Stepwise and cumulative constants

A cumulative or overall constant, given the symbol β, is the constant for the formation of a complex from reagents. For example, the cumulative constant for the formation of ML₂ is given by

${\displaystyle {\ce {{M}+ 2L <=> ML2}}}$;     ${\displaystyle \beta _{1,2}=\mathrm {\frac {[ML_{2}]}{[M][L]^{2}}} }$

The stepwise constants, K₁ and K₂ refer to the formation of the complexes one step at a time.

${\displaystyle {\ce {{M}+ L <=> ML}}}$;     ${\displaystyle K_{1}=\mathrm {\frac {[ML]}{[M][L]}} }$

${\displaystyle {\ce {{ML}+ L <=> ML2}}}$;     ${\displaystyle K_{2}=\mathrm {\frac {[ML_{2}]}{[ML][L]}} }$

It follows that

${\displaystyle \beta _{1,2}=K_{1}K_{2}\,}$

A cumulative constant can always be expressed as the product of stepwise constants. Conversely, any stepwise constant can be expressed as a quotient of two or more overall constants. There is no agreed notation for stepwise constants, though a symbol such as K^L
_ML is sometimes found in the literature. It is good practice to specify each stability constant explicitly, as illustrated above.

Hydrolysis products

The formation of a hydroxo complex is a typical example of a hydrolysis reaction. A hydrolysis reaction is one in which a substrate reacts with water, splitting a water molecule into hydroxide and hydrogen ions. In this case the hydroxide ion then forms a complex with the substrate.

${\displaystyle \mathrm {M+OH} \leftrightharpoons \mathrm {M(OH)} }$;      ${\displaystyle K=\mathrm {\frac {[M(OH)]}{[M][OH]}} }$

In water the concentration of hydroxide is related to the concentration of hydrogen ions by the self-ionization constant, K_w.

${\displaystyle K_{w}=\mathrm {[H]} ^{+}\mathrm {[OH]} ^{-1}}$

The expression for hydroxide concentration is substituted into the formation constant expression

${\displaystyle K={\frac {[\mathrm {M} (\mathrm {OH} )]}{[\mathrm {M} ]K_{\mathrm {w} }[\mathrm {H} ]^{-1}}}}$

${\displaystyle \beta _{1,-1}^{*}=KK_{\mathrm {w} }={\frac {[\mathrm {M} (\mathrm {OH} )]}{[\mathrm {M} ][\mathrm {H} ]^{-1}}}}$

In general, for the reaction

${\displaystyle \mathrm {M} +n\mathrm {OH} \leftrightharpoons \mathrm {M(OH} )_{n}}$      ${\displaystyle \log \beta _{1,-n}^{*}\approx \log K-14n}$

In the older literature the value of log K is usually cited for an hydrolysis constant. The log β^* value is usually cited for an hydrolysed complex with the generic chemical formula M_pL_q(OH)_r.

Acid–base complexes

A Lewis acid, A, and a Lewis base, B, can be considered to form a complex AB.

${\displaystyle \mathrm {A} +\mathrm {B} \leftrightharpoons \mathrm {AB} }$    ${\displaystyle ,K=\mathrm {\frac {[AB]}{[A][B]}} }$

There are three major theories relating to the strength of Lewis acids and bases and the interactions between them.

1. Hard and soft acid–base theory (HSAB).^[10] This is used mainly for qualitative purposes.
2. Drago and Wayland proposed a two-parameter equation which predicts the standard enthalpy of formation of a very large number of adducts quite accurately. −ΔH^⊖ (A − B) = E_AE_B + C_AC_B. Values of the E and C parameters are available.^[11]
3. Guttmann donor numbers: for bases the number is derived from the enthalpy of reaction of the base with antimony pentachloride in 1,2-Dichloroethane as solvent. For acids, an acceptor number is derived from the enthalpy of reaction of the acid with triphenylphosphine oxide.^[12]

For more details see: acid–base reaction, acid catalysis, Extraction (chemistry)

Ionic strength dependence

The thermodynamic equilibrium constant, K^⊖, for the equilibrium

${\displaystyle \mathrm {M+L} \leftrightharpoons \mathrm {ML} }$

can be defined^[15] as

${\displaystyle K^{\ominus }=\mathrm {\frac {\{ML\}}{\{M\}\{L\}}} }$

where {ML} is the activity of the chemical species ML etc. K^⊖ is dimensionless since activity is dimensionless. Activities of the products are placed in the numerator, activities of the reactants are placed in the denominator. See activity coefficient for a derivation of this expression.

Since activity is the product of concentration and activity coefficient (γ) the definition could also be written as

${\displaystyle K^{\ominus }=\mathrm {\frac {[ML]}{[M][L]}} \times {\frac {\gamma _{\mathrm {ML} }}{\gamma _{\mathrm {M} }\gamma _{\mathrm {L} }}}=\mathrm {\frac {[ML]}{[M][L]}} \times \Gamma }$

where [ML] represents the concentration of ML and Γ is a quotient of activity coefficients. This expression can be generalized as

${\displaystyle \beta _{pq...}^{\ominus }={\frac {[\mathrm {M} _{p}\mathrm {L} _{q}...]}{[\mathrm {M} ]^{p}[\mathrm {L} ]^{q}...}}\times \Gamma }$

Dependence of the stability constant for formation of [Cu(glycinate)]⁺ on ionic strength (NaClO₄)^[16]

To avoid the complications involved in using activities, stability constants are determined, where possible, in a medium consisting of a solution of a background electrolyte at high ionic strength, that is, under conditions in which Γ can be assumed to be always constant.^[15] For example, the medium might be a solution of 0.1 mol dm⁻³ sodium nitrate or 3 mol dm⁻³ sodium perchlorate. When Γ is constant it may be ignored and the general expression in theory, above, is obtained.

All published stability constant values refer to the specific ionic medium used in their determination and different values are obtained with different conditions, as illustrated for the complex CuL (L = glycinate). Furthermore, stability constant values depend on the specific electrolyte used as the value of Γ is different for different electrolytes, even at the same ionic strength. There does not need to be any chemical interaction between the species in equilibrium and the background electrolyte, but such interactions might occur in particular cases. For example, phosphates form weak complexes with alkali metals, so, when determining stability constants involving phosphates, such as ATP, the background electrolyte used will be, for example, a tetralkylammonium salt. Another example involves iron(III) which forms weak complexes with halide and other anions, but not with perchlorate ions.

When published constants refer to an ionic strength other than the one required for a particular application, they may be adjusted by means of specific ion theory (SIT) and other theories.^[17]

Temperature dependence

All equilibrium constants vary with temperature according to the Van 't Hoff equation^[18]

${\displaystyle {\frac {d(\ln K)}{dT}}={\frac {\Delta {\mathit {H}}_{\mathrm {m} }^{\ominus }}{RT^{2}}}}$

Alternatively

${\displaystyle {\frac {d\ln K}{d(1/T)}}=-{\frac {\Delta H^{\ominus }}{R}}}$

R is the gas constant and T is the thermodynamic temperature. Thus, for exothermic reactions, where the standard enthalpy change, ΔH^⊖, is negative, K decreases with temperature, but for endothermic reactions, where ΔH^⊖ is positive, K increases with temperature.

The chelate effect

Cu²⁺ complexes with methylamine (left) end ethylene diamine (right)

Consider the two equilibria, in aqueous solution, between the copper(II) ion, Cu²⁺ and ethylenediamine (en) on the one hand and methylamine, MeNH₂ on the other.

${\displaystyle \mathrm {Cu^{2+}+en\rightleftharpoons [Cu(en)]^{2+}} }$

${\displaystyle Cu^{2+}+2MeNH_{2}\rightleftharpoons [Cu(MeNH_{2})_{2}]^{2+}}$

In the first reaction the bidentate ligand ethylene diamine forms a chelate complex with the copper ion. Chelation results in the formation of a five-membered ring. In the second reaction the bidentate ligand is replaced by two monodentate methylamine ligands of approximately the same donor power, meaning that the enthalpy of formation of Cu–N bonds is approximately the same in the two reactions. Under conditions of equal copper concentrations and when then concentration of methylamine is twice the concentration of ethylenediamine, the concentration of the bidentate complex will be greater than the concentration of the complex with 2 monodentate ligands. The effect increases with the number of chelate rings so the concentration of the EDTA complex, which has six chelate rings, is much higher than a corresponding complex with two monodentate nitrogen donor ligands and four monodentate carboxylate ligands. Thus, the phenomenon of the chelate effect is a firmly established empirical fact: under comparable conditions, the concentration of a chelate complex will be higher than the concentration of an analogous complex with monodentate ligands.

The thermodynamic approach to explaining the chelate effect considers the equilibrium constant for the reaction: the larger the equilibrium constant, the higher the concentration of the complex.

${\displaystyle \mathrm {[Cu(en)]=\beta _{11}[Cu][en]|} }$

${\displaystyle \mathrm {[Cu(MeNH_{2})_{2}]=\beta _{12}[Cu][MeNH_{2}]} ^{2}}$

When the analytical concentration of methylamine is twice that of ethylenediamine and the concentration of copper is the same in both reactions, the concentration [Cu(en)]²⁺ is much higher than the concentration [Cu(MeNH₂)₂]²⁺ because β₁₁ ≫ β₁₂.

The difference between the two stability constants is mainly due to the difference in the standard entropy change, ΔS^⊖. In the reaction with the chelating ligand there are two particles on the left and one on the right, whereas in equation with the monodentate ligand there are three particles on the left and one on the right. This means that less entropy of disorder is lost when the chelate complex is formed than when the complex with monodentate ligands is formed. This is one of the factors contributing to the entropy difference. Other factors include solvation changes and ring formation. Some experimental data to illustrate the effect are shown in the following table.^[19]

Equilibrium	log β	ΔG⊖ /kJ mol−1	ΔH⊖ /kJ mol−1	−TΔS⊖ /kJ mol−1
Cd2+ + 4 MeNH2 ⇌ Cd(MeNH 2)2+ 4	6.55	−37.4	−57.3	19.9
Cd2+ + 2 en ⇌ Cd(en)2+ 2	10.62	−60.67	−56.48	−4.19

An EDTA complex

These data show that the standard enthalpy changes are indeed approximately equal for the two reactions and that the main reason why the chelate complex is so much more stable is that the standard entropy term is much less unfavourable, indeed, it is favourable in this instance. In general it is difficult to account precisely for thermodynamic values in terms of changes in solution at the molecular level, but it is clear that the chelate effect is predominantly an effect of entropy. Other explanations, including that of Schwarzenbach,^[20] are discussed in Greenwood and Earnshaw.^[19]

diethylenetriamine

The chelate effect increases as the number of chelate rings increases. For example, the complex [Ni(dien)₂)]²⁺ is more stable than the complex [Ni(en)₃)]²⁺; both complexes are octahedral with six nitrogen atoms around the nickel ion, but dien (diethylenetriamine, 1,4,7-triazaheptane) is a tridentate ligand and en is bidentate. The number of chelate rings is one less than the number of donor atoms in the ligand. EDTA (ethylenediaminetetracetic acid) has six donor atoms so it forms very strong complexes with five chelate rings. Ligands such as DTPA, which have eight donor atoms are used to form complexes with large metal ions such as lanthanide or actinide ions which usually form 8- or 9-coordinate complexes. 5-membered and 6-membered chelate rings give the most stable complexes. 4-membered rings are subject to internal strain because of the small inter-bond angle is the ring. The chelate effect is also reduced with 7- and 8- membered rings, because the larger rings are less rigid, so less entropy is lost in forming them.

Deprotonation of aliphatic –OH groups

2-aminoethanol

Tris

Removal of a proton from an aliphatic –OH group is difficult to achieve in aqueous solution because the energy required for this process is rather large. Thus, ionization of aliphatic –OH groups occurs in aqueous solution only in special circumstances. One such circumstance is found with compounds containing the H₂N–C–C–OH substructure. For example, compounds containing the 2-aminoethanol substructure can form metal–chelate complexes with the deprotonated form, H₂N–C–C–O⁻. The chelate effect supplies the extra energy needed to break the O–H bond.

An important example occurs with the molecule tris. This molecule should be used with caution as a buffering agent as it will form chelate complexes with ions such as Fe³⁺ and Cu²⁺.

The macrocyclic effect

It was found that the stability of the complex of copper(II) with the macrocyclic ligand cyclam (1,4,8,11-tetraazacyclotetradecane) was much greater than expected in comparison to the stability of the complex with the corresponding open-chain amine.^[21] This phenomenon was named the macrocyclic effect and it was also interpreted as an entropy effect. However, later studies suggested that both enthalpy and entropy factors were involved.^[22]

An important difference between macrocyclic ligands and open-chain (chelating) ligands is that they have selectivity for metal ions, based on the size of the cavity into which the metal ion is inserted when a complex is formed. For example, the crown ether 18-crown-6 forms much stronger complexes with the potassium ion, K⁺ than with the smaller sodium ion, Na⁺.^[23]

In hemoglobin an iron(II) ion is complexed by a macrocyclic porphyrin ring. The article hemoglobin incorrectly states that oxyhemoglogin contains iron(III). It is now known that the iron(II) in hemoglobin is a low-spin complex, whereas in oxyhemoglobin it is a high-spin complex. The low-spin Fe²⁺ ion fits snugly into the cavity of the porphyrin ring, but high-spin iron(II) is significantly larger and the iron atom is forced out of the plane of the macrocyclic ligand.^[24] This effect contributes the ability of hemoglobin to bind oxygen reversibly under biological conditions. In Vitamin B12 a cobalt(II) ion is held in a corrin ring. Chlorophyll is a macrocyclic complex of magnesium(II).

Cyclam	Porphine, the simplest porphyrin.

Structures of common crown ethers: 12-crown-4, 15-crown-5, 18-crown-6, dibenzo-18-crown-6, and an aza-crown ether

Geometrical factors

Successive stepwise formation constants K_n in a series such as ML_n (n = 1, 2, ...) usually decrease as n increases. Exceptions to this rule occur when the geometry of the ML_n complexes is not the same for all members of the series. The classic example is the formation of the diamminesilver(I) complex [Ag(NH₃)₂]⁺ in aqueous solution.

${\displaystyle \mathrm {Ag^{+}+NH_{3}\rightleftharpoons [Ag(NH_{3})]^{+};K_{1}={\frac {[Ag(NH_{3})]}{[Ag][NH_{3}]}}} }$

${\displaystyle \mathrm {Ag(NH_{3})^{+}+NH_{3}\rightleftharpoons [Ag(NH_{3})_{2}]^{+};K_{2}={\frac {[Ag(NH_{3})_{2}]}{[Ag(NH_{3}][NH_{3}]}}} }$

In this case, K₂ > K₁. The reason for this is that, in aqueous solution, the ion written as Ag⁺ actually exists as the four-coordinate tetrahedral aqua species [Ag(H₂O)₄]⁺. The first step is then a substitution reaction involving the displacement of a bound water molecule by ammonia forming the tetrahedral complex [Ag(NH₃)(H₂O)₃]⁺. In the second step, all the aqua ligands are lost and a linear, two-coordinate product [H₃N–Ag–NH₃]⁺ is formed. Examination of the thermodynamic data^[25] shows that the difference in entropy change is the main contributor to the difference in stability constants for the two complexation reactions.

Other examples exist where the change is from octahedral to tetrahedral, as in the formation of [CoCl₄]²⁻ from [Co(H₂O)₆]²⁺.

Classification of metal ions

Ahrland, Chatt and Davies proposed that metal ions could be described as class A if they formed stronger complexes with ligands whose donor atoms are nitrogen, oxygen or fluorine than with ligands whose donor atoms are phosphorus, sulfur or chlorine and class B if the reverse is true.^[26] For example, Ni²⁺ forms stronger complexes with amines than with phosphines, but Pd²⁺ forms stronger complexes with phosphines than with amines. Later, Pearson proposed the theory of hard and soft acids and bases (HSAB theory).^[27] In this classification, class A metals are hard acids and class B metals are soft acids. Some ions, such as copper(I), are classed as borderline. Hard acids form stronger complexes with hard bases than with soft bases. In general terms hard–hard interactions are predominantly electrostatic in nature whereas soft–soft interactions are predominantly covalent in nature. The HSAB theory, though useful, is only semi-quantitative.^[28]

The hardness of a metal ion increases with oxidation state. An example of this effect is given by the fact that Fe²⁺ tends to form stronger complexes with N-donor ligands than with O-donor ligands, but the opposite is true for Fe³⁺.

Effect of ionic radius

The Irving–Williams series refers to high-spin, octahedral, divalent metal ion of the first transition series. It places the stabilities of complexes in the order

Mn < Fe < Co < Ni < Cu > Zn

This order was found to hold for a wide variety of ligands.^[29] There are three strands to the explanation of the series.

1. The ionic radius is expected to decrease regularly for Mn²⁺ to Zn²⁺. This would be the normal periodic trend and would account for the general increase in stability.
2. The crystal field stabilisation energy (CFSE) increases from zero for manganese(II) to a maximum at nickel(II). This makes the complexes increasingly stable. CFSE returns to zero for zinc(II).
3. Although the CFSE for copper(II) is less than for nickel(II), octahedral copper(II) complexes are subject to the Jahn–Teller effect which results in a complex having extra stability.

Another example of the effect of ionic radius the steady increase in stability of complexes with a given ligand along the series of trivalent lanthanide ions, an effect of the well-known lanthanide contraction.

Supramolecular complexes

Supramolecular complexes are held together by hydrogen bonding, hydrophobic forces, van der Waals forces, π-π interactions, and electrostatic effects, all of which can be described as noncovalent bonding. Applications include molecular recognition, host–guest chemistry and anion sensors.

A typical application in molecular recognition involved the determination of formation constants for complexes formed between a tripodal substituted urea molecule and various saccharides.^[32] The study was carried out using a non-aqueous solvent and NMR chemical shift measurements. The object was to examine the selectivity with respect to the saccharides.

An example of the use of supramolecular complexes in the development of chemosensors is provided by the use of transition-metal ensembles to sense for ATP.^[33]

Anion complexation can be achieved by encapsulating the anion in a suitable cage. Selectivity can be engineered by designing the shape of the cage. For example, dicarboxylate anions could be encapsulated in the ellipsoidal cavity in a large macrocyclic structure containing two metal ions.^[34]