7.4 Electrolytes, Interionic Attractions & the van 't Hoff Factor

7.4 Electrolytes, Interionic Attractions & the van 't Hoff Factor

Quick Summary: Solutes behave differently in solution depending on whether they dissociate into mobile ions. Strong electrolytes dissociate completely, weak electrolytes ionize partially, and nonelectrolytes dissolve as neutral molecules. The van 't Hoff factor ($i$) quantifies the ratio of actual particles in solution to formula units dissolved. In real solutions, electrostatic attractions between oppositely charged ions cause ion pairing, reducing the effective particle count ($i_{\text{measured}} < i_{\text{ideal}}$). These deviations intensify with higher ionic charges and higher concentrations, converging toward ideal integer values only at infinite dilution.

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1. Classification of Solutes in Aqueous Media

Solutes are classified by their ability to produce mobile charge carriers (ions) that conduct electric current:

Strong Electrolytes

Substances that dissociate virtually 100% into mobile hydrated ions ($\alpha \approx 1.0$), conducting current strongly:

• Strong Acids: $\text{HCl}$, $\text{HBr}$, $\text{HI}$, $\text{HNO}_3$, $\text{HClO}_4$, $\text{H}_2\text{SO}_4$ (first dissociation).
• Strong Soluble Bases: Group 1 hydroxides ($\text{NaOH}$, $\text{KOH}$) and heavy Group 2 hydroxides ($\text{Ba(OH)}_2$).
• Soluble Salts: All common sodium, potassium, ammonium, and nitrate salts ($\text{NaCl}$, $\text{KNO}_3$, $\text{CaCl}_2$).

Weak Electrolytes

Substances that ionize only partially in water, establishing dynamic chemical equilibria (typically only a few percent ionized at ordinary concentrations), conducting current weakly:

• Weak Acids: $\text{CH}_3\text{COOH}$ (acetic acid), $\text{HF}$, $\text{HNO}_2$.
• Weak Bases: $\text{NH}_3$ (ammonia) and organic amines ($\text{CH}_3\text{NH}_2$).
• Equilibrium: $\text{CH}_3\text{COOH}(aq) + \text{H}_2\text{O}(l) \rightleftharpoons \text{H}_3\text{O}^+(aq) + \text{CH}_3\text{COO}^-(aq)$.

Nonelectrolytes

Molecular substances that dissolve as intact neutral molecules without generating ions ($\alpha = 0$), yielding nonconductive solutions:

• Examples: Glucose ($\text{C}_6\text{H}_{12}\text{O}_6$), sucrose ($\text{C}_{12}\text{H}_{22}\text{O}_{11}$), urea ($\text{CO(NH}_2)_2$), ethanol ($\text{C}_2\text{H}_5\text{OH}$).

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2. The van 't Hoff Factor ($i$): Ideality vs Reality

Colligative properties depend on the total concentration of particles in solution. The van 't Hoff factor ($i$) is defined as: $i = \frac{\text{actual moles of particles in solution}}{\text{moles of solute formula units dissolved}} = \frac{\text{measured colligative property}}{\text{expected property for nonelectrolyte of same molality}}$

Theoretical (Ideal) Values ($i_{\text{ideal}}$)

Assuming complete dissociation and zero interaction between ions, $i_{\text{ideal}}$ equals the integer number of ions produced per formula unit:

• Nonelectrolytes (glucose, urea): $i_{\text{ideal}} = 1$
• $\text{NaCl}(s) \to \text{Na}^+ + \text{Cl}^- \implies i_{\text{ideal}} = 2$
• $\text{CaCl}_2(s) \to \text{Ca}^{2+} + 2\text{ Cl}^- \implies i_{\text{ideal}} = 3$
• $\text{FeCl}_3(s) \to \text{Fe}^{3+} + 3\text{ Cl}^- \implies i_{\text{ideal}} = 4$
• $\text{Al}_2(\text{SO}_4)_3(s) \to 2\text{ Al}^{3+} + 3\text{ SO}_4^{2-} \implies i_{\text{ideal}} = 5$

Weak Electrolyte Ionization Factor

For a solute dissociating into $n$ ions with degree of ionization $\alpha$: $i = 1 + (n - 1)\alpha$ For a $0.10\text{ m}$ weak acid ($n = 2$) that is $4.0\%$ ionized ($\alpha = 0.040$): $i = 1 + (2 - 1)(0.040) = 1.04$.

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3. Interionic Attractions & Debye-Hückel Theory

In real solutions, observed van 't Hoff factors fall below theoretical values ($i_{\text{measured}} < i_{\text{ideal}}$). Debye and Hückel demonstrated that ions are not distributed randomly; each cation is surrounded by an average cloud of anions, and vice versa. Electrostatic attractions govern interactions: $F_{\text{electrostatic}} \propto \frac{|q_+ \cdot q_-|}{r^2}$

Ion Pairing

Coulombic attraction causes oppositely charged ions to associate temporarily into ion pairs (e.g., $[\text{Mg}^{2+}\text{SO}_4^{2-}]^0$). An ion pair migrates as a single kinetic entity, reducing the effective count of independent particles and dampening colligative effects.

Factors Driving Non-Ideality

1. Ionic Charge Product ($|q_+ \cdot q_-|$): Multivalent ions experience substantially stronger coulombic attraction:
   • $\text{NaCl}$ ($+1 / -1$, product 1): at $0.10\text{ m}$, $i_{\text{measured}} = 1.87$ (6.5% below ideal).
   • $\text{MgSO}_4$ ($+2 / -2$, product 4): at $0.10\text{ m}$, $i_{\text{measured}} = 1.21$ (39.5% below ideal). The quadrupled charge product causes severe ion pairing.
2. Concentration: Higher concentration places ions closer together, magnifying electrostatic attractions and lowering $i$. At infinite dilution ($m \to 0$), interionic distances approach infinity, ion pairing vanishes, and $i_{\text{measured}} \to i_{\text{ideal}}$.

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4. Ideal vs Measured van 't Hoff Factors Summary

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5. Comparative Colligative Calculations & Ranking

Colligative shifts scale with the effective particle molality ($m_{\text{eff}} = i \cdot m$):

• Freezing Point: $\Delta T_f = i K_f m$. Greater $m_{\text{eff}}$ produces greater depression $\Delta T_f \implies$ lower freezing point.
• Boiling Point: $\Delta T_b = i K_b m$. Greater $m_{\text{eff}}$ produces greater elevation $\Delta T_b \implies$ higher boiling point.
• Vapor Pressure: $\Delta P = X_{\text{solute}} P^\circ \implies$ greater $m_{\text{eff}}$ lowers vapor pressure.
• Osmotic Pressure: $\Pi = i M R T \implies$ greater $m_{\text{eff}}$ raises osmotic pressure.

Worked Example: Freezing Point Ranking

Problem: Rank $0.050\text{ m}$ aqueous solutions of sucrose, $\text{NaCl}$, $\text{CaCl}_2$, and $\text{Al(NO}_3)_3$ in order of decreasing freezing point (highest freezing point first), assuming ideal dissociation.

Step 1: Identify ideal van 't Hoff factors

• Sucrose: nonelectrolyte $\implies i = 1$
• $\text{NaCl} \to \text{Na}^+ + \text{Cl}^- \implies i = 2$
• $\text{CaCl}_2 \to \text{Ca}^{2+} + 2\text{ Cl}^- \implies i = 3$
• $\text{Al(NO}_3)_3 \to \text{Al}^{3+} + 3\text{ NO}_3^- \implies i = 4$

Step 2: Calculate effective particle molalities and depressions ($\Delta T_f = i K_f m$, $K_f = 1.86^\circ\text{C/m}$)

• Sucrose: $i \cdot m = 0.050\text{ m} \implies \Delta T_f = 0.093^\circ\text{C} \implies T_f = -0.093^\circ\text{C}$
• $\text{NaCl}$: $i \cdot m = 0.100\text{ m} \implies \Delta T_f = 0.186^\circ\text{C} \implies T_f = -0.186^\circ\text{C}$
• $\text{CaCl}_2$: $i \cdot m = 0.150\text{ m} \implies \Delta T_f = 0.279^\circ\text{C} \implies T_f = -0.279^\circ\text{C}$
• $\text{Al(NO}_3)_3$: $i \cdot m = 0.200\text{ m} \implies \Delta T_f = 0.372^\circ\text{C} \implies T_f = -0.372^\circ\text{C}$

Step 3: Rank decreasing freezing point (highest to lowest) $\text{Sucrose } (-0.093^\circ\text{C}) > \text{NaCl } (-0.186^\circ\text{C}) > \text{CaCl}_2 (-0.279^\circ\text{C}) > \text{Al(NO}_3)_3 (-0.372^\circ\text{C})$