🧪 Solutions & Colligative Properties

Solutions chemistry bridges molecular thermodynamic interactions with observable bulk colligative phenomena. Advanced examination questions test positive/negative deviations from Raoult's law, vapor-liquid equilibrium curves ($P-x-y$), abnormal molar masses, and degree of association/dissociation ($\alpha$).

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1. 🍶 Vapor Pressure & Raoult's Law

1.1 Ideal Binary Solutions

For two volatile miscible liquids $A$ and $B$:

$$P_A=P_A^{\circ }{x}_A,P_B=P_B^{\circ }{x}_B$$$${\mathbf{P}}_{\text{total}}\mathbf{=}{\mathbf{P}}_{\mathbf{A}}\mathbf{+}{\mathbf{P}}_{\mathbf{B}}\mathbf{=}{\mathbf{P}}_{\mathbf{A}}^{\circ }{\mathbf{x}}_{\mathbf{A}}\mathbf{+}{\mathbf{P}}_{\mathbf{B}}^{\circ }\mathbf{(}\mathbf{1}\mathbf{-}{\mathbf{x}}_{\mathbf{A}}\mathbf{)}\mathbf{=}{\mathbf{P}}_{\mathbf{B}}^{\circ }\mathbf{+}\mathbf{(}{\mathbf{P}}_{\mathbf{A}}^{\circ }\mathbf{-}{\mathbf{P}}_{\mathbf{B}}^{\circ }\mathbf{)}{\mathbf{x}}_{\mathbf{A}}$$

Composition of Vapor Phase in Equilibrium with Liquid:

$$y_A=\frac{P_A}{P_{\text{total}}}=\frac{P_A^{\circ }{x}_A}{P_{\text{total}}},y_B=\frac{P_B}{P_{\text{total}}}=\frac{P_B^{\circ }{x}_B}{P_{\text{total}}}$$$$\frac{\mathbf{1}}{{\mathbf{P}}_{\text{total}}}\mathbf{=}\frac{{\mathbf{y}}_{\mathbf{A}}}{{\mathbf{P}}_{\mathbf{A}}^{\circ }}\mathbf{+}\frac{{\mathbf{y}}_{\mathbf{B}}}{{\mathbf{P}}_{\mathbf{B}}^{\circ }}$$

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2. 📊 Non-Ideal Solutions & Azeotropes

Property	Ideal Solutions	Positive Deviation ($Delta{H}_{\text{mix}}>0$)	Negative Deviation ($Delta{H}_{\text{mix}}<0$)
Intermolecular Forces	$A-B=A-A=B-B$	$A-B<A-A,B-B$ (Weaker)	$A-B>A-A,B-B$ (Stronger, e.g. H-bonding)
Vapor Pressure	$P=P_A^{\circ }{x}_A+P_B^{\circ }{x}_B$	$P>P_{\text{ideal}}$	$P<P_{\text{ideal}}$
Enthalpy of Mixing	$\mathrm{\Delta }{H}_{\text{mix}}=0$	$\mathrm{\Delta }{H}_{\text{mix}}>0$ (Endothermic)	$\mathrm{\Delta }{H}_{\text{mix}}<0$ (Exothermic)
Volume of Mixing	$\mathrm{\Delta }{V}_{\text{mix}}=0$	$\mathrm{\Delta }{V}_{\text{mix}}>0$ (Expansion)	$\mathrm{\Delta }{V}_{\text{mix}}<0$ (Contraction)
Azeotropic Behavior	No azeotrope formed	Minimum Boiling Azeotrope (e.g. $95.6\mathrm{\%}$ Ethanol + Water)	Maximum Boiling Azeotrope (e.g. $68\mathrm{\%}HN{O}_3+32\mathrm{\%}{H}_{2}O$)
Classic Examples	Benzene + Toluene, $n$-Hexane + $n$-Heptane	Ethanol + Acetone, $C{S}_2+\text{Acetone}$	Chloroform + Acetone, $HCl+H_{2}O$

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3. 🎯 The 4 Colligative Properties

Colligative properties depend strictly on the number of solute particles in solution, not on their chemical identity.

3.1 Relative Lowering of Vapor Pressure (RLVP)

$$\frac{P^{\circ }-P}{P^{\circ }}=x_{\text{solute}}=\frac{n_2}{n_1+n_2}⟹\frac{P^{\circ }-P}{P}=\frac{n_2}{n_1}=\frac{w_2/M_2}{w_1/M_1}$$

3.2 Elevation of Boiling Point (Ebullioscopy)

$$\mathrm{\Delta }{T}_b=T_b-T_b^{\circ }=i\cdot K_b\cdot m$$$$K_b=\frac{R{T}_b^{\circ 2}{M}_{\text{solvent}}}{1000\mathrm{\Delta }{H}_{\text{vap}}}=\frac{R{T}_b^{\circ 2}}{1000L_v}(K_b(\text{water})=0.52\text{ K kg/mol})$$

3.3 Depression of Freezing Point (Cryoscopy)

$$\mathrm{\Delta }{T}_f=T_f^{\circ }-T_f=i\cdot K_f\cdot m$$$$K_f=\frac{R{T}_f^{\circ 2}{M}_{\text{solvent}}}{1000\mathrm{\Delta }{H}_{\text{fus}}}=\frac{R{T}_f^{\circ 2}}{1000L_f}(K_f(\text{water})=1.86\text{ K kg/mol})$$

3.4 Osmotic Pressure ($\mathrm{\Pi }$)

$$\mathrm{\Pi }=i\cdot C\cdot R\cdot T=i\left(\frac{n}{V}\right)RT$$

• Isotonic Solutions: ${\mathrm{\Pi }}_1={\mathrm{\Pi }}_2⟹i_1{C}_1=i_2{C}_2$ (at equal $T$).
• Reverse Osmosis: When applied external pressure $P_{\text{ext}}>\mathrm{\Pi }$, pure solvent flows from concentrated solution to pure solvent side through semi-permeable membrane.

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4. 🧮 Van 't Hoff Factor ($i$) & Degree of Association / Dissociation

$$i=\frac{\text{Observed Colligative Property}}{\text{Theoretical Colligative Property}}=\frac{\text{Theoretical Molar Mass}}{\text{Observed Molar Mass}}=\frac{\text{Total Moles After Dissociation/Association}}{\text{Initial Moles}}$$

1. For Dissociation ($A_n\to nA$):$$\alpha =\frac{i-1}{n-1}⟹\mathbf{i}\mathbf{=}\mathbf{1}\mathbf{+}\mathbf{(}\mathbf{n}\mathbf{-}\mathbf{1}\mathbf{)}\alpha$$Example: For $K_4[Fe(CN{)}_6]⟹n=5$. If $\alpha =80\mathrm{\%}=0.8⟹i=1+4(0.8)=4.2$.
2. For Association ($nA\to A_n$, e.g. Benzoic acid dimerizing in Benzene):$$\alpha =\frac{1-i}{1-1/n}=\frac{n(1-i)}{n-1}⟹\mathbf{i}\mathbf{=}\mathbf{1}\mathbf{-}(\mathbf{1}\mathbf{-}\frac{\mathbf{1}}{\mathbf{n}})\alpha$$For complete dimerization ($n=2,\alpha =1$) $⟹i=0.5$.