Solutions and Colligative Properties

A solution is a homogenous mixture of two or more substances with variable compositions.
Mixtures exhibiting more than one phase are classified as heterogeneous (e.g., sand and water).
A solution consists of a solvent (component in large proportion) and a solute (component in small proportion).
Common examples include air (oxygen in nitrogen), soft drinks ( $CO_2$ , sugar, and flavouring in water), and alloys.

Solutions are classified by the number of components: binary (two), ternary (three), and quaternary (four).
Solutions exist in three physical states: Gaseous, Solid, and Liquid.
Gaseous solutions: All gas mixtures are homogeneous (e.g., Air). Components can be separated by phase changes such as low temperature distillation.
Solid solutions: Common in minerals and gemstones like emeralds, sapphires, rubies, and amethysts. Examples include alloys such as Brass (zinc-copper) and Bronze (tin and copper).
Liquid solutions: Formed by dissolving gases, liquids, or solids in a liquid solvent.

Nature of substances: Highly soluble gases include Ammonia and Sulphur(iv)oxide in water.
Temperature: Solubility of gases decreases with an increase in temperature. This is relevant in degassing processes and aquatic ecology (e.g., eutrophication reducing dissolved oxygen).
Pressure: Henry's Law states that the amount of gas dissolved at a constant temperature is proportional to its pressure: $\text{Solubility} \propto \text{Pressure}$ .

Molarity ( $M$ ): Number of moles of solute per $dm^3$ (litre) of solution.
Molality ( $m$ ): Number of moles of solute dissolved in $1000\,g$ ( $1\,kg$ ) of solvent.
Normality ( $N$ ): Gramme-equivalent weight of solute per $dm^3$ of solution. For $H_2SO_4$ , $1.0\,g$ equivalent is calculated as $\frac{98.02}{2} = 49.04\,g$ .
Mole Fraction ( $X$ ): Ratio of the moles of one component to the total moles in the solution. For component 1: $X_1 = \frac{n_1}{n_1 + n_2}$ .
Weight Percent: Grams of solute per $100\,g$ of solution.

Colligative properties depend solely on the number of solute particles, not their identity.
Vapour Pressure Lowering: Adding a non-volatile solute reduces the saturated vapour pressure because solute molecules exert an inward pull and block the escape route of solvent molecules.
Raoult's Law: The relative lowering of vapour pressure is proportional to the mole fraction of the solute: $\frac{P^o - P}{P^o} = \frac{n}{N + n} = X_n$ .
Vapour pressure of the solution ( $P$ ) is directly proportional to the mole fraction of the solvent ( $X_N$ ): $P = P^o \times X_N$ .
Molecular weight ( $M_2$ ) can be determined using $\frac{P^o - P}{P^o} = \frac{m_2 \times M_1}{m_1 \times M_2}$ for dilute solutions.

Elevation of Boiling Point ( $\Delta T_b$ ): A solution boils at a higher temperature than the pure solvent ( $101325\,Nm^{-2}$ ). The rise is proportional to molality: $\Delta T_b = K_e \times (\frac{1000 \times m_2}{M_2 \times m_1})$ .
$K_e$ is the molal elevation constant or ebullioscopic constant.
Depression of Freezing Point ( $\Delta T_f$ ): A solute lowers the freezing point of the solvent. The depression is proportional to the relative lowering of vapour pressure: $\Delta T_f = K_f \times (\frac{1000 \times m_2}{M_2 \times m_1})$ .
$K_f$ is the molal depression constant or cryoscopic constant.

Osmosis is the movement of solvent molecules through a semi-permeable membrane from a region of lower solute concentration (higher vapour pressure) to higher solute concentration (lower vapour pressure).
Osmotic Pressure ( $\pi$ ) is the hydrostatic pressure required to stop osmosis.
Van't Hoff Equation: $\pi = \frac{nRT}{V} = \frac{m_2 RT}{M_2 V}$ , where $R = 8.314\,JK^{-1}mol^{-1}$ .
Applications: Determination of molecular weights for polymers (polystyrene, Polypropylene, starch, proteins) and desalinization of sea water via reverse osmosis.
Relation to Vapour Pressure: For dilute solutions, $\frac{P - P^1}{P} = K\pi$ .