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Lesson 5 · 30 min

Complex ions and formation constants

Ammonia molecules bind to Cu²⁺ one after another, turning the solution from pale to deep blue. Complexation dissolves silver chloride, fixes photographs, and lets aluminium hydroxide dissolve in both acid and base.

In this lesson

  • Describe complex ions as Lewis acid–base adducts and write stepwise and cumulative formation constants
  • Calculate the distribution of complex ions at any ligand concentration from formation constants
  • Use K = Ksp·β to judge whether a ligand can dissolve a precipitate
  • Explain why amphoteric hydroxides dissolve in both acid and strong base

Drip ammonia into pale-blue copper sulfate solution and a pale-blue, fluffy precipitate appears; keep adding and it vanishes, leaving a deep royal-blue solution. When a photographic film is developed, the fixer (sodium thiosulfate) washes away the silver bromide that light never touched. Both are the work of complex ions: a metal ion surrounded by a shell of molecules or ions, forming a new, soluble species.

Predict first

Three tubes hold 0.10 mmol of AgCl, AgBr and AgI precipitate; 5.0 mL of 6 M ammonia is added to each. What happens?

Pick an answer and lock it in before you look. A wrong guess teaches more than no guess.

Complex ions and formation constants

In a complex ion the central metal ion is a Lewis acid (electron-pair acceptor) and the ligands — NHX3\ce{NH3}, HX2O\ce{H2O}, ClX−\ce{Cl-}, CNX−\ce{CN-} … — are Lewis bases, each bonding through a lone pair. In water the metal ion is already surrounded by water molecules, so forming a complex really means ligands replacing water one at a time:

[Cu(HX2O)X6]X2+→NHX3[Cu(NHX3)(HX2O)X5]X2+→NHX3⋯→NHX3[Cu(NHX3)X4(HX2O)X2]X2+\ce{[Cu(H2O)6]^2+ ->[NH3] [Cu(NH3)(H2O)5]^2+ ->[NH3] \cdots ->[NH3] [Cu(NH3)4(H2O)2]^2+}

Each step has its own equilibrium constant KnK_n (the stepwise formation constant); their product is the cumulative formation constant βn\beta_n. For CuX2+\ce{Cu^2+}–NHX3\ce{NH3} the log⁡K\log K values are 4.04, 3.43, 2.80 and 1.48 — falling step by step, as fewer waters remain to be replaced and the bound ligands crowd each other.

βn=K1K2⋯Kn=[MLn][M][L]nαn=βn[L]n1+∑i=1Nβi[L]i\beta_n = K_1K_2\cdots K_n = \frac{[\mathrm{ML}_n]}{[\mathrm{M}][\mathrm{L}]^n} \qquad \alpha_n = \frac{\beta_n[\mathrm{L}]^n}{1 + \sum_{i=1}^{N}\beta_i[\mathrm{L}]^i}
Cumulative constants and distribution fractions: knowing the free ligand [L] fixes the share of every form.
[L][\mathrm{L}]
free ligand concentration (not the total added)
αn\alpha_n
fraction of the metal present as MLₙ
β4(Cu(NHX3)X4X2+)\beta_4(\ce{Cu(NH3)4^2+})
1011.75=5.6×101110^{11.75} = 5.6\times10^{11}
Stepwise constants · coupled to precipitation and acid–base

Copper–ammine complexes, step by step

0.05 M CuSO₄ beaker (colour computed from absorption spectra)
pH
6.51
Free NH₃
1.81 × 10⁻⁴ M
Cu(OH)₂ precipitate
99.6 %
Average ligand number n̄
1.05
Dominant form
[Cu(NHX3)]X2+\ce{[Cu(NH3)]^2+} · 49 %
Stepwise formation constants
lg K1 = 4.04 · lg K2 = 3.43 · lg K3 = 2.80 · lg K4 = 1.48 · lg β₄ = 11.75
10⁻⁷10⁻⁶10⁻⁵10⁻⁴0.0010.010.11Free NH₃ / (mol/L)00.20.40.60.81Fraction of copper
  • Cu²⁺
  • Cu(NH₃)²⁺
  • Cu(NH₃)₂²⁺
  • Cu(NH₃)₃²⁺
  • Cu(NH₃)₄²⁺
00.511.522.53NH₃ added / (mol/L)00.20.40.60.81Fraction / pH ÷ 14
  • Fraction as Cu(OH)₂ precipitate
  • pH ÷ 14
Left: fractions of each form from Martell & Smith’s stepwise constants (log K = 4.04, 3.43, 2.80, 1.48 at 25 °C). Right and the beaker: NH₃ (and optionally NH₄Cl) added to 0.05 M CuSO₄, solving the copper and nitrogen mass balances (NH₃/NH₄⁺, pKa 9.25), the Cu(OH)₂ solubility product (Ksp = 4.8 × 10⁻²⁰) and the charge balance together. The solution colour comes from each form’s absorption band ([Cu(H₂O)₆]²⁺ near 800 nm, [Cu(NH₃)₄]²⁺ near 605 nm, the others evenly between them in wavenumber, about 1000 cm⁻¹ per NH₃) through CIE colorimetry; the pale blue of the precipitate is an observed colour. In a real beaker the first pale-blue solid is often a basic copper sulfate; here it is treated as Cu(OH)₂. Activities are taken as concentrations.
Add ammonia slowly: a few tenths of a mole per litre raise the pH and throw down pale-blue Cu(OH)₂; carry on to 1–3 mol/L and the precipitate gradually redissolves as the colour deepens from pale to royal blue, λmax moving from 800 nm to 605 nm. Then try a little NH₄Cl — ammonium holds down [OH⁻] and the precipitate vanishes much sooner.

Worked example

Copper in 0.10 M free ammonia

With free NHX3\ce{NH3} at 0.10 M, in which forms is the copper present, and in what proportions? (log⁡β\log\beta = 4.04, 7.47, 10.27, 11.75)

Dissolving precipitates by complexation

A ligand pulls free metal ions out of solution, dropping QQ below KspK_\mathrm{sp}, so the precipitate dissolves. Adding the dissolution and complexation equilibria gives an overall constant that is simply the product:

AgCl(s)+2 NHX3⇌[Ag(NHX3)X2]X++ClX−K=Ksp β2=3.0×10−3\ce{AgCl(s) + 2NH3 <=> [Ag(NH3)2]+ + Cl-} \qquad K = K_\mathrm{sp}\,\beta_2 = 3.0\times10^{-3}

In 1 M ammonia, s2/(1−2s)2=3.0×10−3s^2/(1-2s)^2 = 3.0\times10^{-3} gives s=0.050s = 0.050 M — over three thousand times the pure-water value. Thiosulfate binds AgX+\ce{Ag+} even more tightly (log⁡β2=13.67\log\beta_2 = 13.67) and dissolves even AgBr quickly: the chemistry of photographic fixing.

Coupled Ksp × β equilibria

Dissolving by complexation, and amphoteric hydroxides

0.01 M 02468101214pH10⁻¹⁰10⁻⁹10⁻⁸10⁻⁷10⁻⁶10⁻⁵10⁻⁴0.0010.010.1110Dissolved metal in equilibrium with the solid / (mol/L)
  • Al(OH)₃
  • Zn(OH)₂
  • Cu(OH)₂
  • Mg(OH)₂
Al³⁺Zn²⁺Cu²⁺Mg²⁺
  • Al(OH)X3\ce{Al(OH)3} amphoteric At 0.01 M: precipitates from pH 3.5 · redissolves above pH 9.5
  • Zn(OH)X2\ce{Zn(OH)2} amphoteric At 0.01 M: precipitates from pH 7.2 · redissolves above pH 10.9
  • Cu(OH)X2\ce{Cu(OH)2} dissolves in acid only At 0.01 M: precipitates from pH 5.3 · stays solid in strong base
  • Mg(OH)X2\ce{Mg(OH)2} dissolves in acid only At 0.01 M: precipitates from pH 9.4 · stays solid in strong base
Silver-halide mode solves the solubility product, the stepwise silver–ligand formation constants (Martell & Smith: Ag–NH₃ log K = 3.31, 3.91; Ag–S₂O₃²⁻ 8.82, 4.85 at 20 °C) and the ligand mass balance together (ligand bound in complexes lowers the free ligand); NH₃ protonation and activity coefficients are ignored. Amphoteric mode plots the total metal dissolved in equilibrium with the solid: Al(OH)₃ includes Al(OH)²⁺ and Al(OH)₄⁻, Zn(OH)₂ includes Zn(OH)⁺, Zn(OH)₃⁻ and Zn(OH)₄²⁻; hydroxo complexes of Cu(OH)₂ and Mg(OH)₂ are left out (Cu(OH)₂ in fact dissolves slightly in very strong base). The solid drawn in each tube is proportional to the computed undissolved fraction.
Amphoteric hydroxides: the Al(OH)₃ and Zn(OH)₂ curves are U-shaped, redissolving in strong base as Al(OH)X4X−\ce{Al(OH)4-} and Zn(OH)X4X2−\ce{Zn(OH)4^2-}; Cu(OH)₂ and Mg(OH)₂ stay solid. Drag the pH from 7 to 14 and watch the four tubes.

Amphoteric hydroxides combine both equilibria: in acid, Al(OH)X3+3 HX+→AlX3++3 HX2O\ce{Al(OH)3} + \ce{3H+} \mathrel{\ce{->}} \ce{Al^3+} + \ce{3H2O}; in strong base, OHX−\ce{OH-} itself acts as a ligand, and Al(OH)X3(s)+OHX−⇌Al(OH)X4X−\ce{Al(OH)3(s)} + \ce{OH-} \mathrel{\ce{<=>}} \ce{Al(OH)4-} has K=Kspβ4≈0.3K = K_\mathrm{sp}\beta_4 \approx 0.3, so at pH 14 ([OHX−]=1[\ce{OH-}] = 1 M) about 0.3 M of aluminium stays dissolved as Al(OH)X4X−\ce{Al(OH)4-}. Magnesium and iron hydroxides form no stable hydroxo anions, and strong base leaves them untouched — exactly what qualitative analysis uses to separate aluminium and zinc from iron and copper.

Concept check

Which of these hydroxides dissolves in excess NaOH solution?

Try the calculation

Find the equilibrium constant of AgBr(s)+2 SX2OX3X2−⇌[Ag(SX2OX3)X2]X3−+BrX−\ce{AgBr(s)} + \ce{2S2O3^2-} \mathrel{\ce{<=>}} \ce{[Ag(S2O3)2]^3-} + \ce{Br-}. (Ksp(AgBr)=5.0×10−13K_\mathrm{sp}(\ce{AgBr}) = 5.0\times10^{-13}, β2=4.7×1013\beta_2 = 4.7\times10^{13})

Key takeaways

  • A complex ion joins a metal ion (Lewis acid) and ligands (Lewis bases); ligands replace water one by one, with K₁ > K₂ > …
  • βₙ = K₁K₂…Kₙ; the distribution depends on the free ligand, and a solution usually holds a mixture of complexes.
  • For precipitate + ligand, K = Ksp·β: AgCl dissolves in ammonia, AgBr in thiosulfate, and AgI barely in ammonia.
  • Amphoteric hydroxides (Al, Zn) dissolve in strong base as hydroxo anions; those of Mg, Fe and Cu do not.
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Data and references

  1. A. E. Martell & R. M. Smith, Critical Stability Constants (Plenum, 1974–1989), via D. Harvey, Analytical Chemistry 2.1, Appendix 12.
  2. J. Bjerrum, Metal Ammine Formation in Aqueous Solution (P. Haase, Copenhagen, 1941).
  3. C. E. Housecroft & A. G. Sharpe, Inorganic Chemistry, 5th ed. (2018), Ch. 7 and 20 (complex formation; electronic spectra of Cu(II)).
  4. D. C. Harris, Quantitative Chemical Analysis, 9th ed. (2016), Ch. 6 (complex formation) and Ch. 11 (EDTA and formation constants).