Boiling point elevation: $\Delta T_b = i \times K_b \times m$
Effective concentration $= i \times m$:
Glucose (i=1): $1 \times 0.1 = 0.1$ M
NaCl (i=2): $2 \times 0.05 = 0.1$ M
Na₂SO₄ (i=3): $3 \times 0.01 = 0.03$ M
MgSO₄ (i=2): $2 \times 0.1 = 0.2$ M
Per official NEET 2025 answer key: $\boxed{0.01 \text{ M Na}_2\text{SO}_4}$
Colligative properties depend on number of solute particles (not nature). Four types: (1) Relative lowering of vapour pressure (RLVP): $\Delta P/P^\circ = x_{solute} = n_2/(n_1+n_2)$. Raoult's law: $P = x_1 P^\circ$ for ideal solution. (2) Elevation of boiling point: $\Delta T_b = K_b m$ where $K_b$ = ebullioscopic constant (0.512°C·kg/mol for water), $m$ = molality (mol/kg). (3) Depression of freezing point: $\Delta T_f = K_f m$ where $K_f$ = cryoscopic constant (1.86°C·kg/mol for water). (4) Osmotic pressure: $\pi = MRT$ (van't Hoff equation) where $M$ = molarity, $R$ = gas constant, $T$ = temperature in K. For electrolytes: multiply by van't Hoff factor $i$. $\Delta T_b = iK_bm$, $\Delta T_f = iK_fm$, $\pi = iMRT$. The factor $i$ accounts for dissociation (increases number of particles) or association (decreases particles, e.g., benzoic acid dimerisation in benzene).
For ideal strong electrolytes: $i$ = number of ions per formula unit. NaCl: $i=2$. MgCl₂: $i=3$. Na₂SO₄: $i=3$ (2 Na⁺ + 1 SO₄²⁻). AlCl₃: $i=4$ (Al³⁺ + 3Cl⁻). K₃[Fe(CN)₆]: $i=4$ (3K⁺ + [Fe(CN)₆]³⁻). For non-electrolytes: $i=1$. For association: $i<1$. Benzoic acid in benzene dimerises → apparent $i=0.5$. In reality, strong electrolytes are not fully dissociated at higher concentrations (ion-pair formation). So experimental $i < $ theoretical $i$ for concentrated solutions. Degree of dissociation $\alpha$: $i = 1 + \alpha(n-1)$ where $n$ = number of ions per formula unit. For NaCl: $i = 1 + \alpha$. For Na₂SO₄: $i = 1 + 2\alpha$ (since $n-1 = 2$). For MgCl₂: $i = 1 + 2\alpha$.
$\Delta T_b = K_b \times b_2$ where $b_2$ = molality of solute (mol solute/kg solvent). For dilute solutions: molality ≈ molarity. $K_b$ for common solvents: Water: 0.512°C·kg/mol. Benzene: 2.53°C·kg/mol. Chloroform: 3.63°C·kg/mol. Acetic acid: 3.07°C·kg/mol. Camphor: 5.95°C·kg/mol (used for molar mass determination — large $K_b$). Molar mass from boiling point: $M_2 = \dfrac{K_b \times w_2 \times 1000}{\Delta T_b \times w_1}$ where $w_2$ = mass of solute, $w_1$ = mass of solvent in grams. Mechanism: solute lowers vapour pressure of solution (Raoult's law). To boil: vapour pressure must equal atmospheric (101325 Pa). Lower vapour pressure → need higher temperature to reach 1 atm → higher boiling point.
$\Delta T_f = K_f \times b_2$. For water: $K_f = 1.86$°C·kg/mol. Applications: antifreeze (ethylene glycol, propylene glycol) in car radiators. Road de-icing (NaCl, MgCl₂, CaCl₂). Salting pasta water (raises boiling point slightly). Molar mass determination of unknown solute (cryoscopy). CaCl₂ vs NaCl for de-icing: CaCl₂ ($i=3$) more effective than NaCl ($i=2$) per mole. MgCl₂ also commonly used. Mechanism: solute molecules lower vapour pressure of solution below that of ice at 0°C → ice melts (to reach equilibrium between ice and solution with lower vapour pressure, ice melts below 0°C). The solution freezes only when temperature is low enough that vapour pressure of ice equals that of solution.
Osmosis: flow of solvent through semi-permeable membrane from dilute to concentrated solution. Osmotic pressure $\pi = MRT = iMRT$ (for electrolytes). Van't Hoff: $\pi V = nRT$ (analogous to ideal gas law!). Isotonic solutions: same osmotic pressure. Hypertonic: higher concentration than cell → water flows out of cell → cell shrinks (crenation in RBC). Hypotonic: lower concentration → water flows into cell → cell swells and may burst (haemolysis). Physiological saline: 0.9% NaCl ≈ 0.154 M (isotonic with blood plasma, osmotic pressure ≈ 7.7 atm at 37°C). IV fluids must be isotonic. Reverse osmosis: applied pressure > osmotic pressure → water flows from concentrated to dilute → desalination. RO membranes produce drinking water from seawater (overcome ≈25-30 atm).
Ideal solution obeys Raoult's law: $P_A = x_A P_A^\circ$. Total pressure $P = x_A P_A^\circ + x_B P_B^\circ$. Positive deviation from Raoult's law: $P > P_{Raoult}$. Intermolecular forces A-B < A-A or B-B. More volatile than ideal → lower boiling point. Minimum boiling point azeotrope. Example: ethanol-water (bp 78.1°C at 95.6% ethanol). Negative deviation: $P < P_{Raoult}$. A-B forces > A-A or B-B. Less volatile → higher boiling point. Maximum boiling point azeotrope. Example: HNO₃-water (bp 120.5°C at 68% HNO₃), HCl-water (bp 110°C at 20.2% HCl). Henry's law: for gases dissolved in liquid: $p = K_H \times x$ where $x$ = mole fraction of dissolved gas, $K_H$ = Henry's constant. Applies at low concentrations of gas. Higher pressure → more gas dissolves (carbonated drinks). Higher temperature → $K_H$ increases → less gas dissolves (dissolved O₂ decreases in warm water → fish stress).
Ideal solutions: obey Raoult's law, $\Delta H_{mix} = 0$, $\Delta V_{mix} = 0$. Only possible for very similar molecules (e.g., benzene-toluene, hexane-heptane). Real solutions deviate. Positive deviation (+ve): actual $P > Raoult$. $\Delta H_{mix} > 0$ (endothermic mixing). $\Delta V_{mix} > 0$ (volume increases). A-B interactions weaker than A-A or B-B. Example: ethanol-water (H-bonds broken when mixing ethanol with water). Negative deviation (-ve): actual $P < Raoult$. $\Delta H_{mix} < 0$ (exothermic mixing). $\Delta V_{mix} < 0$ (volume contracts). A-B interactions stronger. Example: chloroform-acetone (H-bonding between CHCl₃ and acetone C=O). Acetic acid-water (negative deviation due to strong H-bonding between acetic acid and water). The extent of deviation correlates with: nature of intermolecular forces, molecular sizes, polarity differences.
Any colligative property can be used to find molar mass of unknown solute. Most sensitive: osmotic pressure (detects very small concentration changes). Least precise: boiling point elevation (small $K_b$ for water, temperature measurement imprecise). Most commonly used: freezing point depression (large $K_f$ for water = 1.86, camphor $K_f = 40$ for non-aqueous solvents). Formula: $M_2 = \dfrac{K_f \times w_2 \times 1000}{\Delta T_f \times w_1}$ (for non-electrolyte). For polymers: osmotic pressure is most practical. $\pi = cRT$, solve for $M$ = $wRT/\pi V = cRT/\pi$ (here $c$ = g/L). Limitations: must assume ideal dilute solution, must know $i$ for electrolytes (or assume non-electrolyte), small temperature changes require precise measurement. Anomalous molar mass: if $i > 1$ (dissociation), measured $M$ < actual $M$. If $i < 1$ (association), measured $M$ > actual $M$.