17.3 Solutions, Concentration Units, Colligative Properties & Raoult's Law

Solutions, Concentration Units, Colligative Properties & Raoult's Law

In the Intermediate Chemistry Part I syllabus and the University of Engineering and Technology (UET) ECAT entry test, physical solutions and colligative properties represent high-frequency quantitative topics. Engineering entrance questions test conversions between concentration scales under density constraints, ideal vs. non-ideal vapor pressure behaviors under Raoult's Law, fractional distillation limitations of azeotropes, and calculations of molecular weights using ebullioscopic, cryoscopic, and osmotic pressure equations. Candidates must master these algebraic formulations, their temperature dependencies, and the van 't Hoff ionization factor ($i$) to handle multi-step numerical questions rapidly.

---

1. Physical Nature & Classification of Solutions

A solution is defined thermodynamically as a completely homogeneous physical mixture of two or more chemically non-reacting substances whose proportions can be varied within certain limits. Homogeneity signifies that physical and chemical properties (refractive index, density, concentration) are uniform throughout every microscopic domain of the system:

• Solute: The substance dispersed in the smaller proportion, which undergoes dissolution.
• Solvent: The dispersing medium present in the larger proportion, which determines the physical state of the resulting solution (water is the universal aqueous solvent).

Depending on the physical state (gas, liquid, solid) of solute and solvent, there are nine possible binary solution types:

1. Gas in Gas: Air (oxygen, nitrogen, noble gases).
2. Gas in Liquid: Carbonated soda water ($\text{CO}_2$ in $\text{H}_2\text{O}$), dissolved oxygen in natural streams.
3. Gas in Solid: Hydrogen gas adsorbed/occluded into palladium metal ($\text{H}_2/\text{Pd}$).
4. Liquid in Gas: Water vapor or humidity in air, aerosol mists.
5. Liquid in Liquid: Ethanol in water, antifreeze (ethylene glycol in water).
6. Liquid in Solid: Mercury amalgams with sodium or silver ($\text{Hg}/\text{Ag}$ dental amalgams).
7. Solid in Gas: Dust or smoke particles dispersed in air, sublimed iodine vapor in nitrogen.
8. Solid in Liquid: Table salt or glucose dissolved in water.
9. Solid in Solid: Metallic alloys (brass: copper + zinc; bronze: copper + tin; steel: iron + carbon).

---

2. Quantitative Concentration Units & Temperature Invariance

Concentration expresses the relative quantity of solute dissolved in a specified mass or volume of solvent or total solution. The major concentration units utilized in engineering chemistry include:

1. Percentage Composition

• Percentage Weight/Weight ($\%, w/w$): Mass of solute in grams dissolved per $100\text{ g}$ of total solution: $\% (w/w) = \frac{\text{Mass of solute (g)}}{\text{Total mass of solution (g)}} \times 100$
• Percentage Weight/Volume ($\%, w/v$): Mass of solute in grams dissolved per $100\text{ cm}^3$ ($100\text{ mL}$) of solution: $\% (w/v) = \frac{\text{Mass of solute (g)}}{\text{Total volume of solution (cm}^3)} \times 100$
• Percentage Volume/Volume ($\%, v/v$): Volume of liquid solute in $\text{cm}^3$ per $100\text{ cm}^3$ of solution: $\% (v/v) = \frac{\text{Volume of solute (cm}^3)}{\text{Total volume of solution (cm}^3)} \times 100$

2. Molarity ($M$)

Molarity ($M$) is the number of moles of solute dissolved in exactly one cubic decimeter ($1\text{ dm}^3 = 1\text{ liter} = 1000\text{ cm}^3$) of solution:

$M = \frac{\text{Moles of solute } (n_2)}{\text{Volume of solution in dm}^3 (V)} = \frac{W_2 \times 1000}{M_2 \times V(\text{cm}^3)}$

where $W_2$ is mass of solute (g), $M_2$ is molar mass of solute ($\text{g/mol}$), and $V(\text{cm}^3)$ is volume in cubic centimeters.

ECAT Temperature Trap: Because liquids expand upon heating and contract upon cooling, total solution volume $V$ increases slightly with temperature (an expansion, not a direct proportionality). Consequently, Molarity ($M$) changes with temperature. A solution measured at $20^\circ\text{C}$ will possess a lower molarity when heated to $60^\circ\text{C}$.

3. Molality ($m$)

Molality ($m$) is the number of moles of solute dissolved in exactly one kilogram ($1\text{ kg} = 1000\text{ g}$) of pure solvent:

$m = \frac{\text{Moles of solute } (n_2)}{\text{Mass of pure solvent in kg } (W_1)} = \frac{W_2 \times 1000}{M_2 \times W_1(\text{g})}$

ECAT Axiom: Because mass is strictly invariant with respect to thermal expansion and pressure, molality is completely independent of temperature. For precision thermodynamics and colligative property studies across temperature variations, molality is the mandatory concentration scale.

4. Mole Fraction ($X$)

The mole fraction ($X$) of any component in a mixture is the ratio of the number of moles of that component to the total number of moles of all components present in the system:

$X_A = \frac{n_A}{n_A + n_B} \quad \text{and} \quad X_B = \frac{n_B}{n_A + n_B}$

• The sum of mole fractions of all components is identically equal to unity: $\sum X_i = X_A + X_B = 1$.
• Mole fraction is a dimensionless, temperature-independent quantity.

5. Parts Per Million (ppm)

Used for extraordinarily dilute solutions (e.g., trace water pollutants, toxic heavy metals, atmospheric contaminants):

$\text{ppm} = \frac{\text{Mass of solute}}{\text{Mass of solution}} \times 10^6$

For dilute aqueous solutions with density $\rho \approx 1.0\text{ g/cm}^3$, $1\text{ ppm} = 1\text{ mg of solute per } 1\text{ dm}^3\text{ of solution} = 1\text{ mg/L}$.

Dilution and Mixing Formulas

• Dilution Law: When pure solvent is added to dilute a solution from volume $V_1$ to $V_2$, the absolute moles of solute remain constant: $M_1 V_1 = M_2 V_2$
• Mixing Non-reacting Solutions: If two solutions of the same solute are combined: $M_{\text{final}} = \frac{M_1 V_1 + M_2 V_2}{V_1 + V_2}$

---

3. Vapor Pressure & Raoult's Law

The vapor pressure of a liquid is the pressure exerted by its vapor in dynamic thermodynamic equilibrium with its liquid phase inside a closed container at a specified temperature. As temperature increases, the kinetic energy of liquid molecules rises exponentially, accelerating the evaporation rate and dramatically increasing vapor pressure.

Case 1: Non-Volatile, Non-Electrolyte Solute in a Volatile Solvent

When a non-volatile solute (e.g., sucrose, urea, glucose) is dissolved in a pure volatile solvent (water):

• Solute particles occupy a proportional fraction of the liquid surface area.
• Fewer solvent molecules reside at the liquid-vapor interface per unit area.
• The rate of solvent evaporation decreases, while condensation rate remains unaffected.
• Consequently, the vapor pressure of the solution ($P$) is strictly lower than that of the pure solvent ($P^\circ$).

In 1887, François-Marie Raoult quantified this phenomenon:

Raoult's Law (Form 1): The vapor pressure of a solvent above a dilute solution ($P$) is directly proportional to the mole fraction of the solvent ($X_1$) in the solution: $P = P^\circ X_1$

Because $X_1 + X_2 = 1 \implies X_1 = 1 - X_2$ (where $X_2$ is solute mole fraction):

$P = P^\circ(1 - X_2) = P^\circ - P^\circ X_2$

Rearranging gives the lowering of vapor pressure ($\Delta P = P^\circ - P$):

$\Delta P = P^\circ X_2$

Dividing by the vapor pressure of pure solvent gives the relative lowering of vapor pressure:

$\frac{\Delta P}{P^\circ} = X_2 = \frac{n_2}{n_1 + n_2}$

Key Principle: The relative lowering of vapor pressure ($\frac{\Delta P}{P^\circ}$) is identically equal to the mole fraction of the non-volatile solute, and is entirely independent of temperature.

Case 2: Binary Mixtures of Volatile Liquids

For a homogeneous solution of two miscible, volatile liquids $A$ and $B$:

$P_A = P_A^\circ X_A \quad \text{and} \quad P_B = P_B^\circ X_B$

According to Dalton's Law of Partial Pressures, the total vapor pressure of the mixture ($P_{\text{total}}$) is:

$P_{\text{total}} = P_A + P_B = P_A^\circ X_A + P_B^\circ X_B = P_A^\circ X_A + P_B^\circ (1 - X_A)$

In the vapor phase above the liquid, the mole fractions ($Y_A, Y_B$) are governed by partial pressures:

$Y_A = \frac{P_A}{P_{\text{total}}} = \frac{P_A^\circ X_A}{P_{\text{total}}} \quad \text{and} \quad Y_B = \frac{P_B}{P_{\text{total}}}$

According to Konovalov's Rule, the vapor phase is always richer in the more volatile component (the component with higher pure vapor pressure and lower boiling point).

---

4. Ideal vs. Non-Ideal Solutions & Azeotropic Mixtures

Ideal Solutions

An ideal solution is one that perfectly obeys Raoult's Law across all possible concentrations and temperatures:

1. Thermodynamic Criteria:
   • $\Delta H_{\text{mixing}} = 0$ (No heat is absorbed or evolved upon mixing).
   • $\Delta V_{\text{mixing}} = 0$ (Total volume equals the strict sum of component volumes; no expansion or contraction).
2. Molecular Basis: Intermolecular attractive forces between unlike molecules ($A-B$) are identical in magnitude and character to forces between like molecules ($A-A$ and $B-B$).
3. Examples: Benzene + toluene; n-hexane + n-heptane; bromoethane + chloroethane; chlorobenzene + bromobenzene.

Non-Ideal Solutions & Deviations from Raoult's Law

Real liquid mixtures deviate from Raoult's Law due to differing intermolecular force magnitudes:

1. Positive Deviation ($\Delta H_{\text{mix}} > 0, \Delta V_{\text{mix}} > 0$):
   • Unlike attractions ($A-B$) are weaker than like attractions ($A-A$ or $B-B$).
   • Molecules escape into the vapor phase more easily than in pure states.
   • Observed vapor pressures are higher than predicted: $P_{\text{total}} > P_A^\circ X_A + P_B^\circ X_B$.
   • Examples: Ethanol + water; ethanol + cyclohexane; acetone + carbon disulfide ($\text{CS}_2$).
   • Minimum Boiling Azeotrope: At a specific intermediate composition, the vapor pressure curve exhibits a maximum, causing the boiling point curve to display a minimum. For example, a mixture of $95.6\%$ ethanol and $4.4\%$ water boils at a constant temperature of $78.1^\circ\text{C}$ (lower than pure ethanol at $78.3^\circ\text{C}$ and pure water at $100^\circ\text{C}$). It cannot be concentrated further by fractional distillation.
2. Negative Deviation ($\Delta H_{\text{mix}} < 0, \Delta V_{\text{mix}} < 0$):
   • Unlike attractions ($A-B$) are stronger than like attractions ($A-A$ or $B-B$). This frequently occurs when hydrogen bonding forms between dissimilar molecules.
   • Molecules escape into the vapor phase with greater difficulty.
   • Observed vapor pressures are lower than predicted: $P_{\text{total}} < P_A^\circ X_A + P_B^\circ X_B$.
   • Examples: Acetone + chloroform (strong hydrogen bond forms: $(\text{CH}_3)_2\text{C}=\text{O} \cdots \text{H}-\text{CCl}_3$); hydrochloric acid + water; nitric acid + water.
   • Maximum Boiling Azeotrope: The vapor pressure curve exhibits a minimum, producing a maximum boiling point. For example, $20.2\%\text{ HCl}$ and $79.8\%\text{ H}_2\text{O}$ boils at a constant temperature of $108.6^\circ\text{C}$ (higher than pure water at $100^\circ\text{C}$ and pure $\text{HCl}$ at $-85^\circ\text{C}$).

---

5. The Four Colligative Properties

Colligative properties (from Latin colligatus, meaning "bound together") are physical properties of dilute solutions that depend solely on the total number of solute particles (molecules or ions) present in a given mass of solvent, and are completely independent of the chemical identity, size, or charge of the solute particles.

The four fundamental colligative properties are:

1. Relative Lowering of Vapor Pressure

As derived from Raoult's Law, for a dilute solution ($n_2 \ll n_1$):

$\frac{\Delta P}{P^\circ} = X_2 = \frac{n_2}{n_1 + n_2} \approx \frac{n_2}{n_1} = \frac{W_2 / M_2}{W_1 / M_1} = \frac{W_2 M_1}{M_2 W_1}$

Rearranging to determine the unknown molar mass of solute ($M_2$):

$M_2 = \frac{W_2 \cdot M_1 \cdot P^\circ}{W_1 \cdot \Delta P}$

2. Elevation of Boiling Point (Ebullioscopy)

The boiling point is the temperature at which the vapor pressure of a liquid equals external atmospheric pressure. Because a non-volatile solute lowers vapor pressure, the solution must be heated to a higher temperature than the pure solvent before its vapor pressure matches atmospheric pressure:

$\Delta T_b = T_b - T_b^\circ$

where $T_b$ is the boiling point of the solution and $T_b^\circ$ is that of pure solvent. For dilute solutions:

$\Delta T_b = K_b \cdot m$

where $m$ is molality and $K_b$ is the molal boiling point elevation constant (or ebullioscopic constant):

• $K_b$ is an intrinsic property of the solvent alone, defined as the boiling point elevation produced when $1\text{ mole}$ of non-volatile solute is dissolved in $1000\text{ g}$ ($1\text{ kg}$) of pure solvent.
• Units of $K_b$: $\text{K}\cdot\text{kg}\cdot\text{mol}^{-1}$ or $^\circ\text{C}\cdot\text{kg}\cdot\text{mol}^{-1}$. For water, $K_b = 0.52^\circ\text{C}\cdot\text{kg}\cdot\text{mol}^{-1}$.
• Substituting molality $m = \frac{W_2 \times 1000}{M_2 \times W_1}$: $\Delta T_b = K_b \cdot \frac{W_2 \times 1000}{M_2 \times W_1} \implies M_2 = \frac{K_b \times W_2 \times 1000}{\Delta T_b \times W_1}$
• Experimentally measured using Landsberger's method.

3. Depression of Freezing Point (Cryoscopy)

The freezing point is the temperature at which the vapor pressure of the liquid phase equals the vapor pressure of its solid phase. Because dissolving a solute depresses the liquid vapor pressure, the solution vapor pressure curve intersects the solid solvent curve at a lower temperature:

$\Delta T_f = T_f^\circ - T_f$

where $T_f^\circ$ is the freezing point of pure solvent and $T_f$ is that of the solution. For dilute solutions:

$\Delta T_f = K_f \cdot m$

where $K_f$ is the molal freezing point depression constant (or cryoscopic constant):

• Units of $K_f$: $\text{K}\cdot\text{kg}\cdot\text{mol}^{-1}$. For water, $K_f = 1.86^\circ\text{C}\cdot\text{kg}\cdot\text{mol}^{-1}$.
• Substituting molality: $\Delta T_f = K_f \cdot \frac{W_2 \times 1000}{M_2 \times W_1} \implies M_2 = \frac{K_f \times W_2 \times 1000}{\Delta T_f \times W_1}$
• Experimentally measured using Beckmann's freezing point apparatus.
• Practical Applications: Ethylene glycol used as automotive engine antifreeze/coolant; sodium chloride ($\text{NaCl}$) and calcium chloride ($\text{CaCl}_2$) spread on icy roads to melt ice by lowering its freezing point.

4. Osmotic Pressure ($\pi$)

Osmosis is the spontaneous net flow of solvent molecules through a semi-permeable membrane (SPM) from a region of pure solvent (or dilute solution) into a more concentrated solution. Osmotic pressure ($\pi$) is defined as the minimum excess external hydrostatic pressure that must be applied to the solution compartment to prevent the net inward osmotic flow of solvent.

Jacobus Henricus van 't Hoff formulated the osmotic pressure law for dilute solutions, which mirrors the ideal gas equation:

$\pi V = n_2 R T \implies \pi = \left(\frac{n_2}{V}\right) R T = C R T$

where $C = n_2 / V$ is molar concentration ($\text{mol/dm}^3$), $R = 0.0821\text{ dm}^3\cdot\text{atm}\cdot\text{K}^{-1}\cdot\text{mol}^{-1}$, and $T$ is temperature in Kelvin. Expanding $n_2 = W_2 / M_2$:

$\pi = \frac{W_2 R T}{M_2 V} \implies M_2 = \frac{W_2 R T}{\pi V}$

ECAT High-Yield Fact: Osmotic pressure measurement (via Berkeley and Hartley's method) is the most accurate technique for determining the molar masses of polymers, synthetic plastics, proteins, and biological macromolecules. Because these macromolecules possess huge molar masses ($>10,000\text{ g/mol}$), their dilute solutions produce unmeasurably small boiling point elevations ($\approx 10^{-4\ \circ}\text{C}$) or freezing point depressions, but produce substantial, easily readable osmotic pressures of several millimeters of mercury ($>10\text{ mm Hg}$) at room temperature.

---

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

When calculating molar masses using colligative equations, experimental values frequently deviate from theoretical values. Jacobus van 't Hoff introduced the van 't Hoff factor ($i$) to account for non-ideal particle behavior:

$i = \frac{\text{Observed Colligative Property}}{\text{Calculated (Theoretical) Colligative Property}} = \frac{\text{Theoretical (Normal) Molar Mass}}{\text{Observed (Experimental) Molar Mass}} = \frac{\text{Actual Total Moles of Particles in Solution}}{\text{Moles of Solute Formula Units Dissolved}}$

1. Electrolytic Dissociation ($i > 1$)

When an electrolyte dissociates into ions, the total number of dissolved particles increases, magnifying the colligative effect and causing the apparent experimental molar mass to be lower than normal:

For an electrolyte $A_x B_y \rightleftharpoons x A^{y+} + y B^{x-}$ producing $n = x + y$ ions per formula unit with a degree of dissociation $\alpha$:

$\text{Initial moles: } 1 \quad \longrightarrow \quad \text{Equilibrium moles: } (1 - \alpha) + n\alpha = 1 + (n - 1)\alpha$

$i = 1 + (n - 1)\alpha \implies \alpha = \frac{i - 1}{n - 1}$

• For a strong $1:1$ electrolyte like $\text{NaCl}$ ($n = 2$) with complete dissociation ($\alpha = 1$): $i = 2$. Freezing point depression is doubled: $\Delta T_f = 2 K_f m$.
• For $\text{CaCl}_2$ ($n = 3, \alpha = 1$): $i = 3$. Freezing point depression is tripled: $\Delta T_f = 3 K_f m$.

2. Molecular Association ($i < 1$)

When solute molecules associate (polymerize) into clusters, the total number of particles decreases, diminishing the colligative property and causing the apparent molar mass to be higher than normal:

For dimerization of carboxylic acids (acetic acid or benzoic acid) in non-polar solvents like benzene through intermolecular cyclic hydrogen bonding ($2\text{RCOOH} \rightleftharpoons (\text{RCOOH})_2$):

$\text{Initial moles: } 1 \quad \longrightarrow \quad \text{Equilibrium: } (1 - \alpha) + \frac{\alpha}{n} = 1 - \alpha\left(1 - \frac{1}{n}\right)$

For dimerization ($n = 2$):

$i = 1 - \frac{\alpha}{2} \implies \alpha = 2(1 - i)$

If association is complete ($\alpha = 1$), $i = 0.5$, and the apparent molar mass of acetic acid in benzene is measured as $120\text{ g/mol}$ (double its formula mass of $60\text{ g/mol}$).

Modified Colligative Equations

• $\frac{\Delta P}{P^\circ} = i X_2$
• $\Delta T_b = i K_b m$
• $\Delta T_f = i K_f m$
• $\pi = i C R T$

---

7. Concentration Units & Colligative Equations Reference Table

---

8. Hydration, Hydrolysis and Colloids

• Hydration: water molecules surround dissolved ions (oriented by ion–dipole attraction) and release energy. Some salts crystallise with water of crystallization, such as $\text{CuSO}_4\cdot 5\text{H}_2\text{O}$ (blue) and $\text{Na}_2\text{CO}_3\cdot 10\text{H}_2\text{O}$.
• Hydrolysis: ions of some salts react with water and change the pH. $\text{NH}_4\text{Cl}$ (salt of a strong acid and weak base) gives an acidic solution, $\text{Na}_2\text{CO}_3$ and $\text{CH}_3\text{COONa}$ (salts of a weak acid and strong base) give basic solutions, and $\text{NaCl}$ (strong acid, strong base) does not hydrolyse.
• Solubility curves: the solubility of most solids rises with temperature (for example $\text{KNO}_3$ rises steeply), that of $\text{NaCl}$ changes very little, and gases become less soluble as temperature rises.

True solutions, colloids and suspensions differ in particle size: