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