The density of 3 molal solution of NaOH is $1.110\ \text{g mL}^{-1}$. Calculate the molarity of the solution.

The density of 3 molal solution of NaOH is 1.110g mL⁻¹. Calculate the molarity of the solution.

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1.110\ \text{g mL}^{-1}

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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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Take $1$ kg of solvent (holding $3$ mol $NaOH$ ), find the solution mass, convert to volume with the given density, then divide moles by volume: molarity $= \mathbf{2.97\ M}$ .

1. Interpret 3 molal

A $3$ molal solution has $3$ mol $NaOH$ dissolved in $1$ kg $=1000$ g of water.

2. Mass of the solution

Molar mass of $NaOH = 23+16+1 = 40\ \text{g mol}^{-1}$ , so mass of $NaOH = 3\times40 = 120$ g.

\text{mass of solution}=1000+120=1120\ \text{g}

3. Volume of the solution

Using $\text{density}=1.110\ \text{g mL}^{-1}$ :

V=\frac{1120\ \text{g}}{1.110\ \text{g mL}^{-1}}=1009.0\ \text{mL}=1.009\ \text{L}

4. Molarity …

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Common Mistakes in Molality → Molarity Calculation

Students often struggle with this conversion because it requires careful handling of mass, volume, and the distinction between molality and molarity. Here are the most frequent errors and how to avoid them.

Mistake 1: Confusing molality with molarity

The error:

Students treat 3 molal as 3 moles per litre of solution, which is actually molarity.

Why it happens:

Both terms sound similar and both use "moles of solute," but:

Molality ( $m$ ) = moles of solute per kg of solvent
Molarity ( $M$ ) = moles of solute per L of solution

How to avoid:

Always write the definition before starting:

$m = \frac{\text{moles of solute}}{\text{mass of solvent (in kg)}}$
$M = \frac{\text{moles of solute}}{\text{volume of solution (in L)}}$

Mistake 2: Using density to directly convert molality to molarity

The error:

Students multiply: $3 \times 1.110 = 3.33\ \text{M}$ — this is wrong.

Why it happens:

Density is mass/volume, but molality uses solvent mass, not solution mass.

How to avoid:

Remember the stepwise logic:

Assume 1 kg of solvent (this is the standard starting point for molality)
Find moles of solute from molality
Find mass of solute
Find total mass of solution = mass of solvent + mass of solute
Use density to find volume of solution
Finally, molarity = moles of solute / volume of solution (in L)

Mistake 3: Forgetting to convert units

The error:

Using density in g/mL but volume in litres without conversion.

Example of wrong work:

Volume = $\frac{\text{mass of solution (g)}}{\text{density (g/mL)}}$ → then using this volume directly in litres.

How to avoid:

Always convert mL to L:

$1\ \text{mL} = 10^{-3}\ \text{L}$
So if volume comes out as $V\ \text{mL}$ , then $V\ \text{mL} = \frac{V}{1000}\ \text{L}$

Mistake 4: Using wrong molar mass or forgetting it entirely

The error:

Students skip finding the mass of NaOH, or use an incorrect molar mass.

Why it happens:

They forget that to find the mass of solute from moles, you must know the molar mass.

How to avoid:

Molar mass of NaOH = $23 + 16 + 1 = 40\ \text{g/mol}$ …

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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 density of 3 molal solution of NaOH is $1.110\ \text{g mL}^{-1}$ . Calculate the molarity of the solution.

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? …

Method: Density–Molality to Molarity Conversion

Steps

Common Mistakes in Molality → Molarity Calculation

Mistake 1: Confusing molality with molarity

Mistake 2: Using density to directly convert molality to molarity

Mistake 3: Forgetting to convert units

Mistake 4: Using wrong molar mass or forgetting it entirely

Q.The density of 3 molal solution of NaOH is 1.110 g mL−11.110\ \text{g mL}^{-1}1.110 g mL−1. Calculate the molarity of the solution.

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Q.The density of 3 molal solution of NaOH is $1.110\ \text{g mL}^{-1}$ . Calculate the molarity of the solution.

Unlock everything free for 14 days