Acid-base equilibria

Acids and bases are all about protons moving from one particle to another. This topic turns that idea into numbers: the pH scale, the ionic product of water, the acid dissociation constant K_a, the shapes of titration curves, and the buffers that keep the pH of your blood almost perfectly steady.

Brønsted–Lowry acids and bases

• A Brønsted–Lowry acid is a proton (H⁺) donor.
• A Brønsted–Lowry base is a proton acceptor. It needs a lone pair of electrons to bond to the proton.

An acid can only give away a proton if something accepts it, so acid–base reactions always involve two pairs. When an acid loses its proton, what is left can accept a proton back: it is the acid's conjugate base. Likewise a base that gains a proton becomes its conjugate acid. A conjugate pair differs by exactly one H⁺.

Water acts as a base towards HCl but as an acid towards NH₃: a substance that can do both is called amphoteric. In the last example nitrous acid is forced to act as a base, because nitric acid is the stronger acid; when two acids are mixed, the one with the larger K_a donates the proton.

pH and strong acids

\[ \mathrm{pH} = -\log_{10}[\mathrm{H^+}] \qquad [\mathrm{H^+}] = 10^{-\mathrm{pH}} \]

The pH scale is logarithmic so that hydrogen-ion concentrations ranging over many powers of ten fit onto a scale of roughly 0 to 14. Each fall of one pH unit means ten times more H⁺. Give pH values to two decimal places.

A strong acid is fully dissociated in water, so for a monoprotic strong acid such as HCl or HNO₃, [H⁺] equals the concentration of the acid.

The ionic product of water, K_w

Water itself ionises very slightly: H₂O(l) ⇌ H⁺(aq) + OH⁻(aq). So little water ionises that [H₂O] is effectively constant, and it is folded into the constant to give the ionic product of water:

\[ K_w = [\mathrm{H^+}][\mathrm{OH^-}] = 1.00 \times 10^{-14}\ \mathrm{mol^2\,dm^{-6}}\ \text{at 298 K} \qquad \mathrm{p}K_w = -\log K_w = 14.00 \]

K_w applies to every aqueous solution, so if you know [OH⁻] you can find [H⁺], and the other way round.

Neutral water is not always pH 7

A solution is neutral when [H⁺] = [OH⁻]. At 298 K that gives [H⁺] = √(1.00 × 10⁻¹⁴) = 1.00 × 10⁻⁷ mol dm⁻³ and pH 7.00. The ionisation of water is endothermic, so warming it increases K_w. At 40 °C, K_w = 2.92 × 10⁻¹⁴ mol² dm⁻⁶, so [H⁺] = √(2.92 × 10⁻¹⁴) = 1.71 × 10⁻⁷ mol dm⁻³ and pH = 6.77. That water is still neutral, because [H⁺] and [OH⁻] are still equal; neutral simply has a lower pH at higher temperatures.

pH of strong bases

Weak acids and K_a

A weak acid is only slightly dissociated: the solution is an equilibrium mixture that is mostly undissociated HA. Its strength is measured by the acid dissociation constant, K_a, which is simply K_c for the dissociation:

\[ \mathrm{HA(aq)} \rightleftharpoons \mathrm{H^+(aq)} + \mathrm{A^-(aq)} \qquad K_a = \frac{[\mathrm{H^+}][\mathrm{A^-}]}{[\mathrm{HA}]} \qquad \mathrm{p}K_a = -\log K_a \]

• The units of K_a are mol dm⁻³.
• The larger K_a (the smaller pK_a), the further the equilibrium lies to the right and the stronger the acid. Ethanoic acid has K_a = 1.7 × 10⁻⁵ mol dm⁻³, pK_a = 4.77.
• Strong and weak are not the same as concentrated and dilute. Strength is about how much the acid dissociates; concentration is about how much acid is in each dm³.

Calculating the pH of a weak acid

Two approximations make the calculation simple:

1. [H⁺] = [A⁻], because each HA that dissociates gives one of each. (This ignores the tiny amount of H⁺ from the water.)
2. [HA] at equilibrium ≈ the concentration of acid you started with, because so little of it dissociates.

Then \( K_a = \dfrac{[\mathrm{H^+}]^2}{[\mathrm{HA}]_{\text{initial}}} \), so \( [\mathrm{H^+}] = \sqrt{K_a \times [\mathrm{HA}]_{\text{initial}}} \). The second approximation breaks down for acids that are fairly strong (large K_a) or very dilute, because then a significant fraction of the acid dissociates.

Diluting acids

Diluting a strong acid ten times divides [H⁺] by ten, so the pH rises by exactly 1. Diluting a weak acid ten times raises its pH by only about 0.5: 0.0500 mol dm⁻³ ethanoic acid has pH 3.04, but 0.00500 mol dm⁻³ has pH 3.54. Adding water shifts the dissociation equilibrium to the right, so a larger fraction of the acid dissociates and partly makes up the lost H⁺. Mathematically, [H⁺] is proportional to the square root of the concentration.

Titration curves

A titration curve plots pH against the volume of one solution added to the other. To record one, put a known volume of acid in a beaker with a calibrated pH meter, add the alkali from a burette a little at a time, stirring and recording the pH after each addition, with smaller additions near the end point. The equivalence point is where the acid and base have reacted exactly in the ratio of the equation; on the graph it is the middle of the steep, nearly vertical part.

The pH at equivalence is not always 7 because the salt formed can itself react with water. Sodium ethanoate contains ethanoate ions, the conjugate base of a weak acid, which take protons from water and leave OH⁻: the solution is alkaline. Ammonium chloride contains NH₄⁺, a weak acid, so its solution is acidic.

Half-neutralisation: pH = pK_a

On the curve for a weak acid with a strong base, the pH rises quickly at first and then flattens into a buffer region. Halfway to the equivalence point, exactly half of the acid has been turned into its salt, so [HA] = [A⁻]. These cancel in the K_a expression, leaving K_a = [H⁺], so pH = pK_a. This is the basis of the core practical above.

Choosing an indicator

An indicator is itself a weak acid, HIn, whose undissociated form and conjugate base have different colours: HIn ⇌ H⁺ + In⁻. Adding acid pushes the equilibrium left, giving the colour of HIn; adding alkali removes H⁺ and pulls it right, giving the colour of In⁻. The end point, where the colour changes, is when [HIn] = [In⁻]; it happens over a range of about 2 pH units centred on the indicator's pK_in.

A suitable indicator changes colour completely within the steep part of the titration curve, so that one drop of titrant takes it all the way through its colour change at the equivalence point.

For a weak acid with a weak base there is no steep section, so no indicator gives a sharp end point; use a pH meter instead.

Diprotic acids such as ethanedioic acid, H₂C₂O₄, lose their two protons one at a time, so their curve with sodium hydroxide has two steep sections and two equivalence points. In the same way, adding acid to sodium carbonate gives two steps: CO₃²⁻ + H⁺ → HCO₃⁻, then HCO₃⁻ + H⁺ → H₂O + CO₂.

Buffer solutions

A buffer is a solution whose pH changes very little when small amounts of acid or alkali are added, or when it is diluted.

• Acidic buffer: a weak acid and a salt of that acid, for example ethanoic acid and sodium ethanoate. It holds a pH below 7.
• Basic buffer: a weak base and a salt of that base, for example ammonia and ammonium chloride. It holds a pH above 7.

How an acidic buffer works

The buffer contains large reservoirs of both the weak acid, CH₃COOH, and its conjugate base, CH₃COO⁻, which comes almost entirely from the salt: CH₃COOH ⇌ CH₃COO⁻ + H⁺.

• Add a little acid: the added H⁺ combines with the large supply of ethanoate ions, CH₃COO⁻ + H⁺ → CH₃COOH, so the equilibrium shifts left and almost all the added H⁺ is removed.
• Add a little alkali: the OH⁻ reacts with H⁺ to make water, and the equilibrium shifts right as more ethanoic acid dissociates to replace it. The OH⁻ is also removed directly: CH₃COOH + OH⁻ → CH₃COO⁻ + H₂O.
• Either way the ratio [CH₃COOH] / [CH₃COO⁻] changes only slightly, because both reservoirs are large, so [H⁺] and the pH barely change.

Calculating the pH of a buffer

Rearranging K_a: \( [\mathrm{H^+}] = K_a \times \dfrac{[\mathrm{HA}]}{[\mathrm{A^-}]} \). Take [HA] as the acid added and [A⁻] as the salt added; the tiny amounts that dissociate or react with water make no noticeable difference. Because acid and salt share the same final volume, you can use their moles directly in the ratio.

Buffers in blood

Blood plasma must stay between pH 7.35 and 7.45. The main buffer is carbonic acid and hydrogencarbonate ions: H₂CO₃ ⇌ H⁺ + HCO₃⁻. Extra acid from metabolism is taken up by HCO₃⁻, shifting the equilibrium left; extra alkali removes H⁺, and more H₂CO₃ dissociates to replace it. The carbonic acid level is itself controlled by breathing out carbon dioxide (H₂CO₃ ⇌ H₂O + CO₂). Buffers are also used in foods, shampoos and medicines to keep the pH where the product works best.

Enthalpy change of neutralisation

The standard enthalpy change of neutralisation is the enthalpy change when an acid and an alkali react under standard conditions to form one mole of water. For any strong acid with any strong alkali it is almost the same, about −57 kJ mol⁻¹, because both are fully ionised and the only reaction is H⁺(aq) + OH⁻(aq) → H₂O(l). With a weak acid (or weak base) it is slightly less exothermic, for example about −56 kJ mol⁻¹ for ethanoic acid, because some energy is used up dissociating the acid as the reaction proceeds.

Quick checks

Interactive diagrams