Comprehensive Study Guide for Solutions: Concentration, Solubility, and Colligative Properties

Introduction to Solutions and Mixtures

• In normal life, pure substances are rarely encountered; most substances are mixtures of two or more pure substances.

• The utility and importance of a mixture depend on its specific composition.

• Examples of Alloy Compositions:

  • Brass: A mixture of copper and zinc.

  • German Silver: A mixture of copper, zinc, and nickel.

  • Bronze: A mixture of copper and tin.

• Impact of Concentration (Fluoride Ions):

  • $1\,\text{part per million (ppm)}$ of fluoride ions in water prevents tooth decay.

  • $1.5\,\text{ppm}$ causes teeth to become mottled.

  • High concentrations can be poisonous; for example, sodium fluoride ($NaF$) is used in rat poison.

• Medical Application: Intravenous injections must be dissolved in water containing salts at ionic concentrations that match blood plasma concentrations.

Types of Solutions

• Definition of Solution: A solution is a homogeneous mixture of two or more components.

• Homogeneous Mixture: A mixture whose composition and properties are uniform throughout.

• Solvent: The component present in the largest quantity, which determines the physical state of the solution.

• Solute: One or more components present in the solution other than the solvent.

• Binary Solutions: Solutions consisting of only two components.

• Table 1.1: Classification of Solutions:

  • Gaseous Solutions:

  • Solute: Gas; Solvent: Gas. Example: Mixture of oxygen and nitrogen gases.

  • Solute: Liquid; Solvent: Gas. Example: Chloroform ($CHCl_3$) mixed with nitrogen gas.

  • Solute: Solid; Solvent: Gas. Example: Camphor in nitrogen gas.

  • Liquid Solutions:

  • Solute: Gas; Solvent: Liquid. Example: Oxygen ($O_2$) dissolved in water.

  • Solute: Liquid; Solvent: Liquid. Example: Ethanol dissolved in water.

  • Solute: Solid; Solvent: Liquid. Example: Glucose dissolved in water.

  • Solid Solutions:

  • Solute: Gas; Solvent: Solid. Example: Solution of hydrogen in palladium.

  • Solute: Liquid; Solvent: Solid. Example: Amalgam of mercury with sodium.

  • Solute: Solid; Solvent: Solid. Example: Copper dissolved in gold.

Expressing Concentration of Solutions

• Qualitative Description: Described as dilute (relatively small quantity of solute) or concentrated (relatively large quantity of solute). Quantitative descriptions are preferred to avoid confusion.

• (i) Mass Percentage ($w/w$):

  • Defined as: \text{Mass % of a component} = \frac{\text{Mass of the component in the solution}}{\text{Total mass of the solution}} \times 100

  • Example: $10\,\%$ glucose in water by mass means $10\,\text{g}$ of glucose is dissolved in $90\,\text{g}$ of water, resulting in a $100\,\text{g}$ solution.

  • Commercial application: Bleaching solution contains $3.62\,\text{mass percentage}$ of sodium hypochlorite ($NaOCl$) in water.

• (ii) Volume Percentage ($V/V$):

  • Defined as: \text{Volume % of a component} = \frac{\text{Volume of the component}}{\text{Total volume of solution}} \times 100

  • Example: $10\,\%$ ethanol solution in water means $10\,\text{mL}$ of ethanol dissolved in water to make a total volume of $100\,\text{mL}$.

  • Example: A $35\,\%\, (v/v)$ solution of ethylene glycol (antifreeze) lowers the freezing point of water to $255.4\,K$ ($-17.6^{\circ}C$).

• (iii) Mass by Volume Percentage ($w/V$):

  • The mass of solute dissolved in $100\,\text{mL}$ of solution. Commonly used in medicine and pharmacy.

• (iv) Parts per Million (ppm):

  • Used for solutes present in trace quantities.

  • Defined as: $\text{Parts per million} = \frac{\text{Number of parts of the component}}{\text{Total number of parts of all components of the solution}} \times 10^6$

  • Example: Sea water ($1\,\text{litre}$, weighing $1030\,g$) contains $6 \times 10^{-3}\,g$ of dissolved $O_2$, which is $5.8\,ppm$.

• (v) Mole Fraction ($x$):

  • Symbolized by $x$, it is the ratio of the number of moles of a component to the total number of moles.

  • Formula: $x_A = \frac{n_A}{n_A + n_B}$

  • For $i$ components: $x_i = \frac{n_i}{\sum n_i}$

  • The sum of all mole fractions in a solution is always unity: $x_1 + x_2 + ... + x_i = 1$

  • Example 1.1 Calculation: In a solution of $20\,\%$ ethylene glycol ($C_2H_6O_2$) by mass:

  • Moles of $C_2H_6O_2$: $\frac{20\,g}{62\,g\,mol^{-1}} = 0.322\,mol$

  • Moles of water ($H_2O$): $\frac{80\,g}{18\,g\,mol^{-1}} = 4.444\,mol$

  • $x_{\text{glycol}} = \frac{0.322}{0.322 + 4.444} = 0.068$

  • $x_{\text{water}} = 1 - 0.068 = 0.932$

• (vi) Molarity ($M$):

  • Number of moles of solute dissolved in one litre (or $1\,dm^3$) of solution.

  • Formula: $Molarity = \frac{\text{Moles of solute}}{\text{Volume of solution in litre}}$

  • Example 1.2: $5\,g$ of $NaOH$ in $450\,mL$ solution:

  • Moles of $NaOH = \frac{5\,g}{40\,g\,mol^{-1}} = 0.125\,mol$

  • Molarity $= \frac{0.125\,mol}{0.450\,L} = 0.278\,M$ (or $mol\,L^{-1}$).

• (vii) Molality ($m$):

  • Number of moles of solute per kilogram ($kg$) of the solvent.

  • Formula: $Molality = \frac{\text{Moles of solute}}{\text{Mass of solvent in kg}}$

  • Example 1.3: $2.5\,g$ ethanoic acid ($CH_3COOH$) in $75\,g$ benzene:

  • Molar mass of $C_2H_4O_2 = 60\,g\,mol^{-1}$

  • Moles $= \frac{2.5\,g}{60\,g\,mol^{-1}} = 0.0417\,mol$

  • Mass of benzene $= 0.075\,kg$

  • Molality $= \frac{0.0417\,mol}{0.075\,kg} = 0.556\,mol\,kg^{-1}$

• Temperature Dependence: Mass %, ppm, mole fraction, and molality are independent of temperature. Molarity is a function of temperature because volume changes with temperature.

Solubility

• Definition: The maximum amount of a substance that can be dissolved in a specified amount of solvent at a specified temperature.

• Solubility of a Solid in a Liquid:

  • Rule: "Like dissolves like." Polar solutes (e.g., sodium chloride, sugar) dissolve in polar solvents (water). Non-polar solutes (e.g., naphthalene, anthracene) dissolve in non-polar solvents (benzene).

  • Dissolution: Solute added to solvent, increasing concentration.

  • Crystallisation: Solute particles in solution collide with solid solute and separate out of the solution.

  • Saturated Solution: A solution in solution where no more solute can be dissolved at the same temperature and pressure. It exists in dynamic equilibrium: $\text{Solute} + \text{Solvent} \rightleftharpoons \text{Solution}$.

  • Unsaturated Solution: A solution where more solute can be dissolved.

  • Effect of Temperature:

  • If dissolution is endothermic (\Delta_{sol}H > 0), solubility increases with rising temperature.

  • If dissolution is exothermic (\Delta_{sol}H < 0), solubility decreases with rising temperature.

  • Effect of Pressure: Pressure has no significant effect on the solubility of solids in liquids because solids and liquids are highly incompressible.

• Solubility of a Gas in a Liquid:

  • Solubility of gases increases with an increase in pressure.

  • Henry’s Law: At a constant temperature, the solubility of a gas in a liquid is directly proportional to the partial pressure of the gas above the surface of the liquid or solution.

  • Mathematical Form: $p = K_H x$

  • $p$: partial pressure of the gas in vapour phase.

  • $x$: mole fraction of the gas in solution.

  • $K_H$: Henry’s law constant.

  • Characteristics of $K_H$:

  • Higher $K_H$ values at a given pressure indicate lower solubility.

  • $K_H$ increases with temperature, implying gas solubility decreases as temperature rises.

  • Biological Fact: Aquatic species are more comfortable in cold water due to higher dissolved oxygen levels.

  • Applications of Henry's Law:

  • Soft Drinks: Sealed under high pressure to increase $CO_2$ solubility.

  • Scuba Diving: High pressure underwater increases nitrogen solubility in blood. Rapid ascent causes nitrogen bubbles (forming "bends"), which are painful and dangerous. To prevent this, dive tanks are filled with air diluted with helium ($11.7\%\,\text{He}$, $56.2\%\,\text{N}_2$, $32.1\%\,\text{O}_2$).

  • Anoxia: At high altitudes, low partial pressure of oxygen results in low blood oxygen levels, causing climbers to feel weak and unable to think clearly.

  • Effect of Temperature: Dissolution of gases is typically exothermic (similar to condensation), thus solubility decreases as temperature increases.

Vapour Pressure of Liquid Solutions

• Vapour Pressure: The pressure exerted by vapours over a liquid at equilibrium in a closed vessel.

• Raoult’s Law (Volatile Liquids): For a solution of volatile liquids, the partial vapour pressure of each component ($p_i$) is directly proportional to its mole fraction ($x_i$) in the solution: $p_1 = p_1^0 x_1$ and $p_2 = p_2^0 x_2$

• Dalton’s Law of Partial Pressures: Total pressure ($p_{total}$) is the sum of partial pressures:

  • $p_{total} = p_1^0 + (p_2^0 - p_1^0)x_2$

• Conclusions from Raoult's Law:

  • Total vapour pressure varies linearly with the mole fraction of component 2.

  • Vapour phase is always richer in the more volatile component.

• Composition of Vapour Phase: If $y_1$ and $y_2$ are mole fractions in the vapour phase:

  • $p_1 = y_1 p_{total}$

  • $p_2 = y_2 p_{total}$

  • In general: $p_i = y_i p_{total}$

• Raoult’s Law as a Special Case of Henry’s Law: In Henry's law ($p = K_H x$), if $K_H$ equals $p_1^0$, it becomes Raoult's law.

• Vapour Pressure of Solutions of Solids in Liquids:

  • Adding a non-volatile solute reduces the vapour pressure of the solvent.

  • This happens because solute particles occupy the surface area, reducing the escaping tendency of solvent molecules.

  • The decrease depends only on the quantity of non-volatile solute, not its nature (e.g., $1.0\,mol$ sucrose vs. $1.0\,mol$ urea).

Ideal and Non-Ideal Solutions

• Ideal Solutions:

  • Obey Raoult’s Law over the entire concentration range.

  • Enthalpy of mixing ($\Delta_{mix}H$) is zero.

  • Volume of mixing ($\Delta_{mix}V$) is zero.

  • Molecular level: $A-B$ interactions are equal to $A-A$ and $B-B$ interactions.

  • Examples: n-hexane and n-heptane, bromoethane and chloroethane, benzene and toluene.

• Non-Ideal Solutions:

  • Do not obey Raoult’s law over the entire range.

  • Positive Deviation: $p_{total}$ is higher than predicted. $A-B$ interactions are weaker than $A-A$ or $B-B$. Example: Ethanol + Acetone (acetone breaks ethanol hydrogen bonds).

  • Negative Deviation: $p_{total}$ is lower than predicted. $A-B$ interactions are stronger than $A-A$ or $B-B$. Example: Chloroform + Acetone (forms a new hydrogen bond).

• Azeotropes: Binary mixtures having the same composition in liquid and vapour phase, boiling at a constant temperature.

  • Minimum Boiling Azeotrope: Formed by solutions with large positive deviation (e.g., $95\%\,\text{ethanol}$ by volume).

  • Maximum Boiling Azeotrope: Formed by solutions with large negative deviation (e.g., $68\%\,\text{nitric acid}$ and $32\%\,\text{water}$ by mass; Boiling Point = $393.5\,K$).

Colligative Properties

• Definition: Properties that depend only on the number of solute particles, not their nature.

• 1. Relative Lowering of Vapour Pressure:

  • $\frac{p_1^0 - p_1}{p_1^0} = x_2 = \frac{n_2}{n_1 + n_2}$

  • For dilute solutions: $\frac{p_1^0 - p_1}{p_1^0} = \frac{w_2 \times M_1}{M_2 \times w_1}$

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

  • Boiling point increases when a non-volatile solute is added.

  • $\Delta T_b = T_b - T_b^0$

  • $\Delta T_b = K_b m$

  • $K_b$: Molal Elevation Constant or Ebullioscopic Constant (Unit: $K\,kg\,mol^{-1}$).

  • $M_2 = \frac{1000 \times K_b \times w_2}{\Delta T_b \times w_1}$

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

  • Freezing point decreases when a non-volatile solute is added.

  • $\Delta T_f = T_f^0 - T_f$

  • $\Delta T_f = K_f m$

  • $K_f$: Molal Depression Constant or Cryoscopic Constant.

  • $M_2 = \frac{1000 \times K_f \times w_2}{\Delta T_f \times w_1}$

  • Relationships for $K_f$ and $K_b$:

  • $K_f = \frac{R \times M_1 \times T_f^2}{1000 \times \Delta_{fus}H}$

  • $K_b = \frac{R \times M_1 \times T_b^2}{1000 \times \Delta_{vap}H}$

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

  • Osmosis: Spontaneous flow of solvent molecules through a semi-permeable membrane (SPM) from pure solvent to solution (or from dilute to concentrated solution).

  • Osmotic Pressure ($\Pi$): The excess pressure applied to the solution to stop the flow of solvent.

  • $\Pi = C R T = \frac{n_2}{V} R T$

  • $M_2 = \frac{w_2 R T}{\Pi V}$

  • Advantage: Pressure measurement occurs at room temperature; uses molarity; magnitude is large even for very dilute solutions. Ideal for biomolecules (proteins/polymers).

  • Isotonic Solutions: Solutions with same osmotic pressure at a given temperature. Example: Blood cells are isotonic with $0.9\%\,(w/V)$ $NaCl$ (normal saline).

  • Hypertonic: Concentration > 0.9\% (cell shrinks).

  • Hypotonic: Concentration < 0.9\% (cell swells).

  • Biological Examples: Mangoes shrivel in brine; wilted flowers revive in water; Edema (swelling due to salt-induced water retention).

  • Reverse Osmosis (RO): If pressure greater than $\Pi$ is applied, solvent flows from solution to solvent. Used for desalination of sea water using cellulose acetate membranes.

Abnormal Molar Masses and van’t Hoff Factor

• Abnormal Molar Mass: When experimental molar mass differs from calculated value due to association or dissociation of solute particles.

  • Dissociation: Ionic compounds (like $KCl$) break into ions, increasing the number of particles. Experimental molar mass is lower than true value.

  • Association: Molecules (like acetic acid in benzene) dimerise ($2CH_3COOH \rightleftharpoons (CH_3COOH)_2$), decreasing the number of particles. Experimental molar mass is higher than expected.

• van’t Hoff Factor ($i$):

  • Defined as: $i = \frac{\text{Normal molar mass}}{\text{Abnormal molar mass}}$

  • Also: $i = \frac{\text{Observed colligative property}}{\text{Calculated colligative property}}$

  • Also: $i = \frac{\text{Total moles of particles after association/dissociation}}{\text{Number of moles of particles before association/dissociation}}$

• Values of $i$:

  • i > 1 for dissociation (e.g., $KCl \approx 2$, $K_2SO_4 \approx 3$).

  • i < 1 for association (e.g., ethanoic acid in benzene $\approx 0.5$).

• Modified Colligative Equations:

  • Relative lowering: $\frac{p_1^0 - p_1}{p_1^0} = i \frac{n_2}{n_1}$

  • $\Delta T_b = i K_b m$

  • $\Delta T_f = i K_f m$

  • $\Pi = i C R T$