The boiling point elevation is the phenomenon that the boiling point of a solvent will be higher when another compound is added, meaning that a solution has a higher boiling point than a pure solvent.

The equation for boiling point elevation is:

ΔT_b = i × K_b × m

Where:

• ΔT_b = boiling point elevation
• i = van 't Hoff factor (number of particles the solute dissociates into)
• K_b = ebullioscopic constant (solvent dependent)
• m = molality of the solution (moles solute per kg solvent)

Example: For a 1 molal solution of NaCl in water (i ≈ 2), the boiling point elevation would be ΔT_b = 2 × 0.512 °C·kg/mol × 1 mol/kg = 1.024°C

Freezing point depression is the process in which adding a solute to a solvent decreases the freezing point of the solvent.

The equation for freezing point depression is:

ΔT_f = i × K_f × m

Where:

• ΔT_f = freezing point depression
• i = van 't Hoff factor (number of particles the solute dissociates into)
• K_f = cryoscopic constant (solvent dependent)
• m = molality of the solution (moles solute per kg solvent)

Example: For a 1 molal solution of NaCl in water (i ≈ 2), the freezing point depression would be ΔT_f = 2 × 1.86 °C·kg/mol × 1 mol/kg = 3.72°C

Vapor pressure lowering is the reduction of the vapor pressure of a pure solvent when a non-volatile solute is added.

The equation for vapor pressure lowering (Raoult's Law) is:

P_solution = X_solvent × P°_solvent

Where:

• P_solution = vapor pressure of the solution
• X_solvent = mole fraction of the solvent
• P°_solvent = vapor pressure of the pure solvent

The mole fraction of the solvent is calculated as:

X_solvent = n_solvent / (n_solvent + n_solute)

Example: For a solution with 55.5 moles of water and 1 mole of solute, X_solvent = 55.5 / (55.5 + 1) ≈ 0.982. If P°_water = 23.8 mmHg, then P_solution = 0.982 × 23.8 ≈ 23.4 mmHg

Osmotic pressure is the minimum pressure which needs to be applied to a solution to prevent the inward flow of its pure solvent across a semipermeable membrane.

The equation for osmotic pressure is:

Π = i × M × R × T

Where:

• Π = osmotic pressure (atm)
• i = van 't Hoff factor (number of particles the solute dissociates into)
• M = molarity of the solution (mol/L)
• R = gas constant (0.0821 L·atm/mol·K)
• T = temperature (K)

Example: For a 0.1 M solution of NaCl (i ≈ 2) at 298 K, the osmotic pressure would be Π = 2 × 0.1 × 0.0821 × 298 ≈ 4.89 atm