Solutions and Colligative Properties

Core Review

Concentration Units

Molarity (M) — moles of solute per liter of solution:

M = mol solute / L solution

Molarity is temperature-dependent because solution volume changes with temperature.

Molality (m) — moles of solute per kilogram of solvent:

m = mol solute / kg solvent

Molality is temperature-independent (mass doesn't change with T). Used in colligative property calculations.

Mole fraction (X) — ratio of moles of one component to total moles:

Xₐ = nₐ / (nₐ + n_b + ...)

The sum of all mole fractions in a solution = 1.

Dilution

Adding solvent decreases concentration. Moles of solute remain constant:

M₁V₁ = M₂V₂

where M₁, V₁ = initial concentration and volume; M₂, V₂ = final concentration and volume. Both volumes must use the same unit.

Worked Example: What volume of 12.0 M HCl is needed to prepare 500.0 mL of 1.50 M HCl?

• V₁ = M₂V₂ / M₁ = (1.50 M × 0.5000 L) / 12.0 M = 0.0625 L = 62.5 mL. Add water to 500.0 mL total.

Solubility Rules

For aqueous solutions at 25°C:

Generally SOLUBLE (aq):

• Nitrates (NO₃⁻) — all soluble
• Ammonium (NH₄⁺) — all soluble
• Alkali metal (Group 1) salts — all soluble
• Halides (Cl⁻, Br⁻, I⁻) — soluble EXCEPT with Ag⁺, Pb²⁺, Hg₂²⁺
• Sulfates (SO₄²⁻) — soluble EXCEPT with Ba²⁺, Sr²⁺, Ca²⁺, Pb²⁺, Hg₂²⁺ (CaSO₄ is slightly soluble)

Generally INSOLUBLE (s):

• Carbonates (CO₃²⁻) — insoluble EXCEPT with alkali metals and NH₄⁺
• Phosphates (PO₄³⁻) — insoluble EXCEPT with alkali metals and NH₄⁺
• Hydroxides (OH⁻) — insoluble EXCEPT with alkali metals, NH₄⁺, and slightly soluble with Ca²⁺, Sr²⁺, Ba²⁺
• Sulfides (S²⁻) — insoluble EXCEPT with alkali metals, NH₄⁺, and alkaline earths

Precipitation: Mix two soluble salts; if any product ion combination is insoluble, a precipitate forms.

Colligative Properties

Colligative properties depend ONLY on the number of solute particles, not their identity. The van't Hoff factor (i) accounts for dissociation: i = number of ions per formula unit. For nonelectrolytes, i = 1. For strong electrolytes: NaCl → i ≈ 2 (Na⁺ + Cl⁻); CaCl₂ → i ≈ 3; FeCl₃ → i ≈ 4.

Freezing Point Depression:

ΔT_f = i × K_f × m

ΔT_f = freezing point decrease (°C), K_f = cryoscopic constant (°C/m), m = molality. Water: K_f = 1.86 °C/m.

Boiling Point Elevation:

ΔT_b = i × K_b × m

ΔT_b = boiling point increase (°C), K_b = ebullioscopic constant (°C/m). Water: K_b = 0.512 °C/m.

Osmotic Pressure:

π = i × M × R × T

π = osmotic pressure (atm), M = molarity (mol/L), R = 0.08206 L·atm/mol·K, T = Kelvin. Osmotic pressure is the pressure needed to prevent net water flow across a semipermeable membrane.

Worked Example: What is the freezing point of a 0.500 m solution of CaCl₂ in water? (K_f = 1.86 °C/m)

• CaCl₂ dissociates into 3 ions → i = 3
• ΔT_f = 3 × 1.86 × 0.500 = 2.79°C
• New freezing point = 0.00°C − 2.79°C = −2.79°C

van't Hoff Factor Nuances

• For strong electrolytes at low concentration, i approaches the theoretical value.
• At higher concentrations, ion pairing reduces the effective i (observed i < theoretical i).
• For weak electrolytes, i is between 1 and the theoretical maximum — depends on degree of dissociation.
• The DAT typically uses theoretical i values unless stated otherwise.

Key Equations

Common Traps

• Using molarity (M) instead of molality (m) in ΔT_f and ΔT_b equations.
• Forgetting the van't Hoff factor for ionic solutes (i = 1 only for molecular solutes).
• Confusing K_f and K_b values.
• Applying M₁V₁ = M₂V₂ without consistent volume units.
• Misapplying solubility rules: remember Ag⁺, Pb²⁺, Hg₂²⁺ are the halide exceptions.