Chemistry 351

The van't Hoff Factor: Counting Solute Particles

The colligative properties of solutions depend on the number of dissolved particles rather than on the chemical identity of those particles. For example, freezing-point depression, boiling-point elevation, vapor-pressure lowering, and osmotic pressure all become larger as the number of dissolved particles increases.

For nonelectrolytes such as glucose, each formula unit dissolved in solution remains as a single particle:

\[ \text{C}_6\text{H}_{12}\text{O}_6(s) \rightarrow \text{C}_6\text{H}_{12}\text{O}_6(aq) \]

One mole of glucose therefore produces approximately one mole of dissolved particles.

Ionic compounds behave differently. When many salts dissolve, they dissociate into ions:

\[ \text{NaCl}(s) \rightarrow \text{Na}^+(aq) + \text{Cl}^-(aq) \]

One mole of NaCl can therefore produce approximately two moles of dissolved particles.

To account for this effect, we introduce the van't Hoff factor, denoted by \(i\). The van't Hoff factor represents the effective number of dissolved particles produced by each formula unit of solute.

\[ i = \frac{\text{moles of dissolved particles}} {\text{moles of dissolved solute}} \]

For an ideal solution in which dissociation is complete,

\[ \text{NaCl} \rightarrow \text{Na}^+ + \text{Cl}^- \qquad i=2 \] \[ \text{CaCl}_2 \rightarrow \text{Ca}^{2+} + 2\text{Cl}^- \qquad i=3 \]

Real solutions, however, are not perfectly ideal. Oppositely charged ions attract one another, causing the observed number of independent particles to be somewhat smaller than the ideal prediction.

Theoretical and Experimental van't Hoff Factors

The table below shows several common solutes dissolved in water at approximately 25 °C. The theoretical value assumes complete dissociation, while the experimental value reflects the effects of ion pairing and other nonideal interactions.

Solute	Dissociation	Theoretical \(i\)	Experimental \(i\)
Glucose	No dissociation	1.00	1.00
NaCl	\(\text{Na}^+ + \text{Cl}^-\)	2.00	1.9
KCl	\(\text{K}^+ + \text{Cl}^-\)	2.00	1.9
MgSO4	\(\text{Mg}^{2+} + \text{SO}_4^{2-}\)	2.00	1.3
CaCl2	\(\text{Ca}^{2+} + 2\text{Cl}^-\)	3.00	2.7
AlCl3	\(\text{Al}^{3+} + 3\text{Cl}^-\)	4.00	3.4

The deviation between the theoretical and experimental values becomes larger as ionic charges increase because stronger electrostatic attractions make the ions less independent.

Big picture: Colligative properties depend on the number of dissolved particles. The van't Hoff factor provides a simple correction that accounts for the dissociation of electrolytes and explains why ionic compounds often produce larger freezing-point depressions, boiling-point elevations, and osmotic pressures than nonelectrolytes at the same concentration.

Practice

Key points (one glance)

• Colligative properties depend on the number of dissolved particles rather than on the chemical identity of those particles.
• The van't Hoff factor, \(i\), measures the effective number of dissolved particles produced by each formula unit of solute: \[ i= \frac{\text{moles of dissolved particles}} {\text{moles of dissolved solute}} \]
• Molecular compounds such as glucose, sucrose, and ethanol generally do not dissociate in water and therefore have \[ i \approx 1 \]
• Ionic compounds dissociate into ions, producing larger values of \(i\). For example: \[ \text{NaCl} \rightarrow \text{Na}^+ + \text{Cl}^- \qquad i=2 \] \[ \text{CaCl}_2 \rightarrow \text{Ca}^{2+}+2\text{Cl}^- \qquad i=3 \]
• The theoretical van't Hoff factor is obtained by counting the number of ions produced by complete dissociation.
• Experimental values are usually smaller than theoretical values because oppositely charged ions attract one another and do not behave as completely independent particles.
• The van't Hoff factor is commonly incorporated into colligative-property equations: \[ \Delta T_f=iK_fm \] \[ \Delta T_b=iK_bm \] \[ \pi=iMRT \]

Big picture: The van't Hoff factor provides a simple way to account for the dissociation of electrolytes in solution. By correcting for the actual number of dissolved particles, it allows the same colligative-property equations to be applied to both molecular compounds and ionic compounds.