subject: Stability Constants Of Complexes - China Pdh Multiplexer - Media Converters Manufacturer [print this page] History History
Jannik Bjerrum developed the first general method for the determination of stability constants of metal-ammine complexes in 1941. The reasons why this occurred at such a late date, nearly 50 years after Alfred Werner had proposed the correct structures for coordination complexes, have been summarised by Beck and Nagypl. The key to Bjerrum method was the use of the then recently developed glass electrode and pH meter to determine the concentration of hydrogen ions in solution. Bjerrum recognised that the formation of a metal complex with a ligand was a kind of acid-base equilibrium: there is competition for the ligand, L, between the metal ion, Mn+, and the hydrogen ion, H+. This means that there are two simultaneous equilibria that have to be considered. In what follows electrical charges are omitted for the sake of generality. The two equilibria are
H + L HL
M + L ML
Hence by following the hydrogen ion concentration during a titration of a mixture of M and HL with base, and knowing the acid dissociation constant of HL, the stability constant for the formation of ML could be determined. Bjerrum went on to determine the stability constants for systems in which many complexes may be formed.
M + qL MLq
The following twenty years saw a veritable explosion in the number of stability constants that were determined. Relationships, such as the Irving-Williams series were discovered. The calculations were done by hand using the so-called graphical methods. The mathematics underlying the methods used in this period are summarised by Rossotti and Rossotti. The next key development was the use of a computer program, LETAGROP to do the calculations. This permitted the examination of systems too complicated to be evaluated by means of hand-calculations. Subsequently computer programs capable of handling complex equilibria in general, such as SCOGS and MINIQUAD were developed so that today the determination of stability constants has almost become a outine operation. Values of thousands of stability constants can be found in two commercial databases.
Theory
The formation of a complex between a metal ion, M, and a ligand, L, is in fact usually a substitution reaction. For example, In aqueous solutions, metal ions will be present as aqua-ions, so the reaction for the formation of the first complex could be written as
[M(H2O)n] + L [M(H2O)n-1L] +H2O
The equilibrium constant for this reaction is given by
[L] should be read as "the concentration of L" and likewise for the other terms in square brackets. The expression can be greatly simplified by removing those terms which are constant. The number of water molecules attached to each metal ion is constant. In dilute solutions the concentration of water is effectively constant. The expression becomes
Following this simplification a general definition can be given, For the general equilibrium
pM + qL ... MpLq...
The definition can easily be extended to include any number of reagents. The reagents need not always be a metal and a ligand but can be any species which form a complex. Stability constants defined in this way, are association constants. This can lead to some confusion as pKa values are dissociation constants. In general purpose computer programs it is customary to define all constants as association constants. The relationship between the two types of constant is given in association and dissociation constants.
Stepwise and cumulative constants
A cumulative or overall constant, given the symbol , is the constant for the formation of a complex from reagents. For example, the cumulative constant for the formation of ML2 is given by
The stepwise constants, K1 and K2 refer to the formation of the complexes one step at a time.
It follows that
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 is sometimes found in the literature. It is best always to define each stability constant by reference to an equilibrium expression.
Hydrolysis products
The formation of an hydroxo-complex is a typical example of an hydrolysis reaction. An 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.
M + OH M(OH)
In water the concentration of hydroxide is related to the concentration of hydrogen ions by the self-ionization constant, Kw.
Kw=[H+][OH-]; [OH-] = Kw[H+]-1
The expression for hydroxide concentration is substituted into the formation constant expression
The literature usually gives value of *.
Acid-base complexes
Main article: acid-base equilibrium
A Lewis acid, A, and a Lewis base, B, can be considered to form a complex AB
There are three major theories relating to the strength of Lewis acids and bases and the interactions between them.
Hard and soft acid-base theory (HSAB). This is used mainly for qualitative purposes.
Drago and Wayland proposed a two-parameter equation which predicts the standard enthalpy of formation of a very large number of adducts quite accurately. (A) = EAEB + CACB. Values of the E and C parameters are available
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.
For more details see: acid-base reaction, acid catalysis, acid-base extraction
Thermodynamics
The thermodynamics of metal ion complex formation provides much significant information. In particular it is useful in distinguishing between enthalpic and entropic effects. Enthalpic effects depend on bond strengths and entropic effects have to do with changes in the order/disorder of the solution as a whole. The chelate effect, below, is best explained in terms of thermodynamics.
An equilibrium constant is related to the standard Gibbs free energy change for the reaction
G = -2.303 RT log10 .
R is the gas constant and T is the absolute temperature. At 25C G in kJmol1 = 5.708 log (1 kJmol1 = 1000 Joules per mole). Free energy is made up of an enthalpy term and an entropy term.
G = H TS
The standard enthalpy change can be determined by calorimetry or by using the van 't Hoff equation, though the calorimetric method is preferable. When both the standard enthalpy change and stability constant have been determined, the standard entropy change is easily calculated from the equation above.
The fact that stepwise formation constants of complexes of the type MLn decrease in magnitude as n increases may be partly explained in terms of the entropy factor. Take the case of the formation of octahedral complexes.
[M(H2O)mLn-1] +L [M(H2O)m-1Ln]
For the first step m=6, n=1 and the ligand can go into one of 6 sites. For the second step m=5 and the second ligand can go into one of only 5 sites. This means that there is more randomness in the first step than the second one; S is more positive, so G is more negative and log K1> log K2 . The ratio of the stepwise stability constants can be calculated on this basis, but experimental ratios are not exactly the same because H is not necessarily the same for each step. The entropy factor is also important in the chelate effect, below.
Ionic strength dependence
The thermodynamic equilibrium constant, K, for the equilibrium
M + L ML
can be defined as
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 () the definition could also be written as
where [ML] represents the concentration of ML and is a quotient of activity coefficients. This expression can be generalized as
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 can be assumed to be always constant. For example, the medium might be a solution of 0.1mol/dm3 sodium nitrate or 3mol/dm3 potassium perchlorate. When 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 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
R is the gas constant and T is the thermodynamic temperature . Thus, for exothermic reactions, (the standard enthalpy change, H, is negative) K decreases with temperature, but for endothermic reactions (H 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 fiveembered 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 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] =11[Cu][en]
[Cu(MeNH2)2]= 12[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 11 >> 12.
The difference between the two stability constants is mainly due to the difference in the standard entropy change, S. 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
G
H /kJ mol1
S /kJ mol1
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 in a series such as MLn (n=1, 2 ...) usually decrease as n increases. Exceptions to this rule occur when there is a change of geometry of the complex. The classic example is the formation of ammine complexes of silver(I) in aqueous solution.
Ag+ + NH3 [Ag(NH3)]+;
Ag(NH3)+ + NH3 [Ag(NH3)2]+;
In this case, K2>K1. The reason for this is that the ion written as Ag+ is in fact a tetrahedral, 4-coordinate species with four water molecules acting as ligands. In the first step one of the water molecules is replaced by an ammonia molecule and the complex [Ag(NH3)+ is also tetrahedral. Hovever in the second step the product is a linear, 2-coordinate ion. Examination of the thermodynamic data shows that both enthalpy and entropy effects determine the result.
equilibrium
H /kJmol1
S /JK1mol1
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 CoCl42- 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) anr 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.
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 increa
