Solutions revision notes

Types of Solutions

Solutions are homogeneous mixtures of two or more than two components. The component present in the largest quantity is the solvent, which determines the physical state of the solution, while the other components are solutes. Solutions can be categorized based on the physical state of the solvent and solute into gaseous, liquid, and solid solutions.

• Gaseous Solutions: Gas in Gas (Oxygen and nitrogen), Liquid in Gas (Chloroform in nitrogen), Solid in Gas (Camphor in nitrogen).
• Liquid Solutions: Gas in Liquid (Oxygen in water), Liquid in Liquid (Ethanol in water), Solid in Liquid (Glucose in water).
• Solid Solutions: Gas in Solid (Hydrogen in palladium), Liquid in Solid (Amalgam of mercury with sodium), Solid in Solid (Copper in gold).
• Alloys (Solid Mixtures): Utility depends heavily on composition; for example, brass is a mixture of copper and zinc, German silver contains copper, zinc, and nickel, and bronze is a mixture of copper and tin.

Expressing Concentration of Solutions

The composition of a solution is described quantitatively via various concentration terms.

• Mass percentage (w/w): (Mass of component / Total mass of solution) × 100. Used heavily in industrial chemical applications, such as 3.62% sodium hypochlorite in commercial bleach.
• Volume percentage (V/V): (Volume of component / Total volume of solution) × 100. Standard for liquid-liquid solutions, like a 35% ethylene glycol antifreeze solution which lowers the freezing point of water to 255.4K.
• Mass by volume percentage (w/V): Mass of solute dissolved in 100 mL of solution; commonly used in medicine and pharmacy. Application: Intravenous injections are always dissolved in water containing salts at specific ionic concentrations (0.9% mass/volume NaCl) that perfectly match blood plasma concentrations to prevent cell collapse.
• Parts per million (ppm): Used for trace quantities. $\text{ppm} = (\text{Number of parts of component} / \text{Total parts of all components}) \times 10^6$. Application: A concentration of 1 ppm of fluoride ions in water prevents tooth decay, 1.5 ppm causes teeth to become mottled, and high concentrations are poisonous (e.g., sodium fluoride is used as rat poison).
• Mole fraction ($x$): Moles of a component divided by total moles of all components. For a binary mixture, $x_A = n_A / (n_A + n_B)$. Sum of all mole fractions is exactly 1 ($x_1 + x_2 + ... x_i = 1$).
• Molarity (M): Moles of solute per litre (or cubic decimetre) of solution.
• Molality (m): Moles of solute per kilogram of solvent.

Solubility of Solids & Gases in Liquids

Solubility is the maximum amount of solute that can be dissolved in a specified amount of solvent at a specific temperature and pressure. It follows the principle of "like dissolves like"—polar solutes dissolve in polar solvents, and non-polar in non-polar.

• Solubility of Solids in Liquids:

  • Dynamic Equilibrium: Dissolution and crystallisation occur simultaneously. In a saturated solution, the rate of dissolution equals the rate of crystallisation ($Solute + Solvent \rightleftharpoons Solution$).
  • Effect of Temperature: Follows Le Chatelier's Principle. If dissolution is endothermic ($\Delta_{sol}H > 0$), solubility increases with rising temperature. If exothermic ($\Delta_{sol}H < 0$), solubility decreases.
  • Effect of Pressure: Pressure has no significant effect because solids and liquids are highly incompressible.

• Solubility of Gases in Liquids & Henry's Law:

  • Effect of Pressure: Solubility of gases increases greatly with increasing pressure.
  • Henry's Law: The partial pressure of the gas in the vapour phase ($p$) is proportional to the mole fraction of the gas ($x$) in the solution: $p = K_H x$.
  • Effect of Temperature: Gas dissolution is an exothermic process (similar to condensation), so solubility of gases decreases with an increase in temperature.
  • Applications of Henry's Law:
    1. Sealing soft drinks at high pressure to increase $CO_2$ solubility.
    2. Scuba divers use air diluted with helium (11.7% He, 56.2% $N_2$, 32.1% $O_2$) to avoid "bends"—a painful, toxic bubbling of nitrogen in blood upon surfacing.
    3. At high altitudes, low partial pressure of $O_2$ causes low blood oxygen, leading to weakness and an inability to think clearly (anoxia).

Vapour Pressure of Liquid Solutions & Raoult’s Law

When volatile liquids are mixed, an equilibrium forms between the liquid and vapour phases.

• Raoult's Law: For a solution of volatile liquids, the partial vapour pressure of each component ($p_i$) is directly proportional to its mole fraction in the solution ($x_i$). $p_1 = p_1^0 x_1$ and $p_2 = p_2^0 x_2$.
• Total Vapour Pressure: Following Dalton's Law, $p_{total} = p_1 + p_2 = p_1^0 + (p_2^0 - p_1^0)x_2$.
• Vapour Phase Composition: Mole fractions in the vapour phase ($y_i$) are given by $p_i = y_i p_{total}$.
• Solid in Liquid (Non-volatile Solute): Adding a non-volatile solute decreases the solvent's vapour pressure because solute molecules occupy surface area, reducing the escape rate of solvent molecules.
• Raoult's Law as a Special Case of Henry's Law: The equations $p = K_H x$ and $p_1 = p_1^0 x_1$ are identical in form. Raoult's law is a special case of Henry's law where $K_H = p_1^0$.

Ideal and Non-Ideal Solutions & Azeotropes

Based on Raoult's Law, binary liquid solutions are classified as ideal or non-ideal.

• Ideal Solutions: Obeys Raoult's Law exactly across all concentrations.
  • Conditions: $\Delta_{mix}H = 0$ and $\Delta_{mix}V = 0$.
  • Interactions: Intermolecular forces A-A and B-B are nearly equal to A-B.
  • Examples: n-hexane & n-heptane, bromoethane & chloroethane, benzene & toluene.
• Non-Ideal Solutions (Positive Deviation): Vapour pressure is higher than predicted by Raoult's Law.
  • Interactions: A-B forces are weaker than A-A or B-B forces (molecules escape more easily).
  • Examples: Ethanol & Acetone (acetone breaks ethanol's H-bonds); $CS_2$ & Acetone.
  • Minimum Boiling Azeotrope: Formed by solutions with large positive deviations (e.g., 95% ethanol-water).
• Non-Ideal Solutions (Negative Deviation): Vapour pressure is lower than predicted.
  • Interactions: A-B forces are stronger than A-A or B-B forces.
  • Examples: Phenol & Aniline (stronger H-bonding between the two); Chloroform & Acetone (new H-bond forms between them).
  • Maximum Boiling Azeotrope: Formed by solutions with large negative deviations (e.g., 68% Nitric acid and 32% water).

Colligative Properties & Determination of Molar Mass

Properties that depend only on the number of solute particles relative to total particles, irrespective of their chemical nature, are called colligative properties.

1. Relative Lowering of Vapour Pressure (RLVP)

The lowering of vapour pressure ($\Delta p_1$) depends only on the concentration of solute particles.

• $\Delta p_1 = p_1^0 - p_1 = p_1^0 x_2$.
• RLVP Formula: $\frac{p_1^0 - p_1}{p_1^0} = x_2 = \frac{n_2}{n_1 + n_2}$.
• For dilute solutions ($n_2 \ll n_1$), $\frac{p_1^0 - p_1}{p_1^0} \approx \frac{n_2}{n_1} = \frac{w_2 M_1}{M_2 w_1}$.

2. Elevation of Boiling Point ($\Delta T_b$)

The boiling point of a solution is higher than that of the pure solvent because the vapour pressure must be increased to reach atmospheric pressure.

• Formula: $\Delta T_b = K_b m$ (where $m$ is molality).
• $K_b$ is the Ebullioscopic Constant (Molal Elevation Constant), unit is $K \text{ kg mol}^{-1}$.
• Molar Mass determination: $M_2 = \frac{1000 \times w_2 \times K_b}{\Delta T_b \times w_1}$.

3. Depression of Freezing Point ($\Delta T_f$)

A solution freezes when its vapour pressure matches the vapour pressure of the pure solid solvent. Due to lowered vapour pressure, this occurs at a lower temperature.

• Formula: $\Delta T_f = K_f m$.
• $K_f$ is the Cryoscopic Constant (Molal Depression Constant), unit is $K \text{ kg mol}^{-1}$.
• Molar Mass determination: $M_2 = \frac{1000 \times w_2 \times K_f}{\Delta T_f \times w_1}$.

4. Osmosis and Osmotic Pressure ($\Pi$)

Osmosis is the flow of solvent molecules from a pure solvent (or dilute solution) into a concentrated solution through a semipermeable membrane (SPM).

• Osmotic pressure ($\Pi$) is the excess pressure applied to the solution to just stop the flow of solvent.
• Formula: $\Pi = C R T$ (where C is Molarity).
• Osmotic pressure is uniquely advantageous for determining the molar mass of biomolecules, proteins, and polymers because it is measured at room temperature, uses molarity instead of molality, and provides a measurably large value even for very dilute solutions.
• Isotonic Solutions: Two solutions with the same osmotic pressure at a given temperature. No net osmosis occurs (e.g., 0.9% w/V NaCl matches human blood cells). Hypertonic > 0.9% (cells shrink/plasmolysis), Hypotonic < 0.9% (cells swell/edema).
• Biological & Everyday Examples: Raw mangoes shrivel when placed in concentrated salt water (brine) due to water loss via osmosis. Wilted flowers revive and limp carrots become firm again when placed in fresh water because water moves into their cells. Bacterial action is prevented when preserving meat by salting or fruits by adding sugar; the bacterium loses water via osmosis, shrivels, and dies.
• Reverse Osmosis: If pressure $> \Pi$ is applied to the solution, pure solvent flows out. Used for desalination of seawater, often utilizing a film of cellulose acetate over a support as the SPM, which is permeable to water but strictly impermeable to impurities and ions.

Abnormal Molar Masses & van't Hoff Factor ($i$)

When solutes undergo dissociation (e.g., NaCl, KCl) or association (e.g., dimerization of acetic acid via H-bonding in benzene), the number of particles changes, causing discrepancies in colligative properties and the calculated molar mass.

• van't Hoff factor ($i$):
  • $i = \frac{\text{Normal Molar Mass}}{\text{Abnormal Molar Mass}}$.
  • $i = \frac{\text{Observed Colligative Property}}{\text{Calculated Colligative Property}}$.
  • $i = \frac{\text{Total moles of particles after association/dissociation}}{\text{Total moles of particles before association/dissociation}}$.
• Values of $i$:
  • $i > 1$: Dissociation (e.g., $i \approx 2$ for KCl).
  • $i < 1$: Association (e.g., $i \approx 0.5$ for ethanoic acid in benzene).
  • $i = 1$: Non-electrolytes (no association/dissociation).
• Modified Colligative Equations:
  • RLVP: $\frac{p_1^0 - p_1}{p_1^0} = i \frac{n_2}{n_1}$.
  • Boiling Point: $\Delta T_b = i K_b m$.
  • Freezing Point: $\Delta T_f = i K_f m$.
  • Osmotic Pressure: $\Pi = i C R T$.

Key Concepts & Definitions

Solution:

A homogeneous mixture of two or more components with uniform composition and properties.

Binary Solution:

A solution consisting of exactly two components.

Solute/Solvent:

Component in larger quantity is the solvent; the other is the solute.

Semipermeable Membrane (SPM):

A continuous film with submicroscopic pores allowing only solvent (not solute) molecules to pass (e.g., pig's bladder, parchment, cellophane).

Azeotrope:

A binary mixture having the same composition in liquid and vapour phases and boiling at a constant temperature.

Edema:

Swelling in tissue cells and intercellular spaces due to water retention caused by a high intake of salt (osmosis effect).

Important Rules, Laws & Principles

• Le Chatelier’s Principle: Dictates temperature dependence of solid and gas solubility. Exothermic dissolutions decrease solubility with heat; endothermic dissolutions increase solubility with heat.
• Henry's Law: At constant temperature, the solubility of a gas in a liquid is directly proportional to its partial pressure.
• Dalton's Law of Partial Pressures: Total vapour pressure over a solution is the sum of partial vapour pressures of individual components ($P_{tot} = p_1 + p_2$).
• Raoult's Law: The partial vapour pressure of each volatile component in the solution is directly proportional to its mole fraction.

Formulae & Equations

• Mass %: $\frac{w_{component}}{w_{total}} \times 100$.
• Volume %: $\frac{V_{component}}{V_{total}} \times 100$.
• ppm: $\frac{\text{Parts of component}}{\text{Total parts}} \times 10^6$.
• Mole fraction: $x_A = \frac{n_A}{n_A + n_B}$.
• Molarity (M): $\frac{\text{Moles of solute}}{\text{Volume of solution (L)}}$.
• Molality (m): $\frac{\text{Moles of solute}}{\text{Mass of solvent (kg)}}$.
• Henry's Law: $p = K_H x$.
• Raoult's Law (Volatile): $p_i = p_i^0 x_i$.
• Vapour Phase Mole Fraction: $y_i = \frac{p_i}{p_{total}}$.
• Relative Lowering of Vapour Pressure: $\frac{p_1^0 - p_1}{p_1^0} = x_2$.
• Elevation of Boiling Point: $\Delta T_b = K_b m$.
• Depression of Freezing Point: $\Delta T_f = K_f m$.
• Osmotic Pressure: $\Pi = C R T = \frac{n_2}{V} R T$.
• Thermodynamic Ebullioscopic/Cryoscopic constants: $K_b = \frac{R M_1 T_b^2}{1000 \Delta_{vap}H}$, $K_f = \frac{R M_1 T_f^2}{1000 \Delta_{fus}H}$.
• Modified Colligative Equations (with $i$): $\Delta T_b = i K_b m$, $\Delta T_f = i K_f m$, $\Pi = i C R T$.

⚠️ EXCEPTIONS & ANOMALIES

• Solubility Temperature Anomalies (Le Chatelier's exception): It is a common expectation that heating always dissolves more solid. However, if the dissolution process is exothermic ($\Delta_{sol}H < 0$), the solubility actively decreases with an increase in temperature.
• "Like Dissolves Like" Rule: Sodium chloride and sugar dissolve readily in water, but completely fail to dissolve in benzene. Conversely, naphthalene and anthracene do not dissolve in water but readily dissolve in benzene.
• Azeotropes Defying Distillation: Liquid mixtures are usually separable by fractional distillation due to differing boiling points. Azeotropes are the exception; they boil at a constant temperature with identical liquid and vapour compositions, making fractional distillation completely ineffective.
• Non-Ideal Solutions (Positive Deviation): Normally, mixing two liquids maintains similar intermolecular forces. In mixtures like ethanol and acetone, acetone molecules get between ethanol molecules and break the existing hydrogen bonds, causing unusually weak interactions and unusually high vapour pressures. Similarly, carbon disulphide and acetone show weaker dipolar interactions than their pure states.
• Non-Ideal Solutions (Negative Deviation): In a mixture of phenol and aniline, the new intermolecular hydrogen bonding between the phenolic proton and the lone pair on aniline's nitrogen is unexpectedly stronger than the bonds in the pure liquids. A mixture of chloroform and acetone behaves similarly due to the formation of a new hydrogen bond between them.
• Abnormal Molar Masses (Dissociation): When strong electrolytes like KCl are dissolved in water, we expect a certain molar mass based on the formula. However, the experimentally determined molar mass is always lower than the true value (e.g., observed as ~37.25 g/mol instead of 74.5 g/mol) because the particle count effectively doubles.
• Abnormal Molar Masses (Association): In solvents with a low dielectric constant like benzene, ethanoic acid (acetic acid) dimerizes due to hydrogen bonding. The observed molar mass is therefore exceptionally higher (nearly twice the normal value) than the formula mass.

Previous Year JEE Topics

• Calculation of van't Hoff Factor ($i$) and Degree of Dissociation/Association ($\alpha$): Connecting $\Delta T_f$ or $\Pi$ values to $K_a$ for weak acids (e.g., acetic/fluoroacetic acid).
• Henry's Law Applications: Calculating gas solubility at specific pressures or comparing $K_H$ values.
• Vapour Phase Composition ($y_i$): Direct applications of Dalton's + Raoult's laws, recognizing that the vapor is enriched in the more volatile component.
• Azeotropic Mixtures: Identification of positive vs negative deviation components and resulting min/max boiling azeotropes.

Memory Aids & JEE Traps

[JEE TIP] Trap 1 - Henry's Law Constant ($K_H$): Misconception: A higher Henry’s Law constant ($K_H$) means the gas is more soluble in the liquid. Correct Understanding: Higher $K_H$ means LOWER solubility at a given pressure. From the formula $p = K_H x$, the mole fraction of gas in solution ($x$) is inversely proportional to $K_H$ ($x = p / K_H$).

[JEE TIP] Trap 2 - Gas Solubility & Temperature: Misconception: The solubility of gases in liquids increases when the solution is heated. Correct Understanding: Dissolution of gases is an exothermic process (similar to condensation). Therefore, according to Le Chatelier’s principle, gas solubility decreases with a rise in temperature, and $K_H$ increases.

[JEE TIP] Trap 3 - Azeotrope Deviations: Misconception: Minimum boiling azeotropes are formed by solutions exhibiting negative deviation from Raoult's Law. Correct Understanding: Minimum boiling azeotropes show a large positive deviation from Raoult's Law (higher vapour pressure leads to a lower boiling point). Maximum boiling azeotropes show negative deviation.

[JEE TIP] Trap 4 - Direction of Osmosis: Misconception: In osmosis, solute particles flow from a higher concentration region to a lower concentration region. Correct Understanding: In osmosis, only solvent molecules move through the semipermeable membrane, and they flow from a lower solute concentration (dilute) to a higher solute concentration (concentrated).

[JEE TIP] Trap 5 - Temperature Dependent Concentrations: Misconception: Mole fraction and volume percentage are both independent of temperature. Correct Understanding: Concentration terms involving volume (Molarity, Volume %, Mass by volume %) change with temperature because volume is temperature-dependent. Terms involving strictly mass or moles (Molality, Mass %, Mole fraction, ppm) are independent of temperature.

[JEE TIP] Trap 6 - Vapour Phase Identity: Misconception: The composition of the vapour phase is perfectly identical to the composition of the liquid mixture in an ideal solution. Correct Understanding: The vapour phase will always be richer in the more volatile component. You must use Dalton's Law ($y_i = p_i / P_{total}$) to find the exact vapour phase mole fraction, which differs from the liquid mole fraction ($x_i$).

[JEE TIP] Trap 7 - Isotonicity & Dissociation: Misconception: All 0.1 M aqueous solutions (e.g., 0.1 M glucose and 0.1 M NaCl) will have the exact same osmotic pressure and be isotonic. Correct Understanding: Osmotic pressure is a colligative property dependent on the total number of particles, not just nominal molarity. Because NaCl dissociates ($i \approx 2$), a 0.1 M NaCl solution has nearly double the osmotic pressure of 0.1 M glucose ($i = 1$).

[JEE TIP] Trap 8 - Reverse Osmosis Mechanics: Misconception: Reverse osmosis happens naturally if a very high concentration difference exists across a membrane. Correct Understanding: Reverse osmosis is not natural; it only occurs when an external pressure strictly greater than the osmotic pressure ($\Pi$) is mechanically applied to the solution side.

[JEE TIP] Trap 9 - Constants $K_b$ & $K_f$: Misconception: Ebullioscopic ($K_b$) and Cryoscopic ($K_f$) constants depend on the nature of the solute added to the solution. Correct Understanding: Both $K_b$ and $K_f$ depend only on the nature of the solvent. They are calculated using the solvent's molar mass ($M_1$), freezing/boiling points ($T_f, T_b$), and enthalpies of fusion/vapourisation.

[JEE TIP] Trap 10 - van't Hoff Factor Formula: Misconception: The van't Hoff factor ($i$) is calculated by dividing the experimental (abnormal) molar mass by the theoretical (normal) molar mass. Correct Understanding: The formula is the exact opposite. Because colligative properties are inversely proportional to molar mass, $i =$ Normal (Calculated) molar mass / Abnormal (Observed) molar mass.