Solutions

1. Types of Solutions & Quantitative Concentration Units

• Solution Classification: Homogeneous mixtures of two or more components with uniform composition and physical properties throughout. In binary solutions, the component present in the largest quantity is the solvent, while the other is the solute.
  • Solid Solutions: Brass ($\text{Cu}-\text{Zn}$), Bronze ($\text{Cu}-\text{Sn}$), Hydrogen gas absorbed in Palladium metal ($\text{H}_2/\text{Pd}$), Amalgams of mercury with sodium ($\text{Hg}-\text{Na}$).

Temperature-Independent Units (Mass-Based)

1. Mass Percentage ($\% \text{ w/w}$):

   $$\text{Mass } \% = \frac{\text{Mass of component in solution}}{\text{Total mass of solution}} \times 100$$

2. Parts Per Million (ppm): Useful when solute is present in trace amounts (e.g., fluoride ions in drinking water at $1.0\text{ ppm}$ prevents tooth decay; $1.5\text{ ppm}$ causes enamel mottling; high levels are toxic):

   $$\text{ppm} = \frac{\text{Number of parts of component}}{\text{Total parts of all components}} \times 10^6$$

3. Mole Fraction ($x$): Dimensionless ratio of moles of a component to total moles in solution. For a binary system of components 1 and 2:

   $$x_1 = \frac{n_1}{n_1 + n_2}, \quad x_2 = \frac{n_2}{n_1 + n_2}, \quad x_1 + x_2 = 1$$

4. Molality ($m$): Number of moles of solute dissolved per kilogram ($1000\text{ g}$) of pure solvent:

   $$m = \frac{n_{\text{solute}}}{W_{\text{solvent (kg)}}} = \frac{w_2 \times 1000}{M_2 \times w_1\text{ (g)}}$$

Temperature-Dependent Units (Volume-Based)

1. Molarity ($M$): Number of moles of solute dissolved per litre ($1\text{ dm}^3$) of total solution:

   $$M = \frac{n_{\text{solute}}}{V_{\text{solution (L)}}} = \frac{w_2 \times 1000}{M_2 \times V\text{ (mL)}}$$

2. Conversion from Molarity to Molality:

   $$m = \frac{1000 \times M}{(1000 \times d) - (M \times M_2)}$$

   Where $d$ is solution density in $\text{g mL}^{-1}$ and $M_2$ is solute molar mass.

2. Solubility of Gases & Henry's Law Mechanics

Solubility of a gas in a liquid is governed by pressure and temperature.

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

  $$p = K_H \cdot x$$

  Where $p$ is partial pressure of the gas in vapour phase, $x$ is mole fraction of dissolved gas, and $K_H$ is Henry's law constant.
• Characteristics of $K_H$:
  • $K_H$ is a function of the nature of the gas.
  • At constant pressure, higher $K_H$ indicates lower solubility in the liquid.
  • For gases like $N_2$ and $O_2$, $K_H$ increases as temperature rises, meaning gas solubility decreases at elevated temperatures.

Biological & Industrial Applications

1. Soft Drinks: Carbonated beverage bottles are sealed under high pressure to increase $CO_2$ solubility.
2. Scuba Diving: Under high hydrostatic pressure deep underwater, $N_2$ dissolves in blood. During rapid ascent, $N_2$ forms gas bubbles in capillaries, causing decompression sickness ("the bends"). Diving cylinders use helium dilution (11.7% He, 56.2% $N_2$, 32.1% $O_2$) because He has very low solubility.
3. Anoxia: At high altitudes, low atmospheric partial pressure of $O_2$ leads to reduced oxygen levels in blood, producing fatigue and cognitive impairment (anoxia).

3. Vapour Pressure of Liquid Solutions & Raoult's Law

• Raoult's Law for Volatile Binary Liquids: For a solution of volatile liquids 1 and 2, the partial vapour pressure of each component is directly proportional to its mole fraction in solution:

  $$p_1 = p_1^\circ x_1 \quad \text{and} \quad p_2 = p_2^\circ x_2$$

  Total vapour pressure by Dalton's law of partial pressures:

  $$p_{\text{total}} = p_1 + p_2 = x_1 p_1^\circ + x_2 p_2^\circ = p_1^\circ + (p_2^\circ - p_1^\circ)x_2$$

• Composition of Vapour Phase: If $y_1$ and $y_2$ represent mole fractions of components 1 and 2 in the vapour phase:

  $$y_1 = \frac{p_1}{p_{\text{total}}} \quad \text{and} \quad y_2 = \frac{p_2}{p_{\text{total}}}$$

  The vapour phase is always richer in the component that is more volatile.
• Non-Volatile Solutes: When a non-volatile solid solute is dissolved in a volatile liquid, only solvent molecules enter the vapour phase. Because solute particles occupy part of the liquid surface area, the solvent's rate of vaporization and vapour pressure decrease:

  $$p_{\text{solution}} = p_1^\circ x_1$$

4. Ideal vs Non-Ideal Solutions & Azeotropes

• Azeotropes: Constant-boiling binary liquid mixtures that distill without changes in composition. They cannot be separated into pure components by fractional distillation.
  • Minimum Boiling Azeotrope: 95% Ethanol + 5% Water by volume (boils at $351.15\text{ K}$, lower than pure ethanol).
  • Maximum Boiling Azeotrope: 68% Nitric Acid + 32% Water by mass (boils at $393.5\text{ K}$, higher than pure water).

5. Colligative Properties: RLVP, Boiling Elevation & Freezing Depression

Colligative properties depend exclusively on the number of solute particles relative to the total number of particles in solution, independent of their chemical nature.

1. Relative Lowering of Vapour Pressure (RLVP)

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

For dilute solutions ($n_2 \ll n_1$):

$$\frac{p_1^\circ - p_1}{p_1^\circ} = \frac{w_2 \times M_1}{M_2 \times w_1}$$

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

Addition of a non-volatile solute lowers the solvent's vapour pressure, requiring a higher temperature to match external atmospheric pressure:

$$\Delta T_b = T_b - T_b^\circ = K_b \cdot m = \frac{1000 \cdot K_b \cdot w_2}{M_2 \cdot w_1}$$

Where $K_b$ is the Molal Elevation Constant (Ebullioscopic Constant), measured in $\text{K kg mol}^{-1}$. For water, $K_b = 0.52\text{ K kg mol}^{-1}$.

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

A solution freezes when its vapour pressure equals that of the pure solid solvent:

$$\Delta T_f = T_f^\circ - T_f = K_f \cdot m = \frac{1000 \cdot K_f \cdot w_2}{M_2 \cdot w_1}$$

Where $K_f$ is the Molal Depression Constant (Cryoscopic Constant). For water, $K_f = 1.86\text{ K kg mol}^{-1}$.

• Thermodynamic derivations for $K_b$ and $K_f$:

  $$K_b = \frac{R \cdot M_1 \cdot T_b^2}{1000 \cdot \Delta_{\text{vap}} H} \quad \text{and} \quad K_f = \frac{R \cdot M_1 \cdot T_f^2}{1000 \cdot \Delta_{\text{fus}} H}$$

6. Osmosis, Osmotic Pressure & Desalination

• Osmosis: Spontaneous flow of solvent molecules through a semipermeable membrane (SPM) from pure solvent (or dilute solution) into a concentrated solution.
• Osmotic Pressure ($\Pi$): The excess hydrostatic pressure that must be applied to the solution side to prevent the inward osmotic flow of solvent:

  $$\Pi = C R T = \frac{n_2}{V} R T = \frac{w_2 R T}{M_2 V}$$

• Comparison with Other Colligative Techniques:
  • Osmotic pressure measurements are made at room temperature without heating, preventing thermal breakdown of biomolecules.
  • It employs molarity ($M$) instead of molality ($m$).
  • It produces measurable values even for very dilute solutions of high-molar-mass polymers and proteins.
• Tonicity:
  • Isotonic Solutions: Equal osmotic pressures at identical temperature ($\Pi_1 = \Pi_2$); no net osmosis across SPM.
  • Hypertonic Solution: Higher osmotic pressure than reference fluid; causes cells to lose water and shrivel (e.g., salting meat kills bacteria via plasmolysis).
  • Hypotonic Solution: Lower osmotic pressure than reference fluid; causes solvent to enter cells, leading to swelling.
• Reverse Osmosis (RO): If an external pressure greater than the osmotic pressure ($\text{Pressure} > \Pi$) is applied to the solution, pure solvent flows out through the SPM. Used in municipal desalination of seawater using porous cellulose acetate membranes.

7. Abnormal Molar Mass & The van 't Hoff Factor ($i$)

When a solute undergoes ionic dissociation or molecular association in solution, measured colligative properties deviate from calculated values.

$$\text{van 't Hoff Factor } (i) = \frac{\text{Normal (calculated) molar mass}}{\text{Abnormal (observed) molar mass}} = \frac{\text{Observed colligative property}}{\text{Calculated colligative property}} = \frac{\text{Total particles after change}}{\text{Total particles before change}}$$

Modified Colligative Equations

1. RLVP: $\frac{p_1^\circ - p_1}{p_1^\circ} = i \cdot x_2$
2. Boiling Point Elevation: $\Delta T_b = i \cdot K_b \cdot m$
3. Freezing Point Depression: $\Delta T_f = i \cdot K_f \cdot m$
4. Osmotic Pressure: $\Pi = i \cdot C R T$

Degree of Dissociation ($\alpha$) and Association ($\alpha$)

• For Dissociation ($A_n \rightarrow n A$):

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

  • Complete dissociation ($\alpha = 1$): For $\text{NaCl}, \text{KCl}, \text{MgSO}_4 \implies i = 2$; for $\text{K}_2\text{SO}_4, \text{CaCl}_2 \implies i = 3$; for $\text{K}_4[\text{Fe}(\text{CN})_6], \text{Al}_2(\text{SO}_4)_3 \implies i = 5$.
• For Association ($n A \rightarrow A_n$):

  $$i = 1 - \left(1 - \frac{1}{n}\right)\alpha \quad \implies \quad \alpha = \frac{1 - i}{1 - \frac{1}{n}}$$

  • Complete dimerization of ethanoic/benzoic acid in benzene ($n = 2, \alpha = 1$): $i = 1 - (1 - 0.5) = \mathbf{0.5}$.