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熱力學

金屬離子配合物形成的熱力學提供了許多重要的信息。 特別是它在區分焓和熵效應的時候很有用。焓效應取決於鍵強度，而熵效應取決於溶液中整體的有序和無序的變化。下面的螯合作用用熱力學來解釋是最好的。

平衡常數與該反應的標準吉布斯自由能變有關

ΔG^⊖ = -2.303 RT log₁₀ β.

R是氣體常數而T是熱力學溫度。在25 °C, ΔG^⊖ = (−5.708 kJ mol⁻¹) ⋅ log β。自由能由焓項和熵項組成。

ΔG^⊖ = ΔH^⊖ − TΔS^⊖

標準焓變可以用量熱法或范特霍夫方程確定，但量熱法更好。當標準焓變和穩定常數都已確定，標準熵變可以很容易的用上式計算出來。

ML_n型的配合物的逐級穩定常數隨著n的增大而減小的事實或許可以被熵因素部分解釋。在正八面體的配合物形成時採用這個解釋。

[M(H₂O)_mL_n-1] +L ⇌ [M(H₂O)_m-1L_n]

第一步m = 6，n = 1，而配體可以結合六個位點中的一個。第二步m = 5，而第二個配體只能結合五個位點中的一個。這意味著第一步比第二步有更大的隨機性；ΔS^⊖是正數，變得更大了，而ΔG^⊖是負數，變得更小了，所以log K₁ > log K₂。逐級穩定常數的比值可以在此基礎上算得，但實驗值與理論值並不完全吻合，因為每一步的ΔH^⊖並不一定相同。 熵因素對以下的螯合作用也是很重要的。

離子濃度的影響

熱力學平衡常數K^⊖對於這個平衡

M + L ⇌ ML

可以被定義為 as

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

{ML}是指ML物種的活度。因為活度是無量綱量，所以K^⊖是無量綱量。生成物的活度放在分子，反應物的活度放在分母。這個表達式的推導參見活度係數。

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

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

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

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

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

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. For example, the medium might be a solution of 0.1 mol/dm⁻³ sodium nitrate or 3 mol/dm⁻³ potassium 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.

Temperature dependence

All equilibrium constants vary with temperature according to the van 't Hoff equation

${\frac {\operatorname {d} \ln {\mathit {K}}}{\operatorname {d} T}}={\frac {{\Delta {\mathit {H}}_{m}}^{\ominus }}{RT^{2}}}$

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

Factors affecting the stability constants of complexes

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.

Cu²⁺ + en ⇌ [Cu(en)]²⁺ (1)

Cu²⁺ + 2 MeNH₂ ⇌ [Cu(MeNH₂)₂]²⁺ (2)

In (1) the bidentate ligand ethylene diamine forms a chelate complex with the copper ion. Chelation results in the formation of a five–membered ring. In (2) 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 complex (1) will be greater than the concentration of the complex (2). 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.

[Cu(en)] = β₁₁[Cu][en]

[Cu(MeNH₂)₂] = β₁₂[Cu][MeNH₂]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 equation (1) there are two particles on the left and one on the right, whereas in equation (2) 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.

Equilibrium	log β	ΔG⊖	ΔH⊖ /kJ mol−1	−TΔS⊖ /kJ mol−1
Cd2+ + 4 MeNH2 ⇌ Cd(MeNH2)42+	6.55	−37.4	−57.3	19.9
Cd2+ + 2 en ⇌ Cd(en)22+	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, are discussed in Greenwood and Earnshaw.

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.

Ethylenediamine (en)	Diethylenetriamine (dien)

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. 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.

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⁺.

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 porhyrin ring, but high-spin iron(II) is significantly larger and the iron atom is forced out of the plane of the macrocyclic ligand. 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 diaza-18-crown-6

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.

Ag⁺ + NH₃ ⇌ [Ag(NH₃)]⁺      $K_{1}=\mathrm {\frac {[[Ag(NH_{3})]^{+}]}{[Ag^{+}][NH_{3}]}}$

Ag(NH₃)⁺ + NH₃ ⇌ [Ag(NH₃)₂]⁺      $K_{2}=\mathrm {\frac {[[Ag(NH_{3})_{2}]^{+}]}{[[Ag(NH_{3})]^{+}][NH_{3}]}}$

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(OH₂)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)(OH₂)₃]⁺ (commonly abbreviated as [Ag(NH₃)]⁺). In the second step, the aqua ligands are lost to form a linear, two-coordinate product [H₃N—Ag—NH₃]⁺. Examination of the thermodynamic data shows that both enthalpy and entropy effects determine the result.

equilibrium	ΔH⊖ /kJ mol−1	ΔS⊖ /J K−1 mol−1
Ag+ + NH3 ⇌ [Ag(NH3)]+	−21.4	8.66
[Ag(NH3)]+ + NH3 ⇌ [Ag(NH3)2]+	−35.2	−61.26

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 N, O or F than with ligands whose donor atoms are P, S or Cl and class B if the reverse is true. 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). 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.

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. 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.

Applications

Stability constant values are exploited in a wide variety of applications. Chelation therapy is used in the treatment of various metal-related illnesses, such as iron overload in β-thalassemia sufferers who have been given blood transfusions. The ideal ligand binds to the target metal ion and not to others, but this degree of selectivity is very hard to achieve. The synthetic drug Deferiprone achieves selectivity by having two oxygen donor atoms so that it binds to Fe³⁺ in preference to any of the other divalent ions that are present in the human body, such as Mg²⁺, Ca²⁺ and Zn²⁺. Treatment of poisoning by ions such as Pb²⁺ and Cd²⁺ is much more difficult since these are both divalent ions and selectivity is harder to accomplish. Excess copper in Wilson's disease can be removed by penicillamine or Triethylene tetramine (TETA). DTPA has been approved by the U.S. Food and Drug Administration for treatment of plutonium poisoning.

DTPA is also used as a complexing agent for gadolinium in MRI contrast enhancement. The requirement in this case is that the complex be very strong, as Gd³⁺ is very toxic. The large stability constant of the octadentate ligand ensures that the concentration of free Gd³⁺ is almost negligible, certainly well below toxicity threshold. In addition the ligand occupies only 8 of the 9 coordination sites on the gadolinium ion. The ninth site is occupied by a water molecule which exchanges rapidly with the fluid surrounding it and it is this mechanism that makes the paramagnetic complex into a contrast reagent.

EDTA forms such strong complexes with most divalent cations that it finds many uses. For example, it is often present in washing powder to act as a water softener by sequestering calcium and magnesium ions.

The selectivity of macrocyclic ligands can be used as a basis for the construction of an ion selective electrode. For example, potassium selective electrodes are available that make use of the naturally-occurring macrocyclic antibiotic valinomycin.

Deferiprone	penicillamine	triethylenetetramine, TETA	Ethylenediamine tetracetic acid, EDTA
File:Dtpa structure.png
diethylenetriaminepentacetic acid, DTPA	Valinomycin	tri-n-butylphosphate

An ion-exchange resin such as chelex 100, which contains chelating ligands bound to a polymer, can be used in water softeners and in chromatographic separation techniques. In solvent extraction the formation of electrically-neutral complexes allows cations to be extracted into organic solvents. For example, in nuclear fuel reprocessing uranium(VI) and plutonium(VI) are extracted into kerosene as the complexes [MO₂(TBP)₂(NO₃)₂] (TBP = tri-'n-butyl phosphate). In phase-transfer catalysis, a substance which is insoluble in an organic solvent can be made soluble by addition of a suitable ligand. For example, potassium permanganate oxidations can be achieved by adding a catalytic quantity of a crown ether and a small amount of organic solvent to the aqueous reaction mixture, so that the oxidation reaction occurs in the organic phase.

In all these examples, the ligand is chosen on the basis of the stability constants of the complexes formed. For example, TBP is used in nuclear fuel reprocessing because (among other reasons) it forms a complex strong enough for solvent extraction to take place, but weak enough that the complex can be destroyed by nitric acid to recover the uranyl cation as nitrato complexes, such as [UO₂(NO₃)₄]^2- back in the aqueous phase.

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. 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.

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.

Experimental methods

主條目：Determination of equilibrium constants

The method developed by Bjerrum is still the main method in use today, though the precision of the measurements has greatly increased. Most commonly, a solution containing the metal ion and the ligand in a medium of high ionic strength is first acidified to the point where the ligand is fully protonated. This solution is then titrated, often by means of a computer-controlled auto-titrator, with a solution of CO₂-free base. The concentration, or activity, of the hydrogen ion is monitored by means of a glass electrode. The data set used for the calculation has three components: a statement defining the nature of the chemical species that will be present, called the model of the system, details concerning the concentrations of the reagents used in the titration, and finally the experimental measurements in the form of titre and pH (or emf) pairs.

It is not always possible to use a glass electrode. If that is the case, the titration can be monitored by other types of measurement. Absorbance spectra, fluorescence spectra and NMR spectra are the most commonly used alternatives. Current practice is to take absorbance or fluorescence measurements at a range of wavelengths and to fit these data simultaneously. Various NMR chemical shifts can also be fitted together.

The chemical model will include values of the protonation constants of the ligand, which will have been determined in separate experiments, a value for log K_w and estimates of the unknown stability constants of the complexes formed. These estimates are necessary because the calculation uses a non-linear least-squares algorithm. The estimates are usually obtained by reference to a chemically similar system. The stability constant databases can be very useful in finding published stability constant values for related complexes.

In some simple cases the calculations can be done in a spreadsheet. Otherwise, the calculations are performed with the aid of a general-purpose computer programs. The most frequently used programs are:

• Potentiometric and/or spectrophotometric data: HYPERQUAD, PSEQUAD
• Potentiometric data: BEST
• Spectrophotometric data: SQUAD, SPECFIT,
• NMR data HypNMR, EQNMR

In biochemistry, formation constants of adducts may be obtained from Isothermal titration calorimetry (ITC) measurements. This technique yields both the stability constant and the standard enthalpy change for the equilibrium. It is mostly limited, by availability of software, to complexes of 1:1 stoichiometry.

Critically evaluated data

The following references are for critical reviews of published stability constants for various classes of ligands. All these reviews are published by IUPAC and the full text is available, free of charge, in pdf format.

• ethylenediamine (en)

• Nitrilotriacetic acid (NTA)

• polyamino carboxylic acids (complexones)

• Alpha hydroxy acids and other hydroxycarboxylic acids

• crown ethers

• phosphonic acids

• imidazoles and histamines

• amino acids with polar side-chains

• nucleotides

• acetylacetone

• general

• Chemical speciation of environmentally significant heavy metals with inorganic ligands. Part 1: The Hg²⁺– Cl^–, OH^–, CO₃^2–, SO₄^2–, and PO₄^3– systems.

• Chemical speciation of environmentally significant metals with inorganic ligands Part 2: The Cu²⁺-OH^-, Cl^-, CO₃^2-, SO₄^2-, and PO₄^3- aqueous systems

• Chemical speciation of environmentally significant metals with inorganic ligands Part 3: The Pb²⁺-OH^-, Cl^-, CO₃^2-, SO₄^2-, and PO₄^3- systems

• Chemical speciation of environmentally significant metals with inorganic ligands. Part 4: The Cd²⁺ + OH^–, Cl^–, CO₃^2–, SO₄^2–, and PO₄^3– systems

Further reading

Sigel, Roland K.O.; Skilandat, Miriam; Sigel, Astrid; Operschall, Bert P.; Sigel, Helmut. Chapter 8. Complex formation of cadmium with sugar residues, nucleobases, phosphates, nucleotides and nucleic acids. Astrid Sigel, Helmut Sigel and Roland K. O. Sigel (編). Cadmium: From Toxicology to Essentiality. Metal Ions in Life Sciences 11. Springer. 2013: 191–274. doi:10.1007/978-94-007-5179-8_8. 

Sóvágó, Imre; Várnagy, Katalin. Chapter 9. Cadmium(II) complexes of amino acids and peptides. Astrid Sigel, Helmut Sigel and Roland K. O. Sigel (編). Cadmium: From Toxicology to Essentiality. Metal Ions in Life Sciences 11. Springer. 2013: 275–302. doi:10.1007/978-94-007-5179-8_9. 

Yatsimirsky, Konstantin Borisovich; Vasil'ev,Vladimir Pavlovich. Instability Constants of Complex Compounds. OUP. 1960.  (Translated by D.A. Patterson)

External links

• Stability constants web-site Contains information on computer programs, applications, databases and hardware for experimental titrations.