mol}^{-1}, \qquad M(C) = 12\ \text{g mol}^{-1}$
So every 44 g of $CO_2$ contains 12 g of carbon:
$\text{fraction of C in } CO_2 = \frac{12}{44}$
Step 3 — Mass of carbon in the sample
$m_C = \frac{12}{44}\times m_{CO_2} = \frac{12}{44}\times 0.22 = \frac{2.64}{44} = 0.06\ \text{g}$
Step 4 — Percentage of carbon
$\%\,C = \frac{\text{mass of carbon}}{\text{mass of compound}}\times 100 = \frac{0.06}{0.48}\times 100$ $= \frac{6}{48}\times 100 = 0.125 \times 100 = 12.5\%$
Step 5 — Sanity check

The standard working formula for this experiment is …
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COMEDK 2023Set 2023-E1 markMCQ
$5.8 \mathrm{~g}$
›Reveal solutionSolution

Molar mass: M = mass / n = 5.8 / 0.145 = 40 g/mol

Concept: ideal gas equation, PV = nRT. Same V and same P for both gases, so nT = PV/R = constant:

n_gas x T_gas = n_H2 x T_H2

Hydrogen:

n_H2 = 0.368 / 2 = 0.184 mol, T_H2 = 17 C = 290 K

Unknown gas:

T = 95 C = 368 K, mass = 5.8 g …
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KCET 2021Set B-21 markMCQ
$300$
›Reveal solutionSolution

Apply the unit-cell density formula $\rho = ZM/(a^3N_A)$ with $Z=2$ for a body-centred cubic lattice and solve for $M$ .

Step 1 — The concept.

A crystal's macroscopic density is just the mass of one unit cell divided by its volume. A unit cell of edge $a$ contains $Z$ formula units, each of mass $M/N_A$ :
$\rho = \frac{Z\,(M/N_A)}{a^3} = \frac{Z M}{a^3 N_A}.$
Step 2 — Fix $Z$ for BCC.

BCC has $8$ corner atoms shared by $8$ cells each ( $8\times\tfrac18 = 1$ ) plus $1$ atom fully inside at the body centre. So
$Z = 1 + 1 = 2.$
Step 3 — Convert the edge length to cm (so it matches $\mathrm{g\,cm^{-3}}$ ):
$a = 300\,\text{pm} = 300\times10^{-12}\,\mathrm{m} = 3\times10^{-8}\,\mathrm{cm},$ $a^3 = (3\times10^{-8})^3 = 27\times10^{-24} = 2.7\times10^{-23}\,\mathrm{cm^3}.$
Step 4 — Solve for $M$ . …
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COMEDK 2021Set 2021-B1 markMCQ
$Q. In a given sample of air the ratio between the masses of O2 gas and N2 gas is 6 : 7. What would be the ratio of their moles? (A) Ratio of the molecules of O2:N2 = 2 : 3 (B) Ratio of the molecules of O2:N2 = 1 : 4 (C) Ratio of the molecules of O2:N2 = 2 : 5 (D) Ratio of the molecules of O2:N2 = 3 : 4$
›Reveal solutionSolution

Dividing the given masses by molar masses gives a mole (and molecule) ratio of 3 : 4.

Moles from mass:
$n_{O_2} = \frac{6}{32} = 0.1875, \qquad n_{N_2} = \frac{7}{28} = 0.25$
Ratio:

$\frac{n_{O_2}}{n_{N_2}} = \frac{0.1875}{0.25} = \frac{3}{4}$ …
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←Previous question1.9 — Calculate the atomic mass (average) of chlorine using the following da…→Next question1.11 — What is the concentration of sugar (C₁₂H₂₂O₁₁) in molL⁻¹ if its 20 g a…

Q.In three moles of ethane ( $C_2H_6$ ), calculate 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: …

The Core Idea: Molecular Composition and the Mole

Step-by-Step Calculation

Method: Mole Ratio Method (Using Chemical Formula)

Steps

Mistake 1: Confusing “moles of atoms” with “moles of molecules”

Mistake 2: Forgetting to multiply for hydrogen atoms

Mistake 3: Using Avogadro’s number incorrectly for “number of molecules”

Mistake 4: Mixing up “moles of atoms” and “number of atoms”

Step 1 — The principle (Liebig's combustion method)

Step 2 — The carbon fraction of $CO_2$

Step 3 — Mass of carbon in the sample

Step 4 — Percentage of carbon

Step 5 — Sanity check

Q.In three moles of ethane (C2H6C_2H_6C2​H6​), calculate the following:

The Core Idea: Molecular Composition and the Mole

Step-by-Step Calculation

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Q.In three moles of ethane ( $C_2H_6$ ), calculate the following:

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