Physical ChemistryHigh Yield (2-3 Qs in JEE)

Solutions & Colligative Properties Formulas

Raoult's law, relative lowering of vapor pressure, elevation in boiling point, depression in freezing point, and van 't Hoff factor.

8 Formulas•671+ Questions in Bank

Core Formulas & Mathematical Identities

P_{\text{total}} = P_A^\circ X_A + P_B^\circ X_B, \quad Y_A = \frac{P_A}{P_{\text{total}}} = \frac{P_A^\circ X_A}{P_{\text{total}}}

Partial vapor pressure of volatile components and mole fraction Y_A in vapor phase.

Examiner Tip: Positive deviation has P_total > Raoult's prediction (ΔH_mix > 0, ΔV_mix > 0).

\frac{P_A^\circ - P_A}{P_A^\circ} = i X_B = i \frac{n_B}{n_A + n_B} \approx i \frac{w_B M_A}{M_B w_A}

Vapor pressure reduction caused by dissolving non-volatile solute B.

Examiner Tip: For dilute solutions: (P° - P)/P = i (n_B / n_A) is exact.

\Delta T_b = i K_b m, \quad \Delta T_f = i K_f m \quad \left(m = \text{molality} = \frac{1000 w_B}{M_B w_A}\right)

Thermal colligative shifts dependent strictly on particle concentration.

Examiner Tip: For water: K_b = 0.52 K kg/mol, K_f = 1.86 K kg/mol.

\pi = i C R T = i \left(\frac{n_B}{V}\right) R T

Hydrostatic pressure required to stop inward osmosis through semipermeable membrane.

Examiner Tip: Best colligative property for determining molar mass of polymers and biomolecules.

i = 1 + (n - 1)\alpha \; (\text{Dissociation}), \quad i = 1 - \left(1 - \frac{1}{n}\right)\alpha \; (\text{Association}), \quad i = \frac{M_{\text{theoretical}}}{M_{\text{observed}}}

Ratio of observed colligative property to theoretical calculated colligative property.

Examiner Tip: For complete dimerization (n = 2, α = 1), i = 0.5.

P = K_H \cdot \chi, \quad K_H \propto T \implies \text{Solubility decreases as } T \text{ increases}

Partial pressure of gas above liquid is proportional to mole fraction χ of gas dissolved in the liquid.

Examiner Tip: Aquatic species are more comfortable in cold water because higher dissolved O₂ is present at lower temperatures (smaller K_H).

K_b = \frac{R T_b^2 M_{\text{solvent}}}{1000 \Delta H_{\text{vap}}} = \frac{R T_b^2}{1000 l_v}, \quad K_f = \frac{R T_f^2 M_{\text{solvent}}}{1000 \Delta H_{\text{fus}}} = \frac{R T_f^2}{1000 l_f}

Ebullioscopic and cryoscopic constants determined solely by properties of the pure solvent.

Examiner Tip: K_b and K_f are independent of the nature of the solute dissolved.

\text{Positive Dev: } \Delta H_{\text{mix}} > 0, \; \Delta V > 0 \to \text{Min-Boiling Az.}; \quad \text{Negative Dev: } \Delta H < 0, \; \Delta V < 0 \to \text{Max-Boiling Az.}

Constant-boiling binary liquid mixtures that distill without changing composition in liquid and vapor phases.

Examiner Tip: Acetone + Chloroform shows negative deviation due to intermolecular C-H···O hydrogen bonding.

Frequently tested misconceptions in Solutions & Colligative Properties

•Forgetting to multiply colligative formulas by the van 't Hoff factor i for electrolytes (e.g. i ≈ 2 for NaCl, i ≈ 3 for CaCl₂).
•Molality uses mass of SOLVENT in kg, NOT mass of solution.

4 Topics

›Temperature-independent units: Molality (m), Mole fraction (X), Mass percentage (% w/w), and ppm (by mass). These depend solely on mass, which does not expand or contract with temperature.
›Temperature-dependent units: Molarity (M), Normality (N), and Volume percentage (% v/v). These involve solution volume, which expands as temperature increases, causing M and N to decrease.
›Interconversion formula: m = (1000 × M) / (1000 × d - M × M_solute), where d is the solution density in g/mL and M_solute is the molar mass of the solute in g/mol.

›Ideal Solutions: Obey Raoult's law at all concentrations and temperatures. Intermolecular forces A-B are identical to A-A and B-B interactions. ΔH_mix = 0, ΔV_mix = 0, ΔS_mix > 0, ΔG_mix < 0. Examples: Benzene + Toluene, n-hexane + n-heptane, Chlorobenzene + Bromobenzene.
›Positive Deviation: A-B attractive interactions are weaker than A-A and B-B. Solute particles escape more easily into vapor phase, yielding P_total > P_ideal. ΔH_mix > 0 (endothermic), ΔV_mix > 0 (volume expansion). Form minimum boiling azeotropes at a specific intermediate composition. Examples: Ethanol + Acetone, CS₂ + Acetone, Ethanol + Water.
›Negative Deviation: A-B attractive interactions are stronger than pure liquid forces (e.g., new hydrogen bonds or acid-base adducts). P_total < P_ideal, ΔH_mix < 0 (exothermic), ΔV_mix < 0 (contraction). Form maximum boiling azeotropes. Examples: Acetone + Chloroform (C-H···O=C hydrogen bond), HNO₃ + Water (68% HNO₃ azeotrope), HCl + Water.
›Azeotrope Definition: Binary liquid mixtures having the same composition in liquid and vapor phases and boiling at a single constant temperature without fractional separation.

›Colligative properties depend strictly on the ratio of solute particles to solvent molecules, not on their chemical identity or chemical nature.
›Relative Lowering of Vapor Pressure (RLVP): Non-volatile solute reduces solvent surface area, decreasing the rate of vaporization. For dilute solutions: (P° - P) / P° = X_solute = n_B / (n_A + n_B). For calculations: (P° - P) / P = n_B / n_A = (w_B × M_A) / (M_B × w_A).
›Elevation in Boiling Point (ΔT_b) & Depression in Freezing Point (ΔT_f): Boiling point rises because vapor pressure needs higher temperature to equal atmospheric pressure. Freezing point drops because solute decreases chemical potential of liquid solvent. Constants K_b and K_f are intrinsic solvent properties: K_b = R·T_b²·M_solvent / (1000·ΔH_vap), K_f = R·T_f²·M_solvent / (1000·ΔH_fus).
›Osmotic Pressure (π = iCRT): Best colligative method for high molecular weight macromolecules, proteins, and polymers because π can be measured accurately at room temperature and gives substantial, easily readable hydrostatic heights even at extremely low molarity (10⁻³ to 10⁻⁴ M).

›Definition: i = (Observed Colligative Property) / (Calculated Normal Property) = (Total particles after dissociation or association) / (Initial particles) = M_normal / M_abnormal.
›Degree of Dissociation (α): If 1 molecule dissociates into n ions (A_n → n A): i = 1 + (n - 1)α ⟹ α = (i - 1) / (n - 1). For complete dissociation (α = 1): NaCl (n = 2 ⟹ i = 2), BaCl₂ or K₂SO₄ (n = 3 ⟹ i = 3), K₄[Fe(CN)₆] (n = 5 ⟹ i = 5).
›Degree of Association (α): If n solute molecules associate into an n-mer (n A → A_n): i = 1 - (1 - 1/n)α ⟹ α = (1 - i) / (1 - 1/n). For complete dimerization (e.g. benzoic acid or acetic acid in benzene, n = 2, α = 1): i = 1 - 0.5 = 0.5.
›Isotonic Solutions: Solutions with identical osmotic pressures at the same temperature (π₁ = π₂ ⟹ i₁ C₁ = i₂ C₂). Hypotonic has lower osmotic pressure; hypertonic has higher osmotic pressure (red blood cells shrink in hypertonic saline).