Solutions

Solution: Homogeneous mixture of two or more components. Two parts:

• Solute: Smaller quantity, dissolved
• Solvent: Larger quantity, dissolves the solute

Types of Solutions (based on state):

Concentration Terms (recap from Class 11 with additions):

Important: Molarity changes with $T$ (volume changes); molality does NOT (mass-based).

Inter-conversions:

• $M = (m \times \rho \times 1000)/(1000 + m \cdot M_{solute})$
• $m = (M \times 1000)/(\rho \times 1000 - M \cdot M_{solute})$ (approximation if dilute)

Solubility: Maximum amount of solute that can dissolve in given amount of solvent at given $T$.

Henry's Law (Gas in liquid): Solubility (mole fraction) of a gas in a solvent is proportional to its partial pressure above solution: $$p = K_H \cdot x$$ where $p$ = partial pressure of gas, $K_H$ = Henry's constant.

Higher $K_H$ → lower solubility.

• $K_H$ increases with $T$ (gas less soluble at higher $T$)
• Used to explain: rising bubbles in soft drinks, decompression sickness in divers (bends)

Effect of Temperature on Solubility:

• Solids: usually $\uparrow$ with T (KCl, NaCl, etc.); some decrease ($Ce_2(SO_4)_3$)
• Gases: $\downarrow$ with T (always)
• $\Delta_{sol}H$ governs: endothermic dissolution favored at higher T (Le Chatelier)

Vapour Pressure (VP): Pressure exerted by vapor above liquid in equilibrium. Depends on:

• Nature of liquid
• Temperature (increases with T)
• Independent of amount

Raoult's Law: Partial vapor pressure of any volatile component in solution is equal to the product of its mole fraction and pure component vapor pressure: $$p_A = x_A \cdot p_A^\ominus, \quad p_B = x_B \cdot p_B^\ominus$$ Total pressure (Dalton): $p_{total} = p_A + p_B = x_A p_A^\ominus + x_B p_B^\ominus$.

For solution of non-volatile solute in solvent: Only solvent vaporizes: $$p_{total} = x_{solvent} \cdot p_{solvent}^\ominus$$ This gives the relative lowering of vapor pressure: $$\frac{p^\ominus - p}{p^\ominus} = \frac{n_{solute}}{n_{total}} \approx \frac{n_{solute}}{n_{solvent}} \text{ (dilute)}$$

Ideal Solutions: Obey Raoult's law over the entire composition range. Characteristics:

• $\Delta_{mix}H = 0$
• $\Delta_{mix}V = 0$
• Intermolecular forces A-A ≈ B-B ≈ A-B

Examples: benzene + toluene; n-hexane + n-heptane; bromoethane + chloroethane.

Non-Ideal Solutions: Deviate from Raoult's law.

Positive Deviation: Actual pressure > predicted.

• A-B interactions < A-A or B-B
• $\Delta_{mix}H > 0$ (endothermic mixing)
• $\Delta_{mix}V > 0$
• Examples: ethanol + cyclohexane, acetone + ethanol, $C_6H_6 + CCl_4$, $H_2O + C_2H_5OH$.

Negative Deviation: Actual pressure < predicted.

• A-B interactions > A-A or B-B (especially H-bonding)
• $\Delta_{mix}H < 0$
• $\Delta_{mix}V < 0$
• Examples: acetone + chloroform, $HCl + H_2O$, $HNO_3 + H_2O$.

Azeotropes: Mixtures with constant BP; vapor and liquid have same composition. Cannot be separated by simple distillation.

Two types:

• Minimum boiling azeotrope: From positive deviation (e.g., ethanol $95.6\%$ + water $4.4\%$, BP $78.1°$C)
• Maximum boiling azeotrope: From negative deviation (e.g., $HNO_3 + H_2O$: $68\% HNO_3$, BP $120°$C; $HCl + H_2O$: $20.2\% HCl$, BP $108.5°$C)

Colligative Properties: Properties of solutions that depend on the number of solute particles, not their nature. Four main:

1. Relative lowering of vapor pressure
2. Elevation in boiling point
3. Depression in freezing point
4. Osmotic pressure

All explained by lowering of vapor pressure of solvent due to solute.

1. Relative Lowering of VP: $(p^\ominus - p)/p^\ominus = x_{solute}$.

For dilute solution: $x_{solute} \approx n_{solute}/n_{solvent}$.

2. Elevation in Boiling Point ($\Delta T_b$): When non-volatile solute added, solution boils at higher T than pure solvent.

$$\Delta T_b = T_b - T_b^\ominus = K_b \cdot m$$

• $K_b$ = molal elevation constant (or ebullioscopic constant), unit K·kg/mol
• $m$ = molality of solute
• For water: $K_b = 0.52$ K·kg/mol

3. Depression in Freezing Point ($\Delta T_f$): Solution freezes at lower T than pure solvent.

$$\Delta T_f = T_f^\ominus - T_f = K_f \cdot m$$

• $K_f$ = molal depression constant (or cryoscopic constant), unit K·kg/mol
• For water: $K_f = 1.86$ K·kg/mol

Applications:

• Antifreeze in radiators (ethylene glycol + water)
• De-icing roads (NaCl, CaCl₂)
• Determination of molar mass of unknown solute

Determining Molar Mass: From $\Delta T_b = K_b \cdot (w_2 \times 1000)/(M_2 \times w_1)$ where $w_2$ = mass of solute, $w_1$ = mass of solvent in g.

4. Osmotic Pressure ($\pi$): Pressure needed to prevent osmosis (flow of solvent from low to high concentration through semi-permeable membrane).

$$\pi = CRT = \frac{n}{V}RT$$

• $C$ = molar concentration
• $R$ = $0.0821$ L·atm/mol·K
• $T$ in K

Isotonic: Same $\pi$. Hypertonic: Higher $\pi$. Hypotonic: Lower $\pi$.

Reverse osmosis: Applying pressure greater than $\pi$ forces solvent from high to low concentration; used in water purification.

Abnormal Molar Mass: Sometimes molar mass calculated from colligative properties doesn't match the theoretical (chemical formula) value. Causes:

• Dissociation: Strong electrolytes break into ions; more particles than expected
• Association: Molecules cluster (e.g., acetic acid in benzene dimerizes via H-bonds)

Van't Hoff Factor (i): $$i = \frac{\text{Observed colligative property}}{\text{Calculated colligative property (assuming no dissociation/association)}}$$

Equivalent expressions: $$i = \frac{\text{Total moles of particles after dissociation/association}}{\text{Total moles initially}}$$

$$i = \frac{M_{normal}}{M_{observed}}$$

For Dissociation: If $\alpha$ = degree of dissociation; one molecule gives $n$ ions: $$i = 1 + (n - 1)\alpha$$

If complete dissociation ($\alpha = 1$): $i = n$.

For Association: If $\alpha$ = degree of association; $n$ molecules combine to form one: $$i = 1 - \alpha + \alpha/n = 1 + \alpha(1/n - 1)$$

If complete association: $i = 1/n < 1$.

Example: acetic acid in benzene dimerizes ($n = 2$); $i \to 0.5$ if complete.

Modified Colligative Property Equations (with $i$):

Calculation of Degree of Dissociation: $\alpha = (i - 1)/(n - 1)$

Calculation of Degree of Association: $\alpha = (1 - i)/(1 - 1/n)$