Vapour-Pressure Lowering as a Colligative Effect

Introduction

Vapour-pressure lowering becomes a colligative calculation when masses of solute and solvent are converted to mole amounts. The ideal equation p = xsolvent p° shows that the fraction of solvent particles among liquid components sets the solvent vapour pressure. Two different nonvolatile nonelectrolytes produce the same ideal relative lowering if their effective particle mole fractions match in the same solvent at the same temperature.

Core explanation

Start from a binary solution whose solute has negligible vapour pressure. Convert each measured mass to moles: nsolute = msolute/Msolute and nsolvent = msolvent/Msolvent. Then xsolvent = nsolvent/(nsolvent + nsolute). The ideal solvent pressure is xsolvent p°, and the absolute lowering is Δp = p° − xsolvent p° = p°xsolute. If asked for relative lowering, divide by p° to obtain xsolute. Keeping the exact total-moles denominator avoids an unnecessary dilute approximation.

For 18.0 g water and 18.0 g glucose, take molar masses 18.0 and 180 g mol⁻¹. Water amount is 1.00 mol; glucose amount is 0.100 mol. Thus xwater = 1.00/1.10 ≈ 0.909 and xglucose ≈ 0.0909. If pure water has vapour pressure 3.00 kPa at the chosen temperature, ideal solution water pressure is about 2.73 kPa and the lowering is about 0.273 kPa. The numbers illustrate the equation, not a claim that such a relatively concentrated glucose solution is perfectly ideal.

Replace glucose with another nonvolatile nonelectrolyte but adjust its mass to keep exactly 0.100 mol solute with 1.00 mol water. The ideal mole fractions and pressure lowering remain the same, although measured values may differ because real solutes interact differently with water. If the same mass of a different solute is used instead, its moles may differ because molar mass differs; the pressure effect need not match. “Same number of particles” is the condition, not “same grams.”

An electrolyte requires special care. NaCl dissociates into ions, so a count based on intact NaCl formula units may understate its ideal particle effect. At sufficiently dilute conditions, one might approximate two ions per NaCl unit, but actual solutions show ion interactions. The simple exact binary xsolute formula written for one solute species does not directly accommodate dissociation by just renaming NaCl molecules as ions; use an effective particle model or measured activity. State which approximation the problem requests.

If the solute is volatile, its own vapour contributes to total measured pressure. Even if solvent partial pressure is lowered, total pressure can rise or fall depending on the second vapour. Thus “solute lowers vapour pressure” is properly a statement about the solvent partial pressure or total pressure only when the solute is effectively nonvolatile. A gas-filled headspace adds another possible pressure contribution, so identify exactly what the instrument measured.

Vapour-pressure lowering also depends on temperature through p°. Even at fixed mole fractions, the absolute lowering Δp = xsolute p° changes when pure-solvent pressure changes with temperature. Relative lowering xsolute remains constant in the simple ideal model if composition is unchanged. This is why comparing absolute kilopascal drops at different temperatures does not directly compare particle fractions.

Step-by-step reasoning

1. Verify that the solute is nonvolatile and the intended model is ideal. 2. Convert each component mass to moles using its own molar mass. 3. Calculate xsolvent with both mole amounts in the denominator. 4. Use pure-solvent p° at the stated temperature to calculate p and Δp. 5. Check whether electrolyte dissociation or real-solution effects require qualification.

Visual explanation

Draw a chain: “solute grams → solute moles” and “solvent grams → solvent moles”; merge them into xsolvent = nsolvent/(ntotal), then multiply by p°. Place Δp = p° − p at the end. A second branch shows equal moles of two different nonelectrolytes converging to equal ideal pressure lowering even though their masses differ.

Real-world analogy

If a jar contains 90 blue marbles and 10 red marbles, the blue share is 90%, regardless of whether red marbles are heavy or light. A liquid solvent mole fraction similarly counts chemical amounts, not solute mass share. Real molecules have interactions that marbles lack, so the analogy describes the ideal counting part only.

Real-world example

To estimate water activity in a simple teaching syrup, a student weighs a nonvolatile sugar and water, converts both to moles and predicts a lower ideal water vapour pressure. A food scientist would use measured activity for a concentrated formulation because molecular interactions and high concentration make the ideal estimate less reliable.

Why?

Why can equal-mass additions of two solutes give different pressure lowerings? Their molar masses can differ, so equal grams contain different mole counts. Colligative effects follow effective particle amounts under the ideal dilute model, not the mass alone.

Common misconception

“The percentage pressure lowering equals solute mass percentage.” The ideal relative lowering equals solute mole fraction , which generally differs from mass fraction. Converting masses to moles is essential.

Worked example

At a chosen temperature, p° of pure water is 4.00 kPa. Mix 36.0 g water with 9.00 g of a nonvolatile nonelectrolyte of molar mass 90.0 g mol⁻¹. Water is 2.00 mol and solute is 0.100 mol. xsolute = 0.100/2.10 ≈ 0.04762 and xwater ≈ 0.95238. Ideal p = 0.95238 × 4.00 ≈ 3.81 kPa. The absolute lowering is 0.190 kPa and relative lowering about 4.76%. The mass percentage is 9/45 × 100 = 20%, illustrating the difference between mass and mole fractions.

Quick check

1. Which solute measurement directly sets ideal relative lowering: solute grams or solute mole fraction? Answer: Solute mole fraction under the binary, nonvolatile, ideal model; grams must first be converted to moles.

Exam focus

Use separate molar masses, calculate exact mole fractions and distinguish p, p° and Δp. Report whether a number is absolute pressure, a dimensionless relative lowering or a percentage. Qualify electrolyte or concentrated cases.

Advanced insight

Measured solvent vapour pressure is related to solvent activity. For concentrated solutions, activity is not simply mole fraction, so vapour-pressure measurements can quantify departures from ideality. This connects colligative analysis with practical water-activity measurement and chemical-potential models.

Summary

For an ideal solution with nonvolatile solute, solvent p = xsolvent p° and Δp/p° = xsolute. The mole fractions come from component moles, not masses. Equal effective particle fractions give equal ideal relative lowerings, while volatility and nonideality limit the simple relation.

Practice questions

1. If xsolvent = 0.80 and p° = 5.0 kPa, find ideal p and Δp. Answer: p = 4.0 kPa and Δp = 1.0 kPa. 2. Does a 10% solute mass fraction imply 10% relative pressure lowering? Answer: No. Convert masses to moles to find solute mole fraction first. 3. Why is the same formula insufficient for measured total pressure when solute is volatile? Answer: The solute contributes its own vapour partial pressure to the total.