α and ν → i

Number of ions

Ions per formula unit (ν = ν₊ + ν₋)

110

Degree of dissociation

Fraction of solute that dissociates (α)

0100

i = 1 + \alpha\,(\nu - 1)

Results

Van't Hoff factor

Strong electrolyte (α ≈ 100%)

i as a function of α

i = 1 + α·(ν − 1)

Fundamentals & Explanation

The Van't Hoff factor (i)

Colligative properties (freezing-point depression, boiling-point elevation, osmotic pressure) depend on the number of dissolved particles, not on their nature. The Van't Hoff factor $i$ counts how many effective particles the solute contributes per dissolved formula unit:

i = \frac{\text{actual particles in solution}}{\text{dissolved formula units}}

If a formula unit dissociates into $\nu$ ions and a fraction $\;\alpha$ actually does so (of each mole, $\alpha$ becomes $\;\alpha\nu$ ions and $1-\alpha$ stays undissociated), the particle count gives:

i = 1 + \alpha\,(\nu - 1) \qquad \Rightarrow \qquad \alpha = \frac{i - 1}{\nu - 1}

Non-electrolyte ( $\nu = 1$ ): $i = 1$ .
Weak electrolyte ( $0 < \alpha < 1$ ): $1 < i < \nu$ .
Strong electrolyte ( $\alpha = 1$ ): $i = \nu$ .

Why the measured i falls short of the ideal

In practice the measured $i$ is almost always below the ideal value $\;\nu$ . There are two physical reasons, belonging to different solute types:

1 · Incomplete dissociation

In a weak electrolyte not every unit splits into ions: part stays as molecules. The extent is measured by the degree of dissociation $\alpha < 1$ , tied to the solute's equilibrium.

2 · Ionic non-ideality

In a strong electrolyte dissociation is complete ( $\alpha = 1$ ), yet the ions do not act independently: electrostatic attractions between them lower their effective activity. This is described by the osmotic coefficient $\varphi$ .

They are independent causes: one dominates in weak electrolytes, the other in strong ones.

Ionic activity: the Debye-Hückel law

Ionic non-ideality grows with the ionic strength $I$ , a measure of the charge concentration in solution:

I = \tfrac12\sum_i m_i z_i^{2} = \tfrac12\,(\nu_+ z_+^2 + \nu_- z_-^2)\,m

Two dimensionless coefficients describe the departure from ideality:

The mean activity coefficient $\gamma_\pm$ corrects ionic equilibria and electrochemical potentials. The Debye-Hückel limiting law (DHLL) predicts it for very dilute solutions ( $I \lesssim 0.01\,\text{mol/kg}$ ), and the Davies equation extends it to $I \approx 0.5\,\text{mol/kg}$ :

\log_{10}\gamma_\pm = -A\,|z_+z_-|\sqrt{I} \quad (\text{DHLL})

\log_{10}\gamma_\pm = -A\,|z_+z_-|\left(\tfrac{\sqrt{I}}{1+\sqrt{I}} - 0.3\,I\right) \quad (\text{Davies})

The osmotic coefficient $\varphi$ is the one that corrects the colligative properties: the effective Van't Hoff factor of a strong electrolyte is $i_{\text{eff}} = \nu\,\varphi$ . The formulas for $\varphi$ are:

\varphi = 1 - \tfrac{\ln 10\,\cdot A}{3}\,|z_+z_-|\sqrt{I} \quad (\text{DHLL})

\varphi = 1 - A \ln 10 \, |z_+z_-| \left( \tfrac{\sqrt{I}}{1+\sqrt{I}} - 1 + \tfrac{2}{\sqrt{I}} - \tfrac{2 \ln(1+\sqrt{I})}{I} - 0.15 \, I \right) \quad (\text{Davies})

(The Davies equation for $\varphi$ is an extended version derived via Gibbs-Duhem for higher concentrations).

$\gamma_\pm$ and $\varphi$ are related but not the same: at the limiting law $(1-\varphi) = \tfrac13\,(-\ln\gamma_\pm)$ . That is why colligative properties are corrected with $\varphi$ , not $\gamma_\pm$ . The constant $A = 0.5085$ is for water at 25 °C; it varies with temperature and solvent through the dielectric constant, though in the dilute regime the effect is small.

Where does α come from?

The degree of dissociation is not known a priori; it is obtained from:

Weak-electrolyte equilibrium: from $K_a$ and concentration, $K_a = C\alpha^2/(1-\alpha)$ .
Colligative measurement: $\Delta T_f,\ \Delta T_b\ \text{or}\ \Pi \rightarrow i_{\text{exp}} \rightarrow \alpha$ .
Conductivity (Arrhenius' method): $\alpha = \Lambda_m / \Lambda_m^{\circ}$ .
Tabulated or given in the problem.

The "apparent" α

An $i$ obtained from a measurement lumps incomplete dissociation and ionic non-ideality together. Solving for $\alpha$ from it attributes the whole deviation to dissociation: the result is an apparent $\;\alpha$ , not the pure thermodynamic degree of dissociation.

Conditions

A = 0.5085 (water, 25 °C)

R = 0.08206 L·atm/(K·mol)

Theoretical i (= ν)

NaCl, KCl: 2

CaCl₂, Na₂SO₄: 3

K₂SO₄: 3 · AlCl₃: 4

MgSO₄: 2 (|z₊z₋| = 4)

Academic References

[1] Atkins, P. & de Paula, J. (2018). Atkins' Physical Chemistry (11th ed.). Oxford University Press. (Electrolyte solutions, Debye-Hückel.)
[2] Chang, R. & Goldsby, K. A. (2015). Chemistry (12th ed.). McGraw-Hill. (Colligative properties and the Van't Hoff factor.)
[3] Robinson, R. A. & Stokes, R. H. (2002). Electrolyte Solutions (2nd ed.). Dover. ( $\varphi$ , $\gamma_\pm$ , limiting laws.)
[4] CRC Handbook of Chemistry and Physics (104th ed.). CRC Press. (Activity and osmotic coefficients.)

Ionic Dissociation & Van't Hoff Factor

Results

i as a function of α

Fundamentals & Explanation

The Van't Hoff factor (i)

Why the measured i falls short of the ideal

Ionic activity: the Debye-Hückel law

Where does α come from?

Conditions

Theoretical i (= ν)

Academic References

The Van't Hoff factor (i)

Why the measured i falls short of the ideal

Ionic activity: the Debye-Hückel law

Where does α come from?

Conditions

Theoretical i (= ν)

Academic References