Colligative Properties & Finding Molar Mass 🌟

When you pop a solute into a solvent, you change more than just the taste—you tweak four “together-bound” (colligative) properties that depend only on how many particles are present, not what they are. These are:

• Relative lowering of vapour pressure
• Elevation of boiling point
• Depression of freezing point
• Osmotic pressure

Because each effect scales with particle count, we can use them to work backward and discover the solute’s molar mass. Let’s tour each property with friendly explanations, must-know equations, and exam-ready tips!

1. Relative Lowering of Vapour Pressure 🌬️

Adding a non-volatile solute “dilutes” solvent molecules at the surface, so the escaping tendency—and hence vapour pressure—drops. The key relation is:

\[ \frac{p_1^{0}-p_1}{p_1^{0}} = x_2 \] :contentReference[oaicite:0]{index=0}

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

\[ \frac{p_1^{0}-p_1}{p_1^{0}} \approx \frac{w_2 M_1}{w_1 M_2} \] :contentReference[oaicite:1]{index=1}

If you know masses \(w_1, w_2\) and solvent molar mass \(M_1\), rearrange to find the solute’s molar mass \(M_2\). Easy peasy! 😊

2. Elevation of Boiling Point ☕

A solution must get hotter than the pure solvent before its vapour pressure reaches atmospheric pressure. The rise is:

\[ \Delta T_b = K_b\,m \] :contentReference[oaicite:2]{index=2}

where \(m\) is molality and \(K_b\) is the ebullioscopic (boiling-point-elevation) constant. For a weighed sample:

\[ M_2 = \frac{K_b\,w_2\,1000}{\Delta T_b\,w_1} \] :contentReference[oaicite:3]{index=3}

Example highlight: Dissolving 1.80 g solute in 90 g benzene lifts \(T_b\) by \(0.88\text{ K}\); plugging the numbers gives \(M_2≈58\text{ g mol}^{-1}\). :contentReference[oaicite:4]{index=4}

3. Depression of Freezing Point ❄️

Solute lowers the temperature where solid and liquid solvent coexist. The drop obeys:

\[ \Delta T_f = K_f\,m \] :contentReference[oaicite:5]{index=5}

giving an analogous molar-mass formula:

\[ M_2 = \frac{K_f\,w_2\,1000}{\Delta T_f\,w_1} \] :contentReference[oaicite:6]{index=6}

Tip: Water’s \(K_f\) is \(1.86\ \text{K kg mol}^{-1}\); benzene’s is \(5.12\). Memorise these two and you’ll ace most questions! :contentReference[oaicite:7]{index=7}

4. Osmosis and Osmotic Pressure 💧

Through a semipermeable membrane, solvent sneaks from dilute to concentrated until balanced. The opposing pressure needed to stop the flow is:

\[ \Pi = C R T = \frac{n_2}{V} R T \] :contentReference[oaicite:8]{index=8}

Hence, for a weighed solute:

\[ M_2 = \frac{w_2 R T}{\Pi V} \] :contentReference[oaicite:9]{index=9}

Why it rocks: Even whisper-thin concentrations give sizeable \(\Pi\), so this method shines for proteins and polymers that dislike heat. Kudos, osmosis! 🌱

5. Reverse Osmosis & Water Purification 🚰

Push harder than \(\Pi\) and the solvent flows backward—leaving salts behind. That’s reverse osmosis, the heart of many seawater desalinators using cellulose-acetate membranes. Pure life-saving water! :contentReference[oaicite:10]{index=10}

6. Abnormal Molar Masses & the van’t Hoff Factor ⚡

Electrolytes split into multiple ions, effectively increasing particle count and boosting colligative effects. We introduce \(i\) (van’t Hoff factor):

\[ i = \frac{\text{observed colligative effect}}{\text{calculated (non-electrolyte) effect}} \]

For KCl, ideal dissociation gives \(i≈2\) (two ions), so the expected boiling-point rise doubles—an elegant clue that molar mass derived without \(i\) would look “abnormally” low. :contentReference[oaicite:11]{index=11}

Useful Constants (K_b & K_f) 🔢

See full list in Table 1.3 for extra solvents. :contentReference[oaicite:12]{index=12}

High-Yield NEET Nuggets 🎯

• Remember Raoult’s law form: \(\Delta p/p^0 = x_2\). Quick-fire calculations start here. :contentReference[oaicite:13]{index=13}
• Water’s magic numbers: \(K_b=0.52\), \(K_f=1.86\) → perfect for one-step estimates. :contentReference[oaicite:14]{index=14}
• For molar-mass questions, always write the generic fraction formula first, then substitute—saves marks and time.
• Osmotic pressure excels for macromolecules; expect at least one protein/polysaccharide problem. :contentReference[oaicite:15]{index=15}
• Check dissociation with the van’t Hoff factor so you don’t fall for “abnormal” molar-mass traps! :contentReference[oaicite:16]{index=16}

Keep these formulas on your fingertips and practice a few numericals—colligative questions will feel like a breeze. Happy studying! 😄