Physical Properties of Solutions

Learning Objectives

By the end of the module you should be able to:
- Express concentration of solutions using six different units:
- Percent by mass
- Percent by volume
- Mole fraction
- Molarity
- Molality
- Parts-per-million (ppm)
- Perform stoichiometric calculations for reactions carried out in solution (mole concept, limiting reagent, percent yield, etc.).
- Explain how concentration affects each of the colligative properties (vapor-pressure lowering, boiling-point elevation, freezing-point depression, osmotic pressure).
- Distinguish colligative-property behavior of nonelectrolyte vs. electrolyte solutions (van ’t Hoff factor, ion dissociation, degree of ionization).
- Calculate boiling-point elevation and freezing-point depression from solute concentration.

Fundamental Definition: Solution

A solution is a homogeneous mixture of two or more substances.
Because every portion is identical in composition, physical properties are uniform throughout.

Components of a Solution

Solvent
- Component present in the largest amount.
- Determines the phase (solid, liquid, or gas) of the overall solution.
Solute
- Component(s) present in smaller amount(s).
- Dispersed uniformly at the molecular or ionic level within the solvent.

Types of Solutions (Classified by Phase Combination)

Solid in Solid (alloys)
- Carbon in iron → steel
- Copper in zinc → brass
Liquid in Solid
- Amalgam (Hg in Au)
Gas in Solid
- O₂ trapped in ice
Solid in Liquid
- Sugar syrup (sucrose in water)
- Brine (NaCl in water)
Liquid in Liquid
- Ethanol in water (alcoholic beverages)
Gas in Liquid
- O₂ in water (aerated water)
- CO₂ in water (soft drinks)
Solid in Gas
- Airborne particulate pollution
Liquid in Gas
- Water vapor in air (humidity)
Gas in Gas
- Atmospheric air (N₂ + O₂ + trace gases)

Classification by Degree of Saturation

Unsaturated: contains less solute than the maximum amount possible at a given temperature.
Saturated: contains the maximum possible dissolved solute (dynamic equilibrium between dissolving and crystallizing).
Supersaturated: holds more solute than the equilibrium amount; usually unstable and precipitates excess solute upon disturbance.

Solubility Concept

Solubility = maximum mass of solute that dissolves in a given mass of solvent at a specific temperature.
- Expressed as g solute/100 g solvent or similar.
Illustrated cases (water at room T):
- Unlimited solubility: water + alcohol (miscible in all proportions).
- Limited solubility: water + oil (two layers form).
- No solubility: excess salt beyond saturation precipitates.

Concentration

Qualitative descriptors:
- Dilute: small amount of solute relative to solvent.
- Concentrated: large amount of solute.
Quantitative units covered below.

Common Concentration Units (Overview)

Percent by mass $\%\,(w/w)$
Percent by volume $\%\,(v/v)$
Mole fraction $X$
Molality $m$
Molarity $M$
Parts per million (ppm)

Percent by Mass (w/w)

Definition: $\%\,(w/w)=\frac{m{\text{solute}}}{m{\text{solution}}}\times100$
Notes:
- Both masses must be in the same units (g or kg).
- Often used for solid–liquid mixtures (e.g., saline solutions).
Example 1 (NaCl brine):
- Data: 0.892 g NaCl + 54.6 g H₂O.
- Total mass = $0.892+54.6=55.492\,\text{g}$ .
- $\%\,(w/w)=\frac{0.892}{55.492}\times100=1.61\%$ (answer after full calculation).
Example 2 (coffee sugar):
- Data: 10.0 g sugar + 250 mL H₂O.
- Convert solvent volume to mass: density = 1.00 g/mL → 250 g.
- Total mass = 260 g.
- $\%\,(w/w)=\frac{10.0}{260}\times100=3.85\%$ .

Percent by Volume (v/v)

Definition: $\%\,(v/v)=\frac{V{\text{solute}}}{V{\text{solution}}}\times100$
Used when both solute and solvent are liquids (e.g., alcohol in water).
Example 3 (acetic acid solution):
- 3.5 mL CH₃COOH + 100 mL H₂O → solution volume ≈ 103.5 mL.
- $\%\,(v/v)=\frac{3.5}{103.5}\times100=3.38\%$ .
Example 4 (vinegar dilution): 1.5 mL vinegar + 30 mL H₂O.
- Approx. solution volume = 31.5 mL.
- $\%\,(v/v)=\frac{1.5}{31.5}\times100=4.76\%$ .

Mole Fraction

Symbols: $XA$ (solute), $XB$ (solvent).
Formulae:
- $XA=\frac{nA}{nA+nB}$
- $XB=\frac{nB}{nA+nB}=1-X_A$
Properties:
- Unit-less.
- Sum of all mole fractions in a solution equals 1.
- Essential when working with vapor–pressure relations (Raoult’s Law).
Example 5 (sucrose in water):
- 10.0 g C₁₂H₂₂O₁₁ (M = 342 g mol⁻¹) → $n=0.0292\,\text{mol}$ .
- 50.0 g H₂O (M = 18.0 g mol⁻¹) → $n=2.78\,\text{mol}$ .
- $X{\text{sucrose}}=0.0104$ ; $X{\text{water}}=0.9896$ .
Example 6 (Na₂SO₄ in water):
- 20.25 g Na₂SO₄ (M = 142 g mol⁻¹) → $n=0.142\,\text{mol}$ .
- 65.0 g H₂O → $n=3.61\,\text{mol}$ .
- $X{\text{Na₂SO₄}}=0.038\,,$ $X{\text{H₂O}}=0.962$ .

Molality (m)

Definition: $m=\frac{n{\text{solute}}}{\text{mass}{\text{solvent}}\,(\text{kg})}$
Distinct features:
- Uses kilograms of solvent, not solution volume; independent of temperature.
- Crucial for colligative-property calculations.
Example 7 (aspartame drink):
- 2.00 g C₁₄H₁₈N₂O₅ (M = 294 g mol⁻¹) → $n=0.00680\,\text{mol}$ .
- 250 g H₂O = 0.250 kg.
- $m=\frac{0.00680}{0.250}=0.0272\,m$ (≈ 0.027 mol kg⁻¹).
Example 8 (Na₂CO₃ solution):
- 0.40 g Na₂CO₃ (M = 106 g mol⁻¹) → $n=0.00377\,\text{mol}$ .
- 150 g H₂O = 0.150 kg.
- $m=\frac{0.00377}{0.150}=0.025\,m$ .

Molarity (M)

Definition: $M=\frac{n{\text{solute}}}{V{\text{solution}}\,(\text{L})}$
Temperature dependent (volume expands/contracts).
Widely used in laboratory titrations and stoichiometric calculations.
Example 9 (sucrose):
- 684 g sucrose → $n=2.00\,\text{mol}$ .
- Solution volume = 1.00 L.
- $M=\frac{2.00}{1.00}=2.00\,M$ .
Example 10 (CaCl₂):
- 5.0 g CaCl₂ (M = 111 g mol⁻¹) → $n=0.0450\,\text{mol}$ .
- Volume = 0.25 L.
- $M=\frac{0.0450}{0.25}=0.18\,M$ .

Parts Per Million (ppm)

Environmental-level concentration unit.
Definitions (equivalent in dilute aqueous systems because 1 L ≈ 1 kg):
- $\text{ppm}=\frac{\text{mg solute}}{\text{L solution}}$
- $\text{ppm}=\frac{\text{mg solute}}{\text{kg solution}}$
Example 11 (brine preservative):
- 2.5 mg NaCl in 100 L.
- $\text{ppm}=\frac{2.5}{100}=0.025\,\text{ppm}$ .
Example 12 (salt in water):
- 30 mg NaCl + 105 kg water ≈ 105 kg solution.
- $\text{ppm}=\frac{30}{105}=0.286\,\text{ppm}$ .

Real-World Context & Illustrations

Household brands (Mr. Muscle cleaner, Palmolive dish soap, Ensure protein shakes, soft drinks) are practical examples of solutions where concentration directly influences cleaning efficacy, nutritional content, or flavor.
Orange-juice labels "Not from concentrate" vs. "100 % from concentrate" highlight differences between dilute and concentrated preparations of the same solute (orange solids) in water.

Links to Colligative Properties (Preview)

Boiling-point elevation: $\Delta Tb=Kb\,m\,i$
- $K_b$ = ebullioscopic constant of solvent; $i$ = van ’t Hoff factor.
Freezing-point depression: $\Delta Tf=Kf\,m\,i$
Vapor-pressure lowering & osmotic pressure obey analogous formulae using mole fraction or molarity.
Electrolyte solutions have larger $i$ (due to ionization), amplifying colligative effects versus nonelectrolytes at equal molalities.

Stoichiometry in Solution (Quick Guide)

Steps:
1. Convert given concentrations to moles of solute (using $M, m$ , or other units).
2. Use balanced chemical equations to find mole relationships.
3. Convert moles of products/reactants to masses or volumes as needed.
Key formula: $n=MV$ (for molarity-based problems).

Ethical, Environmental & Practical Implications

ppm measurements critical for monitoring toxic contaminants in drinking water.
Understanding percent by mass guides nutritional labeling (sugar, protein content).
Knowledge of saturation helps avoid precipitation in medical IV solutions or industrial crystallizers.

Quick-Reference Equations (All LaTeX-Ready)

$\%\,(w/w)=\frac{m{solute}}{m{solution}}\times100$
$\%\,(v/v)=\frac{V{solute}}{V{solution}}\times100$
$XA=\frac{nA}{nA+nB}$
$m=\frac{n{solute}}{\text{kg}{solvent}}$
$M=\frac{n{solute}}{\text{L}{solution}}$
$\text{ppm}=\frac{\text{mg solute}}{\text{L or kg solution}}$
Colligative:
- $\Delta Tb=Kb\,m\,i$
- $\Delta Tf=Kf\,m\,i$