Unit 1 – Solutions (Comprehensive Study Notes)

Objectives of the Unit

• After mastering this unit you will be able to:

  • Describe the formation and classification of different kinds of solutions

  • Express solution concentration using a wide variety of quantitative units

  • State & apply Henry’s and Raoult’s laws

  • Distinguish between ideal and non-ideal solutions, and explain deviations

  • Correlate colligative properties with molar mass & diagnose abnormal cases

  • Analyse real-life/industrial/biological implications (e.g.

  • soft-drink carbonation, scuba diving, medicine‐grade saline, desalination)

1.1 Types of Solutions

• Solutions = homogeneous mixtures of $\ge 2$ components, uniform composition & properties

• Terminology

  • Solvent: component present in largest amount; dictates physical phase

  • Solute(s): all other component(s)

  • Binary solution: exactly two components (focus of the text)

• Possible phase-combinations (Table 1.1)

  • Gaseous solutions: e.g. O$<em>2$ + N$</em>2$ (air)

  • Gas–liquid: O$<em>2$ in water; chloroform vapour in N$</em>2$

  • Gas–solid: camphor in N$_2$

  • Liquid solutions: ethanol–water, glucose–water, etc.

  • Solid solutions: H$_2$ in Pd, amalgams, Cu–Au alloys …

1.2 Expressing Concentration of Solutions

• Qualitative terms: ":dilute" vs ":concentrated" (vague)

• Quantitative units (formulas always reference solute unless stated):

  1. Mass % (w/w)
     $\text{Mass \%} = \frac{\text{mass of component}}{\text{total mass of solution}} \times 100$

  2. Volume % (V/V)
     $\text{Volume \%} = \frac{\text{volume of component}}{\text{total volume of solution}} \times 100$

  3. Mass-by-Volume % (w/V): g solute per 100 mL solution (medical usage)

  4. Parts-per-million (ppm)
     $\text{ppm}=\frac{\text{parts of component}}{\text{total parts of solution}} \times 10^6$

  5. Mole fraction $x<em>i$ $x</em>i=\frac{n<em>i}{\sum n</em>i}$ (sum of all $x_i$ = 1)

  6. Molarity $M$
     $M=\frac{n<em>{\text{solute}}}{V</em>{\text{solution (L)}}}$ (temperature-dependent)

  7. Molality $m$
     $m=\frac{n_{\text{solute}}}{\text{mass solvent (kg)}}$ (temperature-independent)

• Practical advice: choose unit based on context (pharmacy prefers w/V; environmental chemists ppm; theorists mole-fraction etc.)

1.3 Solubility

1.3.1 Solubility of a Solid in a Liquid

• Dynamic equilibrium: $\text{Solute}<em>{(s)} + \text{Solvent}</em>{(l)} \rightleftharpoons \text{Solution}$

• Saturated solution: dissolution rate = crystallisation rate

• Factors

  • Nature of solute & solvent ("like dissolves like")

  • Temperature:

  • Endothermic dissolution $(\Delta H_{sol} >0)$ → solubility ↑ with T

  • Exothermic dissolution $(\Delta H_{sol}<0)$ → solubility ↓ with T

  • Pressure: negligible effect for solids (incompressible)

1.3.2 Solubility of a Gas in a Liquid — Henry’s Law

• Statement (popular form): $p = K_H x$

  • $p$ = partial pressure of the gas over solution

  • $x$ = mole fraction of gas in solution

  • $K_H$ (Henry constant) depends on gas–solvent pair & T

• Consequences & applications

  • CO$_2$ bottled under high P (fizzy drinks)

  • Scuba divers use He–O$_2$ mixtures to avoid "bends"

  • Anoxia at high altitudes (lower $p<em>{\text{O}</em>2}$)

  • Cold water holds more O$_2$ (aquatic life preference)

• Empirical data: $K_H$ grows with T ⇒ gases less soluble at higher T

1.4 Vapour Pressure of Liquid Solutions

1.4.1 Binary Volatile Liquid–Liquid Systems — Raoult’s Law

• For ideal behaviour:
  $p<em>1 = x</em>1 p<em>1^0, \; p</em>2 = x<em>2 p</em>2^0$
  $p<em>{\text{total}} = p</em>1 + p<em>2$ leading to $p</em>{\text{total}} = p<em>1^0 + (p</em>2^0 - p<em>1^0) x</em>2$ (linear plot)

• Vapour-phase composition: $y<em>i = \frac{p</em>i}{p_{\text{total}}}$

• Key insight: vapour is richer in the more volatile component

1.4.2 Raoult’s Law as Special Case of Henry’s Law

• Mathematically identical if $K<em>H = p</em>1^0$ and solute is volatile

1.4.3 Solutions of Solids in Liquids (Non-volatile Solute)

• Surface gets diluted by solute → fewer escaping solvent molecules → $p<em>1 < p</em>1^0$

• Generalised Raoult: $p<em>1 = x</em>1 p<em>1^0$ still applies with $x</em>1 \approx 1$

1.5 Ideal and Non-ideal Solutions

1.5.1 Ideal Solutions

• Obey Raoult’s law over entire composition range

• No enthalpy/volume change: $\Delta<em>{mix}H = 0, \; \Delta</em>{mix}V = 0$

• Intermolecular forces A–A ≈ B–B ≈ A–B

• Examples: n-hexane + n-heptane, benzene + toluene

1.5.2 Non-ideal Solutions

• Deviate from Raoult’s law; $\Delta_{mix}H \neq 0$

  • Positive deviation

  • A–B interactions < A–A / B–B

  • $p<em>{total}$ above Raoult line; $\Delta</em>{mix}H>0$ (endothermic)

  • e.g. ethanol + acetone, CS$_2$ + acetone

  • Negative deviation

  • A–B interactions stronger

  • $p<em>{total}$ below Raoult; $\Delta</em>{mix}H<0$ (exothermic)

  • e.g. chloroform + acetone, phenol + aniline

• Azeotropes

  • Constant boiling binary mixtures where $x<em>{liquid}=x</em>{vapour}$

  • Minimum-bp azeotrope (positive deviation): EtOH–H$_2$O (95 % EtOH, 351 K)

  • Maximum-bp azeotrope (negative deviation): HNO$<em>3$–H$</em>2$O (68 % HNO$_3$, 393.5 K)

1.6 Colligative Properties & Determination of Molar Mass

Properties depending only on particle count, not identity.

1.6.1 Relative Lowering of Vapour Pressure (RLVP)

• $\frac{p<em>1^0 - p</em>1}{p<em>1^0} = x</em>2 = \frac{n<em>2}{n</em>1+n<em>2} \approx \frac{n</em>2}{n_1}$

• Molar mass calculation (dilute limit):
  $M<em>2 = \frac{w</em>2 M<em>1}{w</em>1} \times \frac{p<em>1^0}{p</em>1^0-p_1}$

1.6.2 Elevation of Boiling Point

• $\Delta T<em>b = K</em>b m$

• $m = \frac{w<em>2}{M</em>2} \times \frac{1000}{w_1}$

• Hence $M<em>2 = \frac{K</em>b w<em>2 1000}{\Delta T</em>b w_1}$

1.6.3 Depression of Freezing Point

• $\Delta T<em>f = K</em>f m$ (analogous expressions)

• Values: Water $K<em>f=1.86\;\text{K kg mol}^{-1},\;K</em>b=0.52$

1.6.4 Osmosis & Osmotic Pressure

• Semipermeable membrane allows solvent only; net flow until $\pi$ reached

• $\pi = C R T = \frac{n<em>2 R T}{V}$ ⇒ $M</em>2 = \frac{w_2 R T}{\pi V}$

• Biological terms

  • Isotonic (equal $\pi$), hypertonic (higher $\pi$ outside cells), hypotonic (lower $\pi$ outside)

  • Medical saline 0.9 % NaCl ≈ isotonic with blood

1.6.5 Reverse Osmosis & Water Purification

• Apply $P>\pi$ to force solvent from solution → pure water (desalination plants)

• Cellulose acetate or other polymer RO membranes; pressures tens of atm

1.7 Abnormal Molar Masses & van’t Hoff Factor

• Association ↓ particle number (e.g. $2\,\text{CH}<em>3\text{COOH}\rightarrow (\text{CH}</em>3\text{COOH})_2$); observed $M$ ↑

• Dissociation ↑ particles (electrolytes); observed $M$ ↓

• van’t Hoff factor $i$

  • $i = \frac{\text{actual number of particles after diss/assoc}}{\text{formula particles before}}$

  • Also $i = \frac{\text{observed colligative property}}{\text{calculated (ideal) value}} = \frac{M<em>{normal}}{M</em>{observed}}$

  • Modify all colligative equations, e.g. $\Delta T<em>f = i K</em>f m$, $\pi = i CRT$

• Typical $i$ values (very dilute):

  • NaCl, KCl: $i \rightarrow 2$

  • K$<em>2$SO$</em>4$: $i \rightarrow 3$

  • MgSO$_4$: intermediate due to ion pairing

Real-world, Ethical & Practical Connections

• Water fluoridation: ppm levels beneficial; high ppm toxic

• Antifreeze (ethylene glycol) relies on $\Delta T_f$ elevation; environmental toxicity concern

• Blood transfusion compatibility rooted in osmotic pressure (isotonicity)

• Pharmacology measures doses via w/V % & osmotic pressure for large biomolecules (proteins, vaccines)

• Environmental monitoring: ppm pollutants (e.g. CHCl$_3$ carcinogenic threshold)

• Industrial metal alloys (brass, bronze, German silver) properties depend on solution composition of solids

Key Equations (Quick Reference)

• $x<em>i = \frac{n</em>i}{\sum n_i}$; $M = \frac{n}{V}$; $m = \frac{n}{\text{kg solvent}}$

• $p = K_H x$

• $p<em>i = x</em>i p<em>i^0$; $p</em>{tot}=x<em>1 p</em>1^0 + x<em>2 p</em>2^0$

• $\frac{p<em>1^0-p</em>1}{p<em>1^0}=x</em>2$

• $\Delta T<em>b = i K</em>b m$; $\Delta T<em>f = i K</em>f m$

• $\pi = i C R T = i \frac{n R T}{V}$

• $i = \frac{M<em>{normal}}{M</em>{observed}}$ (or reciprocal with CPs)

Common Constants & Data

• $R = 0.083\;\text{L bar mol}^{-1}\text{K}^{-1}$

• Water: $K<em>b = 0.52\;\text{K kg mol}^{-1},\;K</em>f = 1.86\;\text{K kg mol}^{-1}$

• Sample $K<em>H$ at 298 K (water solvent): He 144.9 kbar, N$</em>2$ 76.5 kbar, O$<em>2$ 34.9 kbar, CO$</em>2$ 1.67 kbar

Typical Problem-Solving Strategies

• Always convert masses/volumes to moles first

• For dilute aqueous solutions: assume solvent density $\approx 1\,\text{g mL}^{-1}$ unless specified

• Check if electrolyte → insert $i$

• For gas solubility problems identify pressure, use $K<em>H$ table, apply $p = K</em>H x$ then mole relations

• Colligative property questions often revolve around rearranging formula to isolate $M_2$

• In non-ideal liquid mixtures, sketch Raoult‐law lines then compare actual data to diagnose +/- deviation

(These notes encapsulate every core concept, formula, example, and real-world link discussed across pages 1–30 of the transcript, crafted for rapid revision before exams.)