The vapour pressure of water is 12.3 kPa at 300 K. Calculate vapour pressure of 1 molal solution of a non-vola

Exercises · 1.17

Q. The vapour pressure of water is 12.3 kPa at 300 K. Calculate vapour pressure of 1 molal solution of a non-volatile solute in it.

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Concept understanding — Molality Calculation

Molality: The Concentration That Ignores Temperature

Imagine you're making a cup of sweet tea. You add sugar to hot water, stir, and taste. If you let the tea cool to room temperature, the amount of sugar hasn't changed — but the volume of the liquid has shrunk slightly. If you measured concentration as "grams of sugar per litre of solution," that number would change just because the temperature changed. That's annoying if you're a chemist who needs a reliable, temperature-independent way to describe how much solute is present.

Molality was invented to solve exactly this problem.

The Intuition

Instead of measuring the volume of the solution (which expands and contracts with temperature), molality measures the mass of the solvent. Mass doesn't change with temperature. So molality gives you a concentration that stays the same whether your solution is hot or cold.

Think of it this way:

Molarity = moles of solute per litre of solution (temperature-sensitive)
Molality = moles of solute per kilogram of solvent (temperature-independent)

The solvent is the substance doing the dissolving — usually water. The solute is what gets dissolved — sugar, salt, etc.

The Precise Definition

\text{Molality } (m) = \frac{\text{moles of solute}}{\text{kilograms of solvent}}

The symbol for molality is a lowercase m (not to be confused with M for molarity).

Key points to remember:

The denominator is solvent mass, not solution mass
The unit is mol/kg (often written as simply "m")
It is independent of temperature because mass doesn't change with temperature

Worked Example

Problem: 36 g of glucose ( $C_6H_{12}O_6$ , molar mass = 180 g/mol) is dissolved in 500 g of water. Calculate the molality of the solution.

Step 1: Find moles of solute

\text{Moles of glucose} = \frac{36 \text{ g}}{180 \text{ g/mol}} = 0.2 \text{ mol}

Step 2: Convert solvent mass to kilograms

500 \text{ g} = 0.5 \text{ kg}

Step 3: Apply the formula

m = \frac{0.2 \text{ mol}}{0.5 \text{ kg}} = 0.4 \text{ m}

Note

The answer is 0.4 m (or 0.4 mol/kg). Notice we used the mass of water (500 g), not the mass of the solution (which would be 536 g).

Common Mistake to Avoid

Watch out

Do not use the mass of the solution in the denominator. The formula specifically asks for the mass of the solvent alone. If the problem gives you the total mass of the solution, subtract the mass of the solute to find the solvent mass.

When Do You Use Molality?

Molality is the star in two important situations: …

Why this formula?

Molality Calculation: Why the Formula Works

Molality is a measure of concentration that is temperature-independent — this is its key advantage over molarity. Let's understand why the formula takes the form it does.

The Definition First

Molality ( $m$ ) is defined as:

m = \frac{\text{moles of solute}}{\text{mass of solvent in kg}}

The unit is mol/kg, often written as m (e.g., 0.5 m glucose solution).

Why Mass of Solvent, Not Solution?

This is the critical conceptual point.

The Reasoning

Molarity uses volume of solution → volume changes with temperature (expansion/contraction). So molarity changes with temperature.
Molality uses mass of solvent → mass is invariant with temperature. So molality remains constant regardless of temperature changes.

Key insight: By using the solvent's mass (not the solution's volume), we eliminate temperature dependence. This is why molality is preferred for colligative properties (boiling point elevation, freezing point depression) — these properties depend on the number of solute particles, not on temperature.

Deriving the Formula Step-by-Step

Step 1: Moles of Solute

If you have $w_{\text{solute}}$ grams of solute with molar mass $M_{\text{solute}}$ (g/mol):

\text{moles of solute} = \frac{w_{\text{solute}}}{M_{\text{solute}}}

Step 2: Mass of Solvent in kg

If the solvent mass is $W_{\text{solvent}}$ grams:

\text{mass of solvent in kg} = \frac{W_{\text{solvent}}}{1000}

Step 3: Putting It Together

m = \frac{\frac{w_{\text{solute}}}{M_{\text{solute}}}}{\frac{W_{\text{solvent}}}{1000}}

Simplifying:

m = \frac{w_{\text{solute}} \times 1000}{M_{\text{solute}} \times W_{\text{solvent}}}

The Final Formula (Exam-Ready)

\boxed{m = \frac{w_{\text{solute}} \times 1000}{M_{\text{solute}} \times W_{\text{solvent}}}}

Where:

$w_{\text{solute}}$ = mass of solute in grams
$M_{\text{solute}}$ = molar mass of solute in g/mol
$W_{\text{solvent}}$ = mass of solvent in grams

Why the ×1000 Factor? …

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A non-volatile solute lowers the vapour pressure of water through Raoult's law; for a 1 molal aqueous solution, the mole fraction of water is approximately 0.982, giving a vapour pressure of 12.08 kPa.

When a non-volatile solute dissolves in a solvent, it occupies space at the surface and reduces the number of solvent molecules that can escape into the vapour phase. This phenomenon—vapour pressure lowering—is quantified by Raoult's law, which states that the vapour pressure of the solution is proportional to the mole fraction of the solvent.

The key insight is that molality tells us moles of solute per kilogram of solvent, so we can convert this to mole fractions and then apply Raoult's law directly.

P_{\text{solution}} = \chi_{\text{solvent}} \cdot P^\circ_{\text{solvent}}

where $\chi_{\text{solvent}}$ is the mole fraction of the solvent and $P^\circ_{\text{solvent}}$ is the vapour pressure of the pure solvent.

Step-by-step calculation

Identify what 1 molal means.

A 1 molal solution contains 1 mole of solute dissolved in 1 kg (1000 g) of water. We need to find how many moles of water that corresponds to.
Calculate moles of water.

The molar mass of water is 18 g/mol, so:

n_{\text{water}} = \frac{1000 \text{ g}}{18 \text{ g/mol}} = 55.56 \text{ mol}

Find the mole fraction of water. The total moles in solution = moles of water + moles of solute = $55.56 + 1 = 56.56$ mol.

\chi_{\text{water}} = \frac{n_{\text{water}}}{n_{\text{water}} + n_{\text{solute}}} = \frac{55.56}{56.56} = 0.9823

Apply Raoult's law. …

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Vapour Pressure of a 1 Molal Solution – Raoult’s Law Application

1. Concept First – The Idea Being Tested

This question tests Raoult’s Law for a non-volatile solute. The core idea is simple:

When you dissolve a non-volatile solute (one that doesn’t evaporate) in a solvent, the solute particles block some solvent molecules from escaping into the vapour phase. This lowers the vapour pressure of the solvent.

Why does this happen?

In pure water, all surface molecules can evaporate freely.
When solute is added, some surface sites are occupied by solute particles (which don’t evaporate).
Fewer water molecules are available at the surface to escape → vapour pressure drops.

Raoult’s Law quantifies this:

P_{\text{solution}} = X_{\text{solvent}} \cdot P_{\text{solvent}}^0

Where:

$P_{\text{solution}}$ = vapour pressure of the solution
$X_{\text{solvent}}$ = mole fraction of the solvent
$P_{\text{solvent}}^0$ = vapour pressure of pure solvent (given as 12.3 kPa)

The key challenge here: we are given molality (1 molal), not mole fraction. So we must convert.

2. Step-by-Step Solution

Step 1: Understand what “1 molal” means

A 1 molal solution means:

1 mole of solute dissolved in 1 kg of solvent (water).

So we have:

Moles of solute, $n_{\text{solute}} = 1 \text{ mol}$
Mass of water = 1000 g

Step 2: Find moles of water (solvent)

Molar mass of water ( $H_2O$ ) = 18 g/mol

n_{\text{water}} = \frac{\text{mass of water}}{\text{molar mass}} = \frac{1000}{18} = 55.56 \text{ mol}

Step 3: Calculate mole fraction of water

Mole fraction of solvent (water) is:

X_{\text{water}} = \frac{n_{\text{water}}}{n_{\text{water}} + n_{\text{solute}}}

Substitute:

X_{\text{water}} = \frac{55.56}{55.56 + 1} = \frac{55.56}{56.56}

Calculate:

X_{\text{water}} = 0.9823

Why this step?

Raoult’s Law uses mole fraction, not molality. We must convert because the “blocking effect” depends on the relative number of solute vs solvent particles.

Step 4: Apply Raoult’s Law

P_{\text{solution}} = X_{\text{water}} \cdot P_{\text{water}}^0

Given $P_{\text{water}}^0 = 12.3 \text{ kPa}$ :

P_{\text{solution}} = 0.9823 \times 12.3

P_{\text{solution}} = 12.08 \text{ kPa}

3. Final Answer

The vapour pressure of the 1 molal solution is $\boxed{12.08 \text{ kPa}}$ .

4. Why It Works & Exam Tip

Why it works …

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

COMEDK 2025Set 2025-A1 markMCQ
$4.25 \mathrm{M} \mathrm{H}_2 \mathrm{SO}_4$
›Reveal solutionSolution
Molality is moles of solute per kilogram of solvent. Given molarity and density, we find the mass of 1 L of solution, subtract the mass of H₂SO₄ to get solvent mass, then compute molality. The result is 5.161 m, so option (C) is correct.

Concept & Intuition

Molality (m) depends on the mass of solvent, not the volume of solution. Molarity (M) gives moles per liter of solution, but the solvent mass is hidden inside the density. The trick: take exactly 1 liter of solution, find its total mass from density, subtract the mass of H₂SO₄ (from moles × molar mass), and you have the solvent mass in kg. Then molality = moles / kg solvent.

Step-by-step solution

Interpret the given data

Molarity of H₂SO₄ = 4.25 M → 4.25 moles per liter of solution.

Density of solution = 1.24 g/mL = 1240 g/L (since 1 mL = 1 g water equivalent, but here it’s the whole solution).

Molar mass of H₂SO₄ = 2(1.008) + 32.06 + 4(16.00) = 98.08 g/mol (we’ll use 98.08).

Mass of 1 liter of solution

$\text{Mass of solution} = 1.24 \ \frac{\text{g}}{\text{mL}} \times 1000 \ \text{mL} = 1240 \ \text{g}$

Mass of H₂SO₄ in 1 liter

$\text{Moles of H₂SO₄} = 4.25 \ \text{mol}$ $\text{Mass of H₂SO₄} = 4.25 \times 98.08 = 416.84 \ \text{g}$

Mass of solvent (water) in 1 liter
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COMEDK 2025Set 2025-E1 markMCQ
$Q.The mole fraction of an unknown solute in 1560 g of Benzene is 0.5 . What is the molality of the solution? (M. M of Benzene:78 amu) (A) 12.8 (B) 10.3 (C) 3.25 (D) 16.9$
›Reveal solutionSolution
The key idea is to use the definition of mole fraction to find the moles of solute, then divide by the mass of solvent in kg. The molality is 12.8 m, so the correct option is (A).

Concept and Intuition

Mole fraction tells us the ratio of moles of one component to the total moles in the mixture. Here, the mole fraction of solute is 0.5, meaning the solute and solvent have equal moles. Since we know the mass of benzene (solvent) and its molar mass, we can find the moles of benzene, then the moles of solute, and finally the molality (moles of solute per kg of solvent). The trap is forgetting to convert grams to kilograms for molality.

Step-by-step solution

Find moles of benzene (solvent) Mass of benzene = 1560 g Molar mass of benzene (C₆H₆) = 78 g/mol

$n_{\text{benzene}} = \frac{1560 \text{ g}}{78 \text{ g/mol}} = 20 \text{ mol}$

Use mole fraction to find moles of solute Mole fraction of solute, $X_{\text{solute}} = 0.5$ By definition:

$X_{\text{solute}} = \frac{n_{\text{solute}}}{n_{\text{solute}} + n_{\text{benzene}}}$
Substitute known values:
$0.5 = \frac{n_{\text{solute}}}{n_{\text{solute}} + 20}$
Multiply both sides by $n_{\text{solute}} + 20$ :
$0.5(n_{\text{solute}} + 20) = n_{\text{solute}}$ $0.5 n_{\text{solute}} + 10 = n_{\text{solute}}$ $10 = 0.5 n_{\text{solute}} \quad \Rightarrow \quad n_{\text{solute}} = 20 \text{ mol}$

Calculate molality …
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COMEDK 2024Set 2024-E1 markMCQ
$4.5 \mathrm{~M}$
›Reveal solutionSolution

Take $1\ \mathrm{L}$ of solution: it holds $4.5$ mol $\mathrm{H_2SO_4}$ in $(1280-441)=839\ \mathrm{g}$ water, giving molality $\approx 5.36\ m$ .

Molar mass of $\mathrm{H_2SO_4} = 98\ \mathrm{g/mol}$ . Consider $1\ \mathrm{L}$ of the $4.5\ \mathrm{M}$ solution.

Mass of solution:
$1.28\ \mathrm{g/mL} \times 1000\ \mathrm{mL} = 1280\ \mathrm{g}$
Mass of $\mathrm{H_2SO_4}$ :

$4.5\ \mathrm{mol} \times 98\ \mathrm{g/mol} = 441\ \mathrm{g}$ …
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COMEDK 2023Set 2023-E1 markMCQ
$5 \mathrm{~m}$
›Reveal solutionSolution

Mole fraction of solute: x_solute = 5 / (5 + 55.55) = 5 / 60.55 = 0.0826 ~ 0.082

Concept: molality m = moles of solute per 1 kg (1000 g) of solvent. Convert to mole fraction.

Basis: 1000 g of water.

moles of solute = 5 mol

moles of water = 1000 / 18 = 55.55 mol …
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COMEDK 2021Set 20211 markMCQ
$_2$
›Reveal solutionSolution

Molarity = 0.2 / 1 = 0.2 M

Concept: Molarity = moles of solute / volume of solution in litres.

Molar mass of CaCl2 = 40 + 2(35.5) = 111 g/mol

Moles = 22.2 / 111 = 0.2 mol

Volume = 1 L …
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Q.The vapour pressure of water is 12.3 kPa at 300 K. Calculate vapour pressure of 1 molal solution of a non-volatile solute in it.

Concept understanding — Molality Calculation

Molality: The Concentration That Ignores Temperature

The Intuition

The Precise Definition

Worked Example

Common Mistake to Avoid

When Do You Use Molality?

Why this formula?

Molality Calculation: Why the Formula Works

The Definition First

Why Mass of Solvent, Not Solution?

The Reasoning

Deriving the Formula Step-by-Step

Step 1: Moles of Solute

Step 2: Mass of Solvent in kg

Step 3: Putting It Together

The Final Formula (Exam-Ready)

Why the ×1000 Factor? …

Step-by-step calculation

Vapour Pressure of a 1 Molal Solution – Raoult’s Law Application

1. Concept First – The Idea Being Tested

2. Step-by-Step Solution

Step 1: Understand what “1 molal” means

Step 2: Find moles of water (solvent)

Step 3: Calculate mole fraction of water

Step 4: Apply Raoult’s Law

3. Final Answer

4. Why It Works & Exam Tip

Why it works …

1. Using Molality Directly in Raoult's Law

2. Forgetting What "1 Molal" Means

3. Using the Mole Fraction of Solute Instead of Solvent

4. Forgetting That Water's Molar Mass Gives About 55.5 mol per kg

Q.The vapour pressure of water is 12.3 kPa at 300 K. Calculate vapour pressure of 1 molal solution of a non-volatile solute in it.

Step-by-step calculation

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