Starting Point: Experimental Observation

When a non-volatile solute is dissolved in a solvent, the boiling point rises. The rise is proportional to concentration. This led to the empirical law: $\Delta Tb$ = Kb × m.

Step 1: Express Molality in Terms of Masses

$m = \frac{n_{solute}}{w_{solvent}(\text{kg})} = \frac{w_2/M_2}{w_1/1000}$

where w_{2} = mass of solute (g), $M_{2}$ = molar mass of solute (g/mol), w_{1} = mass of solvent (g).

Step 2: Substitute into $\Delta Tb$ Formula

$\Delta T_b = K_b \times m = K_b \times \frac{w_2 \times 1000}{M_2 \times w_1}$

Step 3: Rearrange to Isolate $M_{2}$

$\Delta T_b \times M_2 \times w_1 = K_b \times w_2 \times 1000$

$M_2 = \frac{K_b \times w_2 \times 1000}{\Delta T_b \times w_1}$

Step 4: Logical Consequences

• If $M_{2}$ is large: Very few moles for the same mass → small m → small $\Delta Tb$. (Polymers/proteins give tiny $\Delta Tb$ — why osmometry is preferred.)
• If association occurs: Observed $\Delta Tb$ is LESS than expected → when we plug in smaller $\Delta Tb$ into formula → calculated $M_{2}$ comes out LARGER than true $M_{2}$.
• If dissociation occurs: More particles → larger $\Delta Tb$ → if we ignore i, calculated $M_{2}$ will be SMALLER than true $M_{2}$. (This is why apparent molar mass of NaCl from freezing point depression appears ~half the true value without including i.)

Generalisation

The SAME reasoning applies to cryoscopy (replacing Kb→Kf, $\Delta Tb$→$\Delta Tf$). For osmometry: $M_{2}$ = w_{2}RT/(π·V) is derived from π = CRT = (n_{2}/V)RT → n_{2} = πV/RT → $M_{2}$ = w_{2}/n_{2} = w_{2}RT/(πV).