Chapter 2 — Solutions

Fig 1 — Educational Diagram: Introduction to Solutions

What is a solution?
A solution is a homogeneous mixture of two or more substances. The component present in larger amount is called the solvent (usually a liquid) and the component(s) present in smaller amount are called solute(s). Examples: salt in water, sugar in tea, alloys like brass.

Types of solutions
- By physical state: solid in liquid (NaCl in water), liquid in liquid (ethanol in water), gas in liquid (CO₂ in soda), solid in solid (alloys).
- By composition: binary (two components), ternary (three components).
- Based on saturation: unsaturated (can dissolve more solute), saturated (equilibrium between dissolving and crystallizing), supersaturated (contains more solute than at equilibrium; metastable).

Concentration measures
Different ways to express how much solute is present in a solution (useful depending on context):

• Molarity (M) — moles of solute per litre of solution: M = n_solute (mol) / V_solution (L).
• Molality (m) — moles of solute per kilogram of solvent: m = n_solute (mol) / mass_solvent (kg). Useful when temperature changes because it does not depend on volume.
• Mole fraction (x) — ratio of moles of a component to total moles: x_i = n_i / Σ n_j (dimensionless).
• Mass percent (w/w %) = (mass_solute / mass_solution) × 100%.
• Volume percent (v/v %) = (volume_solute / volume_solution) × 100% (used for miscible liquids).
• ppm / ppb — parts per million/ billion for very dilute solutions: ppm = (mass_solute / mass_solution) × 10^6.

Ideal and real solutions
An ideal solution obeys Raoult's law for all compositions; interactions between like and unlike molecules are similar. Most real solutions deviate from ideality — positive deviation when solvent–solute interactions are weaker, negative when stronger.

Key laws (brief)
- Raoult's law (for an ideal solution): P_i = x_i · P_i° where P_i is partial vapour pressure of component i, x_i its mole fraction in liquid phase and P_i° the vapour pressure of pure i.
- Henry's law (for gases dissolved in liquids at low concentration): C = k_H · p (concentration of dissolved gas ∝ partial pressure above the liquid).

Factors affecting solubility
Nature of solute & solvent ("like dissolves like"—polar dissolves polar), temperature (solubility of most solids in water increases with temperature; solubility of gases decreases with temperature), pressure (significant for gases: increased pressure increases gas solubility).

Why concentrations matter
Concentration units are used to calculate reaction rates, equilibria, colligative properties (boiling point elevation, freezing point depression, osmotic pressure), and to design practical solutions (pharmaceutical, industrial, laboratory).

Fig 2 — Educational Diagram: Concentration Terms and Expressions

Overview
Concentration terms quantify how much solute is present in a given quantity of solution or solvent. Different expressions are used depending on experimental convenience and on whether the effect of the solute depends on amount of solute per mass of solvent (colligative properties) or per volume of solution (reactivity, stoichiometry).

Common concentration terms (definitions)

• Molarity (M): moles of solute per litre of solution. M = moles solute / L solution.
• Molality (m): moles of solute per kilogram of solvent. m = moles solute / kg solvent. Independent of temperature.
• Mole fraction (x_i): ratio of moles of component i to total moles. x_i = n_i / (Σ n_j). Dimensionless.
• Mass percent (w/w %): (mass solute / mass solution) × 100%.
• Volume percent (v/v %): (volume solute / volume solution) × 100% (used for miscible liquids).
• Mass/volume percent (w/v %): (mass solute (g) / volume solution (mL)) × 100% (common in biology/medicine).
• Parts per million (ppm) and parts per billion (ppb): ppm = (mass solute / mass solution) × 10^6; ppb = ×10^9. Used for very dilute contaminants.
• Normality (N): equivalents of solute per litre of solution. N = equivalents / L solution. For acid–base reactions equivalents = moles × basicity or acidity (or charge for redox).
• Formality (F): molar concentration expressed in terms of formula units per litre of solution (used when species dissociate; initial analytical concentration before dissociation).

Why choose one expression over another?
Use molality for colligative properties (ΔTb, ΔTf, osmotic pressure per mole of solute, independent of T). Use molarity for reaction stoichiometry and volumetric titrations. Use mole fraction for Raoult's law and vapour-pressure relationships. Use ppm/ppb for trace analysis and environmental measurements.

Key practical points
- Molarity depends on temperature because volume changes with temperature. Molality does not.
- For electrolyte solutions, account for dissociation using van't Hoff factor i (effective number of particles). For colligative effects, replace m by i · m (or use effective concentration).

Conversions — important relations (idea)
For 1 L of solution of molarity M and solution density d (g mL^-1):

• Mass of solute = M · M_r (g) (where M_r is molar mass).
• Mass of solution = 1000 · d (g).
• Mass of solvent = 1000·d − M·M_r (g) ⇒ in kg: (1000·d − M·M_r)/1000.
• Molality m = M·1000 / (1000·d − M·M_r).
• If solvent is water (M_w = 18.015 g mol^-1), mole fraction of solute x_solute = M / (M + (mass solvent in g)/18.015).

Colligative relations (use molality)

• Freezing point depression: ΔTf = Kf · m · i
• Boiling point elevation: ΔTb = Kb · m · i
• Osmotic pressure: π = i · M · R · T (for dilute solutions; M is molarity)

Practical tips for lab work
When preparing solutions, decide whether concentration should be expressed per litre of solution (molar) or per mass of solvent (molal). For volumetric analysis and titrations use molarity or normality (N = M × n where n is number of H+ or OH- or electrons per formula unit for the reaction). For measuring colligative properties use molality.

Fig 3 — Educational Diagram: Vapour Pressure of Solutions and Raoult's Law

Vapour pressure is the pressure exerted by the vapour in dynamic equilibrium with its liquid at a given temperature. When a non-volatile or volatile solute is mixed with a solvent, the vapour pressure of the solvent over the solution usually changes.

Raoult's law (ideal solutions)
For an ideal solution, the partial vapour pressure of each volatile component is directly proportional to its mole fraction in the liquid phase. For component A:

p_A = x_A · p_A⁰

where p_A is the partial vapour pressure of A over the solution, x_A is the mole fraction of A in the liquid, and p_A⁰ is the vapour pressure of pure A at the same temperature.

Binary mixture (A + B)
Total vapour pressure: P = p_A + p_B = x_Ap_A⁰ + x_Bp_B⁰ (with x_A + x_B = 1).

Non-volatile solute
If a non-volatile solute is dissolved in a volatile solvent, only the solvent contributes to vapour. Raoult's law gives:

p = x_solvent · p_solvent⁰

Vapour pressure lowering: Δp = p_solvent⁰ − p = p_solvent⁰ · x_solute. Relative lowering: Δp / p_solvent⁰ = x_solute.

Meaning and consequences
- Vapour pressure lowering is a colligative property: at a given temperature it depends on the number of solute particles (mole fraction) and not on their identity (for non-electrolytes and ideal behaviour).
- Lower vapour pressure of a solution means higher boiling point (boiling point elevation) and lower freezing point (freezing point depression).

Ideal vs non-ideal behaviour
- Ideal solution: A–A, B–B and A–B interactions are similar → Raoult's law holds over entire composition range.
- Positive deviation: A–B interactions are weaker than A–A and B–B → observed vapour pressure > Raoult's prediction → lower boiling point; may give a minimum-boiling azeotrope (e.g., ethanol–water azeotrope).
- Negative deviation: A–B interactions stronger than A–A and B–B → observed vapour pressure < Raoult's prediction → higher boiling point; may give a maximum-boiling azeotrope (e.g., chloroform–acetone shows strong specific interaction).

When is Raoult's law applicable?
- Solutions of chemically similar substances (e.g., benzene–toluene), or at low solute concentrations for solute acting as ideal dilute solute. For dilute non-volatile solute in a solvent, the solvent follows Raoult's law while solute follows Henry's law at very low concentrations.

How to use: Measure or use known vapour pressures of pure components (p⁰) and mole fractions to calculate partial and total vapour pressures and predict boiling behaviour or compositions of vapor in distillation problems.

Fig 4 — Educational Diagram: Colligative Properties — Overview

Definition: Colligative properties are physical properties of solutions that depend only on the number (concentration) of solute particles present, and not on their chemical identity (for ideally behaving, non-electrolyte solutes).

Four main colligative properties:

• Vapor pressure lowering
• Boiling point elevation
• Freezing point depression
• Osmotic pressure

How they arise (qualitative idea): Adding a non-volatile solute to a solvent lowers the chemical potential of the solvent in the liquid phase. As a result the solvent's vapour pressure above the solution is lower than that of the pure solvent (Raoult's law). Because vapour pressure is reduced, a higher temperature is needed to reach the external pressure (boiling point elevation) and a lower temperature is needed for solidification (freezing point depression). Lowered chemical potential also leads to osmotic flow (solvent moves from pure solvent to solution).

Important assumptions & corrections: Formulas below are strictly for dilute, ideally behaving solutions. For ionic solutes (electrolytes) dissociation produces more particles; the observed effect is multiplied by the van't Hoff factor i (effective number of particles per formula unit). At higher concentrations deviations occur due to ion pairing and non-ideal interactions (use activities).

Practical uses: Determination of molar mass of solutes (especially polymers using osmotic pressure), antifreeze formulations, de-icing roads (freezing point depression), reverse osmosis and water purification, biological osmotic balance (cells, plasmolysis).

Fig 5 — Educational Diagram: Boiling Point Elevation and Freezing Point Depression

Overview (colligative properties): Boiling point elevation and freezing point depression are colligative properties — they depend only on the number of solute particles dissolved in a solvent, not on their chemical identity. When a non-volatile solute is added to a solvent, the vapor pressure of the solvent is lowered. This causes the boiling point to rise (because a higher temperature is needed for the vapor pressure to reach the external pressure) and the freezing point to fall (because the equilibrium between solid and liquid phases is shifted).

Physical basis:

• Vapor-pressure lowering (Raoult's law): For an ideal solution with a nonvolatile solute, p_solvent = x_solvent · p°_solvent. Adding solute reduces x_solvent and therefore the vapor pressure.
• Boiling point elevation: At constant external pressure, the temperature at which vapor pressure equals the external pressure increases when vapor pressure is reduced by solute.
• Freezing point depression: The chemical potential of the liquid solvent is lowered by the solute, so equilibrium between liquid and solid shifts to lower temperature — the solution must be cooled further to freeze.

Key assumptions and limits: Results below apply to dilute, ideally behaving solutions. For electrolytes, the effective number of particles is multiplied by the van't Hoff factor (i), but ion pairing and non-ideal behavior at higher concentrations reduce i from its ideal integer value.

Derivation outline (sketch):

• Start with Raoult's law for solvent vapor pressure in a dilute solution: p = x_solvent · p°.
• For small solute mole fraction x_solute, x_solvent ≈ 1 - x_solute, so vapor-pressure lowering Δp ≈ x_solute · p°.
• Use Clausius-Clapeyron / Clapeyron relations to relate change in vapor pressure to change in temperature and the enthalpy of vaporization (or fusion). Linearization for small ΔT yields ΔTb proportional to solute molality m.
• Result: ΔTb = Kb · m (and ΔTf = Kf · m) for non-electrolytes; for electrolytes multiply by the van't Hoff factor i: ΔT = i · K · m.

Practical numbers for water: Kb (water) = 0.512 K kg mol⁻¹; Kf (water) = 1.86 K kg mol⁻¹. These are experimentally tabulated values.

Important consequences and notes:

• Boiling point elevation and freezing point depression are proportional to molality (moles solute per kg solvent), so they are independent of temperature and volume effects associated with concentration expressed as molarity.
• Electrolytes produce more particles; for example, ideally NaCl gives i ≈ 2, CaCl2 gives i ≈ 3. Real solutions often show i < ideal because of ion pairing.
• These effects are small for typical household concentrations. For example, adding 1 molal NaCl to water raises its boiling point by about 1.02 K and lowers its freezing point by about 3.72 K (using ideal i = 2).

Applications: De-icing roads, automotive antifreeze, ice-cream making, osmometry and molar mass determinations, preserving cells/tissues (cryobiology) and many environmental impacts (seawater freezing point is lower than pure water).

End of summary for Class 12 CBSE level.

Fig 6 — Educational Diagram: Osmotic Pressure

Definition: Osmotic pressure (π) is the external pressure that must be applied to a solution to stop the net flow of solvent through a semipermeable membrane from the pure solvent into the solution. It is a colligative property — it depends on the number of solute particles, not their identity.

Qualitative explanation: If a semipermeable membrane separates pure solvent (e.g., water) and a solution, solvent molecules tend to move into the solution side (where solvent chemical potential is lower) until equilibrium is reached. To stop this flow, an external pressure π is applied on the solution side. That pressure is the osmotic pressure.

Thermodynamic derivation (concise): At equilibrium, the chemical potential of the solvent in the pure state (μ1° at pressure P) equals the chemical potential of the solvent in the solution under the applied pressure (μ1 at pressure P+π). For a dilute ideal solution:

• μ1(solution, P+π) = μ1°(P) + V̄1 π + RT ln x1
• Setting μ1°(P) = μ1(solution, P+π) and using x1 ≈ 1 − n2/n1 (dilute), we get: V̄1 π ≈ −RT ln x1 ≈ RT (n2/n1).

For 1 L of solution where n1 ≈ (mass of solvent)/M1 and V ≈ volume of solution, this simplifies to the van't Hoff form:

πV = n2RT or equivalently π = CRT where C is molar concentration of solute (mol·L−1) and R is the gas constant. For electrolytes, use the van't Hoff factor i: π = iMRT.

Assumptions & limits: The van't Hoff relation is valid for dilute, ideally behaving solutions. Deviations occur at higher concentrations due to solute-solute and solute-solvent interactions; electrolytes may not fully dissociate (effective i < expected) and activity coefficients become important.

How it’s used: Osmotic pressure is used to determine molar masses of large molecules (proteins, polymers) by measuring π at known concentration (osmometry). It is also important in biology (cell volume regulation), water purification (reverse osmosis), and medicine (infusion fluids, dialysis).

Fig 7 — Educational Diagram: Determination of Molar Mass from Colligative Properties

What are colligative properties? Colligative properties depend only on the number of solute particles (not their nature). Important colligative properties are lowering of vapor pressure, elevation of boiling point, depression of freezing point, and osmotic pressure. These properties can be used experimentally to determine the molar mass (molecular weight) of a solute.

Basic idea: For a known mass of solute dissolved in a known mass (or volume) of solvent, measure a colligative effect (e.g., ΔT_f, ΔT_b, ΔP, or Π). Use the relevant relation to find the number of moles of solute; then molar mass = (mass of solute)/(moles of solute).

Common methods and their relations (for dilute, ideally behaving solutions; non-volatile solute):

• Freezing point depression: ΔT_f = K_f · m (m = molality = mol solute/kg solvent). Measure ΔT_f, compute molality m = ΔT_f / K_f, then moles solute = m · (kg solvent), and molar mass = mass_solute / moles_solute.
• Boiling point elevation: ΔT_b = K_b · m (analogue of freezing case).
• Osmotic pressure: Π = MRT (M = molarity in mol L⁻¹, R = gas constant, T = K). Measure Π, find M = Π/(RT), moles = M · volume (L), then molar mass = mass_solute / moles.
• Vapour pressure lowering (Raoult's law): ΔP = X_solute · P°_solvent and ΔP/P° = X_solute ≈ n_solute / n_solvent for dilute solutions. Using masses and known solvent molar mass one can solve for solute molar mass.

Corrections and assumptions:

• For electrolytes that dissociate or for solutes that associate, include the van't Hoff factor i: effective molality = i · m and ΔT = i K m, Π = i MRT, etc.
• Assume dilute solutions and non-volatile solute (so solvent vapour pressure measured is that of the solvent only) and ideal behaviour.

Typical experimental procedure (freezing/boiling point method):

1. Measure the freezing (or boiling) point of pure solvent.
2. Dissolve a known mass of solute in a known mass of solvent; measure the new freezing (or boiling) point.
3. Compute ΔT = T_pure - T_solution (for freezing) or ΔT = T_solution - T_pure (for boiling).
4. Find molality m = ΔT / K_f (or ΔT / K_b). Compute moles solute = m · kg_solvent. Finally, molar mass = mass_solute / moles_solute.

Why it works (short derivation highlights): Freezing/boiling point shifts arise because solute particles change the chemical potential of the solvent; for dilute solutions these shifts are proportional to molality with proportionality constants (K_f, K_b) that depend on the solvent. Osmotic pressure comes from solvent flow tendency and equals the pressure required to stop solvent flow; for dilute ideal solutions Π = nRT/V = MRT.

Fig 8 — Educational Diagram: Van't Hoff Factor and Abnormal Molar Masses

Van't Hoff factor (i) — definition: Van't Hoff factor i is the ratio of the actual number of solute particles in solution (taking dissociation or association into account) to the number of formula units dissolved. It is used to correct colligative property equations for electrolytes or associating solutes.

Why it matters: Colligative properties (boiling point elevation, freezing point depression, osmotic pressure, vapor-pressure lowering) depend on the number of solute particles, not their identity. If a solute dissociates into ions (electrolyte) or associates into larger units (dimerisation), the effective number of particles changes and the simple non-electrolyte formulas must be multiplied by i.

Derivation and common cases:

- Dissociation: if one formula unit dissociates into v particles (for example NaCl -> Na+ + Cl- has v = 2), and fraction α dissociates, the total number of particles per initial formula unit is (1 − α) + α·v = 1 + α(v − 1). So

i = 1 + α (v − 1)

From this, α = (i − 1)/(v − 1).

- Association (k-mers): if k monomers associate to form one particle and fraction α of monomers associate, the particle count per initial monomer becomes (1 − α) + α/k = 1 − α(1 − 1/k). So

i = 1 − (k − 1)/k · α = 1 − α(1 − 1/k)

For dimerisation (k = 2) this gives i = 1 − α/2.

Effect on colligative properties (dilute solution approximations):

• Freezing point depression: ΔTf = i · Kf · m
• Boiling point elevation: ΔTb = i · Kb · m
• Osmotic pressure: Π = i · M · R · T
• Relative lowering of vapour pressure (for dilute solutions): ΔP/P ≈ i · x2 (x2 = mole fraction of solute)

Abnormal molar masses (apparent molar mass from colligative measurements):

When we determine molar mass from a colligative property assuming the solute gives 1 particle per formula unit (i = 1), but the solute actually dissociates or associates, the experimentally determined molar mass M_exp will differ from the true molar mass M_true. From Π = i M_true R T (n/V) and the naive formula M_exp = (mass · R · T)/(Π · V) (which assumes i = 1) we get

M_exp = M_true / i

Thus:

• If dissociation occurs (i > 1): M_exp < M_true (apparent molar mass is smaller).
• If association occurs (i < 1): M_exp > M_true (apparent molar mass is larger).

Estimating degree of dissociation/association from experiments:

- If M_true (from molecular formula) and M_exp (from a colligative measurement) are known, then i = M_true / M_exp. Use i in the dissociation/association formulas above to find α.

Limitations and real behaviour:

- The ideal values of i (equal to v for complete dissociation or 1/k for complete association) are approached only at infinite dilution. In real solutions inter-ionic attractions, ion-pairing and incomplete dissociation make observed i lower than the ideal value for electrolytes. i depends on concentration, solvent, temperature and ionic strength.

Practical importance / real-life examples: salts in seawater change boiling/freezing points and osmotic pressure; association of weak acids in non-polar solvents (e.g., acetic acid in benzene) affects colligative-based molar-mass determinations; understanding i is essential in biochemical and industrial processes involving osmotic balance.

Summary: always correct colligative equations by the van't Hoff factor i. Use i = 1 + α(v − 1) for dissociation and i = 1 − α(1 − 1/k) for association. Apparent (abnormal) molar mass relates to true molar mass by M_exp = M_true / i.

Fig 9 — Educational Diagram: Non-ideal Solutions and Deviations from Raoult's Law

What are non‑ideal solutions? A non‑ideal solution is one in which the interactions between unlike molecules (A–B) differ significantly from those between like molecules (A–A or B–B). Because of this, the vapour pressures and other thermodynamic properties do not follow Raoult's law exactly.

Raoult's law (ideal behaviour): For an ideal binary liquid solution of components A and B, the partial vapour pressure of each component is proportional to its mole fraction in the liquid:

p_A = x_A p_A^* ,   p_B = x_B p_B^*

and total vapour pressure

p_total = x_A p_A^* + x_B p_B^*

Here p_A^* and p_B^* are the vapour pressures of the pure components at the temperature of interest.

Why deviations occur:

• If A–B interactions are weaker than A–A and B–B, molecules escape more easily → vapour pressure higher than predicted (positive deviation).
• If A–B interactions are stronger than A–A and B–B, molecules are held more tightly → vapour pressure lower than predicted (negative deviation).

Character of deviations and consequences:

• Positive deviation: total vapour pressure is greater than the Raoult's‑law value; boiling point is lower than expected. Can produce a minimum‑boiling azeotrope (constant‑boiling mixture) which cannot be separated by simple distillation beyond the azeotropic composition.
• Negative deviation: total vapour pressure is less than Raoult's‑law value; boiling point is higher than expected. Can produce a maximum‑boiling azeotrope.

Activity and activity coefficient (measure of non‑ideality):

Introduce activity a_A so that p_A = a_A p_A^*. For liquid solutions a_A is often written as a_A = x_A γ_A, where γ_A is the activity coefficient. For an ideal solution γ_A = 1. Deviations correspond to γ_A ≠ 1 (γ_A > 1 for positive deviation, γ_A < 1 for negative deviation).

Vapour composition: The mole fraction of A in vapour (y_A) is

y_A = p_A / p_total = x_A γ_A p_A^* / (x_A γ_A p_A^* + x_B γ_B p_B^*)

Azeotrope condition (brief): An azeotrope is a composition where liquid and vapour compositions are identical (y_A = x_A). In practice, azeotropes appear where the p_total vs composition curve has an extremum (dp_total/dx = 0), giving constant‑boiling behaviour.

Significance / real life:

• Industrial distillation: azeotropes (e.g., ethanol–water) limit the purity obtainable by simple distillation and require special separation methods (azeotropic distillation, extractive distillation, molecular sieves).
• Formulation of mixtures (solvents, fuels): non‑ideal behaviour affects vapour release, flammability, and evaporation rates.

How to detect non‑ideality experimentally: Plot partial or total vapour pressure versus composition and compare with straight lines from Raoult's law; deviations above the line indicate positive deviation, below indicate negative deviation. Measure activity coefficients from p_A/(x_A p_A^*).

Summary: Non‑ideal solutions occur when intermolecular forces change on mixing. Raoult's law is recovered when these forces are similar. Deviations are quantified by activity coefficients and give rise to practical phenomena such as azeotropes which strongly influence separation processes.

Fig 10 — Educational Diagram: Solubility of Gases in Liquids and Henry's Law

Definition (Henry's law - CBSE statement): At a constant temperature, the solubility (expressed as mole fraction) of a gas in a liquid is directly proportional to the partial pressure of the gas above the liquid. Mathematically: p = k_H x, where p is the partial pressure of the gas, x is its mole fraction in the liquid and k_H is Henry's law constant (for that gas–solvent pair at that temperature).

Alternate concentration form: When concentration is used (molarity c, mol·L⁻¹), Henry's law is commonly written as c = k'_H p, where k'_H (often called the Henry's law solubility coefficient) = c/p and has units such as mol·L⁻¹·atm⁻¹. Different authors swap which constant is named k_H, so check the definition used.

Why it holds (brief physical picture): At equilibrium the tendency of gas molecules to escape from the liquid equals the tendency of gas molecules in the gas phase to dissolve. For relatively low concentrations and non-reacting gases, this balance leads to a linear relation between dissolved amount and partial pressure.

Applicability and limitations: Henry's law is valid for dilute solutions of non-reacting gases and at moderate pressures. Deviations occur at high pressures, high concentrations, or when the gas chemically reacts with the solvent (e.g., CO₂ reacts with water to form H₂CO₃), which increases apparent solubility beyond Henry's law prediction.

Factors affecting gas solubility:

• Pressure: Solubility ∝ partial pressure of the gas (Henry's law). Increasing pressure increases solubility linearly (for ideal conditions).
• Temperature: For most gases dissolving in liquids the process is exothermic, so solubility decreases as temperature increases. This is why warm soda goes flat faster.
• Nature of gas and solvent: Nonpolar gases dissolve better in nonpolar solvents; polar gases (or those that react) behave differently. CO₂ is more soluble in water than O₂ because it reacts partially with water.
• Presence of dissolved salts (salting-out): Addition of salts usually decreases gas solubility in water ("salting-out").

Temperature dependence (van 't Hoff type relation): The Henry's law constant is temperature dependent. Using equilibrium thermodynamics one can write a van 't Hoff form:
ln(k_H(T2)/k_H(T1)) = -ΔH_sol/R · (1/T2 - 1/T1)
or equivalently a differential form d ln k_H/dT = ΔH_sol/(R T²), where ΔH_sol is the enthalpy change on dissolution. For exothermic dissolution (ΔH_sol < 0) k_H decreases with increasing T, i.e. solubility falls with rising temperature.

Key conceptual points to remember:

• Henry's law links gas partial pressure and amount dissolved (linear relation at low concentrations).
• Different conventions exist for k_H — check whether p = k_H x or c = k'_H p is used.
• Reactive gases and high-pressure conditions cause deviations. Chemical reactions (e.g., CO₂ + H₂O ⇄ H₂CO₃) increase apparent solubility.

Fig 11 — Educational Diagram: Applications, Problem-Solving Strategies and Experimental Methods

Overview
This topic deals with colligative properties of dilute solutions (vapor pressure lowering, boiling point elevation, freezing point depression and osmotic pressure), their applications, typical problem-solving approaches, and experimental methods used to measure these properties and determine molar masses.

Core concepts

• Colligative properties depend on the number of solute particles, not their identity (for ideal dilute solutions).
• Raoult's law (for ideal solutions): P_solution = x_solvent · P°_solvent. For a non-volatile solute, vapor pressure lowering = P° − P = x_solute · P°.
• Boiling point elevation and freezing point depression are proportional to solute molality (m): ΔT_b = K_b · m and ΔT_f = K_f · m.
• Osmotic pressure π for dilute solutions: π = C · R · T (C = molarity). Equivalent: πV = nRT.
• Electrolytes dissociate; introduce Van't Hoff factor i (effective number of particles). For dissociation, ΔT = iK·m. For partial dissociation, i = 1 + (z − 1)α for a salt that gives z ions and dissociation fraction α.

Problem-solving strategies

• Read carefully and list given data (mass, temperature changes, densities, volumes, molar masses if known).
• Decide which colligative property applies (ΔT_b, ΔT_f, vapor pressure lowering, or π) and pick the appropriate formula.
• Use molality (m = moles solute / kg solvent) for ΔT_b and ΔT_f; use molarity (C) for osmotic pressure. Convert volumes → mass → moles as needed.
• For electrolyte solutions include Van't Hoff factor i. If ionization/dissociation given indirectly, relate i and degree of dissociation α.
• For vapor pressure problems use mole fractions: x_solute = n_solute / (n_solute + n_solvent). For dilute solutions x_solute ≈ n_solute/(n_solvent) when n_solute << n_solvent (use with caution).
• Check units (Kb, Kf are in K·kg·mol−1, R usually 0.08206 L·atm·K−1·mol−1 or 8.314 J·mol−1·K−1) and convert temperatures to K where needed (for π = CRT temperature must be in K).
• Use approximations only when justified (very dilute solutions behave ideally). For concentrated or strongly interacting systems expect deviations from Raoult's law.

Experimental methods & practical procedure

• Ebullioscopy (boiling point elevation): Measure increase in boiling point of solvent after adding known mass of solute. Use ΔT_b = K_b · m to compute molar mass of solute. Apparatus: ebulliometer (boiling point thermometer, sample bulb, heating and reflux to ensure equilibrium).
• Cryoscopy (freezing point depression): Determine depression of freezing point ΔT_f to calculate molar mass using ΔT_f = K_f · m. Apparatus: cryoscope, controlled cooling bath, thermometer.
• Osmometry (osmotic pressure): Place solution and pure solvent separated by semipermeable membrane; measure pressure required to stop osmosis. Use π = CRT to find molar mass (from π and concentration). Practical for macromolecules (proteins, polymers).
• Vapour-pressure lowering and isopiestic methods: Measure vapor pressure of solution relative to pure solvent or equilibrate with a reference solution of known properties. Apply Raoult's law to determine mole fractions and molar mass.
• Typical workflow when determining molar mass experimentally: weigh solute, dissolve in measured solvent mass/volume, measure colligative effect, compute m (or C), then deduce molar mass from n = mass / M and relation between n and measured effect.
• Common sources of error: non-ideality (assume dilute), incomplete dissociation, temperature measurement inaccuracies, volatile solutes (affect vapor pressure), impurities in solvent.

Applications — real life

• Antifreeze in automobile radiators (ethylene glycol lowers freezing point and raises boiling point of coolant).
• Desalination and water purification via reverse osmosis (osmotic pressure concept).
• Blood osmolarity and medical IV solutions (isotonic saline to avoid cell swelling or shrinkage).
• Determination of molar masses of polymers and biomolecules via osmometry or boiling/freezing point methods.
• Carbonated beverages: CO2 solubility follows Henry's law; pressure and temperature control carbonation.

Tips & quick checks

• Molality is temperature-independent (useful for boiling/freezing experiments); molarity is temperature-dependent.
• Use molality for ΔT calculations and molarity for osmotic pressure problems (or convert appropriately).
• At very low concentrations Raoult's and Henry's laws are good approximations; at higher concentrations expect positive/negative deviations.