Doc Brown's Advanced A Level Chemistry Revision Notes

Theoretical–Physical Advanced Level Chemistry – Equilibria – Chemical Equilibrium Revision Notes PART 6.2

6.2 Indicator theory of acid–base titrations, pH curves & pK_ind

What is the theory behind how an acid–alkali indicator works?

Why are some weak acids good indicators in acid–base titrations?

What is the pH range of an indicator?

How does the pH change throughout an weak/strong acid–soluble weak/strong base titration?

Chemical Equilibrium Notes Parts 5 & 6 Index

6.2 Theory of acid–base titration indicators, pH curves and pK_ind

selection of a suitable indicator and how pH changes when acids react with bases e.g. in a titration

• All of Equilibria part 5 "pH and weak–strong acids and bases in aqueous solution" should have been studied prior to tackling Part 6.

• 6.2.1 Acid–base titration indicators are quite often weak acids in which the unionised acid (lets call it HIn) and its 'de–protonated' form, or conjugate base, the anion (In^–), have different colours.

  • One form can be colourless e.g. phenolphthalein in acid–neutral solutions.

  • The equilibrium can be simply expressed as ....

  • HIn^–_{(aq, colour 1)} H⁺_(aq) + In^–_{(aq, colour 2)}

    • Applying Le Chatelier's equilibrium principle:

    • Addition of acid favours the formation of more HIn (colour 1)

      • HIn_(aq) H⁺_(aq) + In^–_(aq)

        • because an increase on the right of [H⁺] causes a shift to left increasing [HIn] to minimise 'enforced' rise in [H⁺].

    • Addition of alkali favours the formation of more I^– (colour 2):

      • HIn_(aq) H⁺_(aq) + In^–_(aq)

        • The increase in [OH^–] causes a shift to right because the reaction

        •  H⁺_(aq) + OH^–_(aq) ==> H₂O_(l)

        • reduces the [H⁺] on the right so more HIn ionises to try to increase the [H⁺] i.e. minimising the change in [H⁺].

• 6.2.2 The colour that is observed will depend on the ratio [HIn]_/[In^–], but at pH extremes i.e. very acid, colour 1 will dominate, or in very alkaline solution, colour 2 will dominate. Therefore the maximum colour 'shade' change from one to the other will occur when [HIn] = [In^–], or [colour 1] = [colour 2].

  • The pH when [HIn] = [In^–] can be calculated from the dissociation constant, K_ind (k_a), for the weak acid indicator.

  • K_ind = [H⁺_(aq)] [In^–_(aq)]_/[HIn_(aq)], but when  [HIn] = [In^–] the equilibrium expression simplifies to ...

  • K_ind = [H⁺_(aq)], so at this point the pH = –log(K_ind)

  • and is referred to as the pK_ind value.

• 6.2.3 pH titration curves and choice of indicator

  • A simple pH curve for an acid–alkali is explained on one of my GCSE acid–alkali notes pages and is well worth reading first, before tackling all the possibilities described and explained below.

  • The greatest change in indicator colour (per volume of reagent added), will occur at the equivalence point in the titration.

  • Therefore you need to choose an indicator with a pK_ind close to the pH at the equivalence point (theory above).

  • In fact acid–base titration indicators are usually effective over a range of several pH units but it is essential for accurate titrations that the colour change is sharp at the equivalence point with a small addition of acidic or alkaline titration solution.

  • Universal indicator is NOT suitable for quantitative analysis and the indicator choices tabulated below are explained via the sets of pH graphs shown further down using Graphs A to D. The effective pH range of the indicator is where there is sufficient colour change to give a good sharp end–point and can be above or below, but close to the pK_ind value of the indicator. Some effective pH ranges for selected indicators are given below.

• Indicator colour change, from acid to alkali	pKind	pH range	example of titration use
  Methyl orange, (red ==> yellow)	3.7	3.1–4.4	weak base – strong acid titration e.g. ammonia, sodium carbonate or sodium hydrogencarbonate titrated with hydrochloric acid
  Bromophenol blue, (yellow ==> blue)	4.0	2.8–4.6	weak base – strong acid titration
  Methyl red, (red ==> yellow)	5.1	4.2–6.3	weak base – strong acid titration
  Bromothymol blue, (yellow ==> blue)	7.0	6.0–7.6	strong acid – strong base titration e.g. hydrochloric acid <=> sodium hydroxide titration
  Phenol red, (yellow ==> red)	7.9	6.8–8.4	strong acid – strong base titration e.g. hydrochloric acid <=> sodium hydroxide titration
  Thymol blue (base form), (yellow ==> blue)	8.9	8.0–9.6	weak/strong acid – strong base titration
  Phenolphthalein, (colourless ==> pinky–red)	9.3	8.3–10.0	weak/strong acid – strong base titration e.g. ethanoic acid titrated with sodium hydroxide

• 6.2.4 The change in pH through various titrations is illustrated and explained to extend the idea of choosing the right indicator.

  • Below are two graphs of sets of curves showing how the pH changes when weak/strong alkalis are added to weak/strong acids (set A) and vice versa (set B).

  • The curves are a bit simplified and approximate, but show how the pH changes in titrations.

  • By putting two graph sections from (1) to (4) together you can construct an approximate pH curve.

  • This is done below each graph for the four acid–base permutations and it is assumed the acids and bases are monobasic/monoprotic.

    • Note:

      1. The end–point = equivalence point or stoichiometric point.

      2. If a buffer calibrated  pH meter is used rather than an indicator, the end–point is obtained from the graph at the mid–point of the steepest inflexion of the titration curve.

      3. The first two graphs (A and B) assume 20cm³ of the acid/alkali is titrated with an alkali/acid of the same concentration e.g. 0.1 or 1.0 mol dm^–3.

      4. For monoprotic/monobasic acids–base titrations there is only one point of steepest inflexion on the pH curve.

      5. However, apart from the strong acid–strong base curves, there are one or two other, but much less steep, points of inflexion due to the formation of a buffer mixture (see determination of K_a of weak acid via titration curve).

      6. The formation of this buffer mixture makes the end–point less sharp because it resists pH change.

      • Graph A

  • 6.2.5 pH curves – Graph A: The pH change when adding soluble base (alkali) to acid

    • 6.2.5a: Curve A1 (1) + (3): Adding a weak base to a strong acid, end point at (i1), approx. pH 3–5.

      • pH change at end–point reasonable sharp e.g. adding ammonia to hydrochloric acid, though this titration is usually done the other way round, for more details see B4 below.

    • 6.2.5b: Curve A2 (1) + (4): Adding a strong base to a strong acid, end point (i2), approx. pH 7.

      • pH change at end–point very sharp e.g. titrating hydrochloric acid with sodium hydroxide.

      • Suitable indicators: bromothymol blue (pK_ind 7.0, range 6.0–7.6), phenol red (pK_ind 7.9, range 6.8–8.4), phenolphthalein (pK_ind 9.3, range 8.3–10.0, ok for any strong acid – strong base titration because the pH change is so sharp at the end–point i.e. the point of inflexion is very sharp for 1–2 drops of alkali over pH 3–10)

        • This titration has the sharpest pH change at the end–point, hence the sharpest indicator colour change of all the titrations due to lack of buffering effects.

      • For all for all these four cases, in terms of H⁺ and OH^– ions: Initially a high/low concentration of H⁺, very/mildly acid, as the OH^– is steadily added, the H⁺ ions are neutralised to water, so the H⁺ concentration steadily falls and the pH rises as the solution becomes less acid. At pH 7, neutral there are very tiny equal concentrations of H⁺ and OH^– (due to the self–ionisation of water). If excess alkali is added the pH steadily rises to around 13 as the concentration of OH^– from the alkali rises ie solution becomes more alkaline.

    • 6.2.5c: Curve A3 (2) + (3): Adding a weak base to a weak acid, end point (i2), approx. pH 7.

      • pH change at end–point not very sharp, not practical for any titration e.g. adding ammonia to ethanoic acid.

      • Suitable indicators: None.

        • This titration gives the lowest rate of change of pH approaching the end–point, hence the poorest end–point to detect with indicator. This is due to strong buffering effect of the mixture of a weak base, weak acid and their salt. (for more details see buffer examples 6.3.1 and 6.3.2)

    • 6.2.5d: Curve A4 (2) + (4): Adding a strong base to weak acid, end point (i3), approx. pH 9.

      • pH change at end–point reasonable sharp e.g. you can titrate weak organic acids like ethanoic acid with sodium hydroxide.

      • Suitable indicators:  phenolphthalein (pK_ind 9.3, range 8.3–10.0), Thymol blue (base, pK_ind 8.9, range 8.0–9.6)

        • The lower rate of change of pH approaching the end–point compared to curve A2 (above) is due to the weak buffering effect of the mixture of a weak acid and the salt of a weak acid–strong base. (for more details see buffer example 6.3.1)

        • Using a pH titration curve to determine the K_a of a weak acid

        • Graph E

        • When a weak acid (HA) is titrated with a strong base (e.g. NaOH) a buffer mixture of A^– and HA exists from soon after the titration starts to near the end–point.

        • Therefore, half–way to the equivalence point e.g. on addition of 10 cm³ of alkali of a 20 cm³ titration (Graph E, curve (2) above), it means in terms of concentrations

        • [NaA_(aq)]_salt = [A^–_(aq)] = [HA_(aq)]_{unreacted acid}

        • Now, the equilibrium expression for a mono basic/protic weak acid is ...

        • Ka =	[H+(aq)] [A–(aq)]
          	––––––––––––
          	[HA(aq)]

        • so, at the half–way point, when [A^–] = [HA], K_a = [H⁺_(aq)],

        • or at half–way point: pH = pK_a and K_a = 10^–pKa.

        • In the 'fictitious' case of the weak acid above the pH is 4.2 at this point,

        • therefore: [H⁺_(aq)] = K_a = 6.3 x 10^–5 mol dm^–3.

    • Graph B

  • 6.2.6 pH curves – Graph B: The pH change when adding acid to a soluble base (alkali)

  • 6.2.6a: Curve B1 (1) + (3): Adding a weak acid to a strong base, end point (i1), approx. pH 9.

    • Reasonably sharp pH change at end–point, but this titration is usually done the other way round, for more details see A4.

  • 6.2.6b: Curve B2 (1) + (4): Adding a strong acid to strong base, end point (i2), approx. pH 7.

    • pH change at end–point very sharp e.g. titrating sodium hydroxide with hydrochloric acid. For more details see A2 above.

    • For all these four cases, in terms of H⁺ and OH^– ions: Initially a high/low concentration of OH^–, very/mildly alkaline, as the H⁺ is steadily added, the OH^– ions are neutralised to water, so the OH^– concentration steadily falls as does the pH as the solution becomes less alkaline. At pH 7, neutral there are very tiny equal concentrations of H⁺ and OH^– ions (due to the self–ionisation of water). If excess acid is added the pH steadily falls to around 1 as the concentration of H⁺ from the acid rises.

  • 6.2.6c: Curve B3 (2) + (3): Adding a weak acid to weak base, end point (i2), approx. pH 7.

    • The pH change at the end–point NOT very sharp and so not suitable for titrations e.g. adding ethanoic acid to ammonia solution (see A3 above for more details).

  • 6.2.6d: Curve B4 (2) + (4): Adding a strong acid to weak base, end point (i3), approx. pH 3–5

    • e.g. titrating ammonia with hydrochloric acid.

    • Suitable indicators: methyl orange (pK_ind 3.7, range 3.1–4.0)

      • The lower rate of change of pH approaching the end–point compared to curve B2 (above) is due to buffering effect of the mixture of a weak base and the salt of a weak base–strong acid. (for more details see buffer example 6.3.2)

• 6.2.7 More complicated pH titration curves

  • 6.2.7a The titration of a weak dibasic acid e.g. 25 cm³ of 0.1 mol dm^–3 ethanedioc acid (oxalic acid) titrated with 0.1 mol dm^–3 sodium hydroxide

    • Graph C

    • There are two inflexion points on the pH curve corresponding to the half and full neutralisation of the dibasic/diprotic acid..

      • HOOC–COOH_(aq) + NaOH_(aq) ==> HCOO–COO^–Na⁺_(aq) + H₂O_(l)

      • ionically: HOOC–COOH_(aq) + OH^–_(aq) ==> HCOO–COO^–_(aq) + H₂O_(l)

      • HCOO–COO^–Na⁺_(aq) + NaOH_(aq) ==> Na^+–OOC–COO^–Na⁺_(aq) + H₂O_(l)

      • ionically: HCOO–COO^–_(aq) + OH^–_(aq) ==> ^–OOC–COO^–_(aq) + H₂O_(l)

    • To detect the 2nd end–point, and hence the acid quantitatively, phenolphthalein indicator (pK_ind 9.3, range 8.3–10.0) is used, since it is essentially a weak acid–strong base titration.

      • I'm not sure if the 1st 'end–point' can be detected e.g. using methyl orange (pK_ind 3.7, range 3.1–4.0) and I doubt if it is of any quantitative use?

    • The lower rate of change of pH approaching the end–point compared to a strong base–strong acid titration is due to the weak buffering effect of the mixture of a weak acid and the salt of a weak acid–strong base. (for more details see  Case study ?)

    • Other acids like propanedioic acid (malonic acid) and butanedioic acid (succinic acid) behave, and be titrated, in the same way.

    • In the case of the tribasic/triprotic phosphoric(V) acid, H₃PO₄, you would get three points of inflexion on the titration curve of added sodium hydroxide versus pH corresponding to the formation of

      • (i) NaH₂PO₄, (ii) Na₂HPO₄ and (iii) Na₃PO₄

      • as each hydrogen of the acid is successively replaced.

  • 6.2.7b The titration of 25cm³ of 0.1 mol dm^–3 sodium carbonate titrated with 0.1 mol dm^–3 hydrochloric acid.

    • Graph D

    • There are two inflexion points on the pH curve.

    • Endpoint (1) corresponds to the 1st stage of neutralisation, the formation of the hydrogencarbonate ion.

      • Na₂CO_3(aq) + HCl_(aq) ==> NaCl_(aq) + NaHCO_3(aq)

      • ionically: CO₃^2–_(aq) + H⁺_(aq) ==> HCO₃^–_(aq)

      • This end–point at around pH 8–9 can be detected with phenolphthalein (pK_ind 9.3, range 8.3–10.0) or Thymol blue – base form (pK_ind 8.9, range 8.0–9.6)

      • If the conical flask is rapidly swirled on adding the acid, you don't see any gas bubbles of carbon dioxide.

    • End–point (2) corresponds to the 2nd stage of neutralisation, the formation of water and carbon dioxide.

      • NaHCO_3(aq) + HCl_(aq) ==> NaCl_(aq) + H₂O_(l) + CO_2(g)

      • ionically: HCO₃^–_(aq) + H⁺_(aq) ==> H₂O_(l) + CO_2(g)

      • This end–point around pH 3–4 can be detected with methyl orange indicator (pK_ind 3.7, range 3.1–4.0) or Bromophenol blue (pK_ind 4.0, 2.8–4.6).

    • Overall the reaction is ...

      • Na₂CO_3(aq) + 2HCl_(aq) ==> 2NaCl_(aq) + H₂O_(l) + CO_2(g)

      • ionically: CO₃^2–_(aq) + 2H⁺_(aq) ==> H₂O_(l) + CO_2(g)

    • Note:

      • A mixture of sodium carbonate and sodium hydrogencarbonate can be analysed using two separate titrations.

      • Titration (i) using phenolphthalein indicator measures the sodium carbonate,

      • and titration (ii) measures the sodium carbonate plus the sodium hydrogencarbonate. So both quantities can be calculated from the titration results.

• 6.2.8 Back titrations

  • Examples include ...

    • Where a known excess of acid is added to a base–alkali and the unreacted acid is 'back titrated' with a standard alkali solution from which the actual amount of base–alkali originally present can be calculated.

    • Where a known excess of alkali is added to an acid and the unreacted alkali is 'back titrated' with a standard acid solution from which the actual amount of acid originally present can be calculated.

• 6.2.9 Examples of acid–base titration questions with all the answers and working.

• –

Equilibria Part 6 sub–index: 6.1 Salt hydrolysis *

6.2 Acid–base indicator theory, pH curves and titrations

6.3 Buffers – definition, formulation and action *

6.4 Buffer calculations *

6.5 Case studies of buffer function

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