R3.1.10 Acid and base strength and constants (Higher Level Only)

Equilibrium Constants and Acid/Base Strength

You are sipping a glass of lemonade on a hot day.
The tangy taste comes from citric acid, a weak acid that partially dissociates in water.
But why does this tanginess feel different from the sharp burn of a strong acid like hydrochloric acid?

The answer lies in how much these acids dissociate and the equilibrium constants that quantify their strength.

Acid Ionization Constant ( $K_{a}$ )

When a weak acid dissolves in water, it establishes an equilibrium between the undissociated acid and its ions:

$H A (a q) ⇌ H^{+} (a q) + A^{-} (a q)$

The equilibrium constant for this reaction is called the acid ionization constant ( $K_{a}$ ):

$K_{a} = \frac{[H^{+}] [A^{-}]}{[H A]}$

[H⁺]: Concentration of hydrogen ions (or hydronium ions, $H_{3} O^{+}$ ).
[A⁻]: Concentration of the conjugate base.
[HA]: Concentration of the undissociated acid.

Hint

A larger $K_{a}$ indicates a stronger acid because more $H A$ dissociates, producing more $H^{+}$ ions.
Conversely, a smaller $K_{a}$ indicates a weaker acid.

Example

Acetic acid ( $C H_{3} C O O H$ ) 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:

$B O H (a q) ⇌ B^{+} (a q) + O H^{-} (a q)$

The base ionization constant ( $K_{b}$ ) is defined as:

$K_{b} = \frac{[O H^{-}] [B^{+}]}{[B O H]}$

[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 ( $N H_{3}$ ) 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.

$p K_{a}$ and $p K_{b}$ : A Logarithmic Perspective

Working with very small $K_{a}$ and $K_{b}$ values can be cumbersome. To simplify comparisons, we use their negative logarithms:

$p K_{a} = - \log_{10} (K_{a}) and p K_{b} = - \log_{10} (K_{b})$

Lower $p K_{a}$ : Indicates a stronger acid (larger $K_{a}$ ).
Lower $p K_{b}$ : Indicates a stronger base (larger $K_{b}$ ).

Analogy

Think of $K_{a}$ and $p K_{a}$ as two sides of the same coin. If $K_{a}$ is like measuring the strength of an acid in grams, $p K_{a}$ is like measuring it in milligrams—both describe the same property, but one scale makes small values easier to interpret.

Example

Acetic acid ( $K_{a} = 1.8 \times 10^{- 5}$ ) has $p K_{a} = - \log (1.8 \times 10^{- 5}) \approx 4.74$ .
Ammonia ( $K_{b} = 1.8 \times 10^{- 5}$ ) has $p K_{b} = - \log (1.8 \times 10^{- 5}) \approx 4.74$ .

Common Mistake

Many students confuse $K_{a}$ and $p K_{a}$ . Remember: as $K_{a}$ increases (stronger acid), $p K_{a}$ decreases.

Relationship Between $K_{a}$ , $K_{b}$ , and $K_{w}$

Acids and bases exist in conjugate pairs.

Example

Acetic acid ( $C H_{3} C O O H$ ) and acetate ( $C H_{3} C O O^{-}$ ) form a conjugate acid–base pair.

The strength of an acid is inversely related to the strength of its conjugate base.

This relationship is quantified by:

$K_{a} \cdot K_{b} = K_{w}$

Here, $K_{w}$ is the ionic product of water:

$K_{w} = [H^{+}] [O H^{-}] = 1.0 \times 10^{- 14} at 25°C .$

For conjugate acid–base pairs:
$p K_{a} + p K_{b} = 14 (at 25°C) .$

Example

For acetic acid ( $p K_{a} = 4.74$ ), its conjugate base (acetate) has $p K_{b} = 14 - 4.74 = 9.26$ .

Note

This relationship highlights the inverse strength of conjugates: the stronger the acid, the weaker its conjugate base.

Reflection

Self review

What does a $p K_{a}$ of 3.0 tell you about an acid's strength?
If the $K_{a}$ of an acid is $1.0 \times 10^{- 6}$ , what is its $p K_{a}$ ?
How are $K_{a}$ and $K_{b}$ related for a conjugate acid–base pair?

Theory of Knowledge

How does the logarithmic nature of $p K_{a}$ and $p K_{b}$ scales influence our perception of acid and base strength?
Are there other fields where logarithmic scales simplify comparisons?

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R3.1.10 Acid and base strength and constants (Higher Level Only)

Equilibrium Constants and Acid/Base Strength

Acid Ionization Constant (Ka)

Base Ionization Constant (Kb)

pKa and pKb: A Logarithmic Perspective

Relationship Between Ka, Kb, and Kw

Reflection

You've read 2/2 free chapters this week.

Introduction to Acid and Base Strength

Acid Ionization Constant ( $K_{a}$ )

Base Ionization Constant ( $K_{b}$ )

$p K_{a}$ and $p K_{b}$ : A Logarithmic Perspective

Relationship Between $K_{a}$ , $K_{b}$ , and $K_{w}$