Degree of Dissociation ($\alpha$)
The fraction of the original substance that has dissociated into its constituent ions or smaller molecules. It tells you how much of a substance actually breaks apart.
Intuition
- Strong acids/bases: $\alpha \approx 1$ — they dissociate completely. Every molecule breaks apart.
- Weak acids/bases: $\alpha \ll 1$ — only a tiny fraction dissociates. Most molecules stay intact.
- Gas-phase dissociation (e.g. N$_2$O$_4$ $\rightleftharpoons$ 2NO$_2$): $\alpha$ depends on pressure and temperature. Dilution = more dissociation.
The core insight: $\alpha$ is not a fundamental constant. It changes with concentration. $K_a$ and $K_b$ are the constants — $\alpha$ is just a derived number that tells you where you stand in a given solution.
Definition
Weak acid: $$\alpha = \frac{[\text{H}3\text{O}^+]{\text{eq}}}{[\text{HA}]_0} \qquad %\alpha = \alpha \times 100%$$
Weak base: $$\alpha = \frac{[\text{OH}^-]_{\text{eq}}}{[\text{B}]_0} \qquad %\alpha = \alpha \times 100%$$
Gas-phase (general): $$\alpha = \frac{\text{moles dissociated}}{\text{initial moles}}$$
The Central Relationship
When $%\alpha < 10%$ (small dissociation assumption holds):
$$\alpha = \sqrt{\frac{K_a}{c}} \quad \text{or} \quad \alpha = \sqrt{\frac{K_b}{c}}$$
This is an inverse square root relationship. Halve the concentration → $\alpha$ increases by $\sqrt{2}$ ($\sim 1.4\times$). Diluting a weak acid makes it dissociate more, but the total [H⁺] still drops.
Why "Small $\alpha$" Matters
If the assumption $%\alpha < 10%$ is true, you can avoid solving a quadratic: $$K_a = \frac{x^2}{c - x} \approx \frac{x^2}{c}$$
If $%\alpha \ge 10%$, the approximation fails and you must use the full quadratic: $$x^2 + K_a x - K_a c = 0$$
Gas-Phase Dissociation
For reactions like N$_2$O$_4$ $\rightleftharpoons$ 2NO$_2$, degree of dissociation relates to $K_p$ and total pressure:
$$K_p = \frac{4\alpha^2}{1-\alpha^2} \cdot P_{\text{total}} \quad \text{(for A $\rightleftharpoons$ 2B)}$$
Derivation from ICE: If 1 mole of A starts with degree of dissociation $\alpha$:
- Moles at equilibrium: A = $1-\alpha$, B = $2\alpha$
- Total moles = $1+\alpha$
- Mole fractions: $X_A = \frac{1-\alpha}{1+\alpha}$, $X_B = \frac{2\alpha}{1+\alpha}$
- Partial pressures: $P_A = X_A P_{\text{total}}$, $P_B = X_B P_{\text{total}}$
From Conductivity (Electrolytic Dissociation)
For weak electrolytes (Arrhenius theory): $$\alpha = \frac{\Lambda_m}{\Lambda_m^\circ}$$
Where $\Lambda_m$ is molar conductivity at a given concentration and $\Lambda_m^\circ$ is limiting molar conductivity (at infinite dilution).
From Colligative Properties (van't Hoff Factor)
The van't Hoff factor $i$ relates the observed colligative effect to the expected effect for a non-electrolyte:
$$i = \frac{\text{observed colligative property}}{\text{expected for non-electrolyte}}$$
For dissociation: $$\alpha = \frac{i - 1}{n - 1}$$
Where $n$ = number of ions produced per formula unit (e.g. $n = 2$ for NaCl, $n = 3$ for CaCl$_2$).
Factors That Affect $\alpha$
| Factor | Effect | Why |
|---|---|---|
| Dilution | $\alpha \uparrow$ | Equilibrium shifts to produce more ions (Le Chatelier) |
| Temperature | $\alpha \uparrow$ (usually) | Dissociation is generally endothermic |
| Common ion | $\alpha \downarrow$ | Added product ions push equilibrium backward |
| Pressure (gases) | $\alpha \uparrow$ at lower $P$ | More volume available for more gas molecules |
Quick Comparison
| Context | Formula for $\alpha$ | Notes |
|---|---|---|
| Weak acid/base (small $\alpha$) | $\displaystyle \alpha = \sqrt{\frac{K_a}{c}}$ | Check $%\alpha < 10%$ |
| Weak acid/base (general) | Solve $K_a = \frac{x^2}{c-x}$ then $\alpha = x/c$ | Quadratic |
| Gas-phase A $\rightleftharpoons$ 2B | $\displaystyle K_p = \frac{4\alpha^2}{1-\alpha^2}P$ | Relates $\alpha$ to $P$ and $K_p$ |
| Conductivity | $\displaystyle \alpha = \frac{\Lambda_m}{\Lambda_m^\circ}$ | Electrolytic |
| van't Hoff factor | $\displaystyle \alpha = \frac{i-1}{n-1}$ | Colligative properties |