Solutions & Colligative Properties

Methods of Expressing Concentration of Solution

Concentration of solution = amount of solute dissolved in a known amount of solvent/solution.

1) Percentage

• Weight/Weight (% w/w): % w/w = (Wt. of solute / Wt. of solution) × 100
• Weight/Volume (% w/v): % w/v = (Wt. of solute / Volume of solution) × 100
• Volume/Volume (% v/v): % v/v = (Vol. of solute / Vol. of solution) × 100
• Volume/Weight (% v/w): % v/w = (Vol. of solute / Wt. of solution) × 100

2) Parts per Million (ppm) & Parts per Billion (ppb)

• ppm = (Mass of solute component / Total mass of solution) × 10⁶
• ppb = (Mass of solute component / Total mass of solution) × 10⁹

3) Strength

• Strength = (Mass of solute in grams) / (Volume of solution in litres)
• Meaning: grams of solute present in 1 litre of solution.

4) Normality (N)

• N = (Number of gram equivalents of solute) / (Volume of solution in litres)
• N = (Weight of solute in g) / (g-equivalent weight × Volume in litres)
• N = (Strength of solution) / (Equivalent weight of solute)
• N = (Wt% × density × 10) / (Equivalent weight)
• Normality equation (dilution): N₁V₁ = N₂V₂
• Mixing (same solute): N = (N₁V₁ + N₂V₂) / (V₁ + V₂)
• Strong acid + strong base (V_a, N_a) with (V_b, N_b):
  • If V_aN_a = V_bN_b → solution is neutral
  • If V_aN_a < V_bN_b → solution is basic
  • If V_aN_a > V_bN_b → solution is acidic

5) Molarity (M)

Molarity = number of moles of solute per litre of solution (or millimoles per mL).

• M = (Number of gram moles of solute) / (Volume of solution in litres)
• M = (Wt. of solute in g per litre of solution) / (Molar mass of solute)
• M = (Wt. of solute in g × 1000) / (Molar mass × Volume of solution in mL)
• M = (10 × specific gravity (density) × Wt.% of solute) / (Molar mass of solute)
• Molarity equation (dilution): M₁V₁ = M₂V₂

Relation Between Molarity and Normality

• Normality = molarity × (Molecular mass / Equivalent mass)
• For acids: Equivalent mass = Molecular mass / Basicity → N(acid) = M × Basicity
• For bases: Equivalent mass = Molecular mass / Acidity → N(base) = M × Acidity

6) Molality (m)

Molality = moles of solute per 1000 g (1 kg) of solvent.

• m = (Number of moles of solute × 1000) / (Weight of solvent in grams)
• m = (Number of gram moles of solute) / (Weight of solvent in kg)
• m = (Wt. of solute × 1000) / (Molar mass × Wt. of solvent in g)

Relation Between Molarity (M) and Molality (m)

• m = (1000 × M) / (100 × d − M × M₁)
• Where: d = density of solution (g/cc), M₁ = molar mass of solute

7) Formality (F)

• Formality = gram formula masses of an ionic solute dissolved per litre of solution.
• F = (Mass of ionic solute in g) / (Gram formula mass × Volume of solution in litres)

8) Mole Fraction (X)

• X_A = n_A / (n_A + n_B)
• X_B = n_B / (n_A + n_B)
• X_A + X_B = 1

9) Mass Fraction

• For a solution containing w_A g of A and w_B g of B:
• Mass fraction of A = w_A / (w_A + w_B)
• Mass fraction of B = w_B / (w_A + w_B)

10) Demal Unit (D)

• 1 demal = 1 mole of solute present in 1 litre of solution at 0°C.

Colligative Properties

Properties of dilute solutions containing a non-volatile solute that depend only on the number of solute particles (concentration), not the nature of solute.

Four Important Colligative Properties

• Relative lowering in vapour pressure
• Osmotic pressure
• Elevation in boiling point
• Depression in freezing point

Lowering in Vapour Pressure

• Vapour pressure: pressure exerted by vapours above a liquid in equilibrium at a given temperature.
• Depends on:
  • Nature of liquid: weaker intermolecular forces → more volatile → higher vapour pressure
  • Temperature: vapour pressure increases with increase in temperature
• Clausius–Clapeyron form (as given): log(P₂/P₁) = [ΔH_vap / (2.303R)] × (1/T₁ − 1/T₂)
• Raoult’s law (statement): Relative lowering of vapour pressure of a solution containing a non-volatile solute equals the mole fraction of solute.
• Relative lowering (as shown): (p° − p) / p° = X_solute = n / (n + N) ⇒ (p° − p)/p = n/N

Azeotropic Mixture

• Boils at constant temperature like a pure liquid and has the same composition in liquid and vapour phases.
• Also called constant boiling mixtures; cannot be separated by fractional distillation.
• Types:
  • Minimum boiling azeotropes: formed by liquid pairs showing positive deviation from ideal behaviour; boiling point lower than either component.
  • Maximum boiling azeotropes: formed by liquid pairs showing negative deviation from ideal behaviour.

Osmosis & Osmotic Pressure (π)

• Osmosis: spontaneous flow of solvent from pure solvent to solution (or lower concentration to higher concentration) through a semi-permeable membrane.
• Osmotic pressure: minimum pressure required to stop osmosis.
• Formula: π = i × C × R × T

Types of Osmosis

• Exo-osmosis: outward flow of water from a cell in concentrated solution (cell shrinks).
• Endo-osmosis: inward flow of water into a cell in dilute medium (cell swells).
• Reverse osmosis: applying pressure > osmotic pressure forces solvent to flow in reverse direction.

Measurement of Osmotic Pressure (names only)

• Pfeffer’s method
• Morse and Frazer’s method
• Berkeley and Hartley’s method
• Townsend’s reverse osmosis method
• De Vries plasmolytic method

Isotonic, Hypertonic & Hypotonic Solutions

• Isotonic (Iso-osmotic): two solutions having same osmotic pressure at same temperature.
  • Primary condition: π₁ = π₂
  • Secondary conditions (for non-associating/non-dissociating solutes): C₁ = C₂ or n₁/V₁ = n₂/V₂ or w₁/(m₁V₁) = w₂/(m₂V₂)
• Hypertonic: higher osmotic pressure than the other solution.
• Hypotonic: lower osmotic pressure than the other solution.
• Solvent flows from hypotonic to hypertonic (lower π to higher π).

Osmotic Pressure of Mixtures (as given)

• Same solute: π = (π₁v₁ + π₂v₂) / (v₁ + v₂)
• Two different solutes: π = π₁ + π₂ = [(n₁i₁ + n₂i₂)RT] / (v₁ + v₂)

Elevation in Boiling Point (Ebullioscopy)

• ΔT_b = i × K_b × m
• Also shown: ΔT_b = T_b − T_b⁰
• K_b (as given): K_b = (R T_b²) / (1000 l_v)

Depression in Freezing Point (Cryoscopy)

• ΔT_f = i × K_f × m
• Also shown: ΔT_f = T_f⁰ − T_f
• K_f (as given): K_f = (R T_f²) / (1000 l_f)

Van’t Hoff Factor (i)

• i = (Normal molecular mass) / (Observed molecular mass)
• i = (Observed value of colligative property) / (Calculated value assuming no association/dissociation)
• i = (No. of particles after association/dissociation) / (No. of particles before association/dissociation)

Degree of Dissociation (α)

• α = (i − 1) / (m − 1)
• Here, m = number of particles obtained after 100% dissociation of 1 molecule of solute.

Degree of Association (α)

• α = (1 − i) / (1 − 1/m)
• Here, m = number of solute particles that associate to form one giant particle.

Ideal vs Non-Ideal Solutions (Raoult’s Law)

Ideal Solutions

• Obey Raoult’s law at every concentration
• ΔH_mix = 0 and ΔV_mix = 0
• P = P_A + P_B = P_A⁰X_A + P_B⁰X_B
• Escaping tendency of A and B is the same in pure liquids and in the solution
• Examples: dilute solutions; benzene + toluene; n-hexane + n-heptane; chlorobenzene + bromobenzene; ethyl bromide + ethyl iodide; n-butyl chloride + n-butyl bromide

Non-Ideal Solutions: Positive Deviation from Raoult’s Law

• Do not obey Raoult’s law
• ΔH_mix > 0 and ΔV_mix > 0
• P_A⁰ > P_A (X_A), and P_B⁰ > P_B (X_B)
• Components escape easily → vapour pressure higher than expected
• Examples: acetone + ethanol; acetone + CS₂; water + methanol; water + ethanol; CCl₄ + toluene; CCl₄ + CHCl₃; acetone + benzene; CCl₄ + CH₃OH; cyclohexane + ethanol

Non-Ideal Solutions: Negative Deviation from Raoult’s Law

• Do not obey Raoult’s law
• ΔH_mix < 0 and ΔV_mix < 0
• P_A⁰ < P_A (X_A), and P_B⁰ < P_B (X_B)
• Escaping tendency lowered → vapour pressure lower than expected
• Examples: acetone + aniline; acetone + chloroform; CH₃OH + CH₃COOH; H₂O + HNO₃; chloroform + diethyl ether; water + HCl; acetic acid + pyridine; chloroform + benzene