Equilibrium Constants and Acid/Base Strength
Acid Ionization Constant ($K_a$)
- When a weak acid dissolves in water, it establishes an equilibrium between the undissociated acid and its ions: $$
HA(aq) \rightleftharpoons H^+(aq) + A^-(aq)
$$ - The equilibrium constant for this reaction is called the acid ionization constant ($K_a$): $$
K_a = \frac{[H^+][A^-]}{[HA]}
$$- [H⁺]: Concentration of hydrogen ions (or hydronium ions, $H_3O^+$).
- [A⁻]: Concentration of the conjugate base.
- [HA]: Concentration of the undissociated acid.
Hint
- A larger $K_a$ indicates a stronger acid because more $HA$ dissociates, producing more $H^+$ ions.
- Conversely, a smaller $K_a$ indicates a weaker acid.
Example
- Acetic acid ($CH_3COOH$) has a $K_a$ of $1.8 \times 10^{-5}$.
- This small value shows that acetic acid is a weak acid, as only a small fraction of its molecules dissociate in water.
- This is why vinegar, which contains acetic acid, has a sour but not overpowering taste.
Base Ionization Constant ($K_b$)
- For weak bases, the dissociation in water can be represented as: $$
BOH(aq) \rightleftharpoons B^+(aq) + OH^-(aq)
$$ - The base ionization constant ($K_b$) is defined as: $$
K_b = \frac{[OH^-][B^+]}{[BOH]}
$$- [OH⁻]: Concentration of hydroxide ions.
- [B⁺]: Concentration of the conjugate acid.
- [BOH]: Concentration of the undissociated base.
Hint
As with acids, a larger $K_b$ corresponds to a stronger base, while a smaller $K_b$ indicates a weaker base.
Example
- Ammonia ($NH_3$) has a $K_b$ of $1.8 \times 10^{-5}$.
- This indicates that ammonia is a weak base, as it only partially ionizes in water.
- This is why ammonia solutions are less caustic compared to strong bases like sodium hydroxide.
Tip
- You can estimate the strength of an acid or base by comparing its $K_a$ or $K_b$ to 1.
- Strong acids/bases typically have $K_a$ or $K_b > 1$, while weak acids/bases have values much smaller than 1.
$pK_a$ and $pK_b$: A Logarithmic Perspective
- Working with very small $K_a$ and $K_b$ values can be cumbersome.
- To simplify comparisons, we use their negative logarithms: $$
pK_a = -\log_{10}(K_a) \quad \text{and} \quad pK_b = -\log_{10}(K_b)
$$ - Lower $pK_a$: Indicates a stronger acid (larger $K_a$).
- Lower $pK_b$: Indicates a stronger base (larger $K_b$).
Example
- Acetic acid ($K_a = 1.8 \times 10^{-5}$) has $pK_a = -\log(1.8 \times 10^{-5}) \approx 4.74$.
- Ammonia ($K_b = 1.8 \times 10^{-5}$) has $pK_b = -\log(1.8 \times 10^{-5}) \approx 4.74$.
Common Mistake
- Many students confuse $K_a$ and $pK_a$.
- Remember: as $K_a$ increases (stronger acid), $pK_a$ decreases.
