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# Complex Ion

Stability constants of metal complexes are defined as an equilibrium constant for the complex formation in the solution.

"Stability constants of metal complexes are also known as formation constant or binding constant."

Stability constant of metal complexes is the measure of the capacity of interaction between the metal and the ligand that together form a complex. Stability constant of metal complexes is usually denoted by β. Stability constant of metal complexes gives the information which is required to calculate the concentration of the complex in the solution.

The method for the determination of the stability constants of metal-amine complexes was first discovered by Jannik Bjerrum in 1941.

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## Define Complex Ion

If a coordination compound carries either a positive or negative charge, they are called complex ions:
Complex ions can be either positive or negative charges. On the basis of the charge, a complex ion can be of two types,
1. Cationic complex ion
2. Anionic complex ion
A complex carrying positive charge is known as cationic complex ion, while a complex ion with negative charge is called an anionic complex ion.

 Anionic Complex Ion Cationic Complex Ion [Fe(CN)6]- , [Cu(NH3)4]2+ [CuCl4]2-[Ag(CN)2]-, [Cu(NH3)4(H2O)2]2+ [Cu(H2O)6]3+ [Co(NH3)6]3+[Ni(NH3)6]2+

Coordination compounds are formed by the association of a metal atom or ion with ligands. Metal atoms or ions are bonded with coordination linkage. A Ligand can be an atom or a molecule or ion which can donate its lone pair of electrons to form a coordination bond with a metal atom or ion.

For example, water molecule is a ligand due to the presence of two lone pair of electrons on the oxygen atom of the molecule, which can be donated to form a coordination bond.

Some other examples of ligands are as follows,

 Formula Name Ligand name :NH3 Ammonia amine H2O Water aqua :Câ‰¡O: Carbon monoxide carbonyl :PH3 Phosphine phosphine :N=O Nitric Oxide nitrosyl NO3- Nitrate nitrato NH2- Amide amido C2O4-2 Oxalate oxalato 2HN-CH2-COO- Glycinate gly :NH2-CH2CH2-:NH2 Ethylehediamine en CO3-2 Carbonate carbonato O-2 Oxide oxo Cl- Chloride Chloro F- Fluoride fluoro :Câ‰¡N:- Cyanide cyano OH- Hydroxide hydroxo :NO2- Nitrite nitro(NO2) :NO2- Nitrite nitrito(ONO-) SO4-2 Sulphate sulfato SCN- Thiocynate thiocyanato S2O3-2 Thiosulfate thiosulfato C5H5N: Pyridine pyridine C5H5 Cyclopentadienyl Cyclopentadienyl H2NCSNH2 Thiourea tu

Generally, transition elements are involved in the formation of complex compounds because they have penultimate inner d-orbitals which accept electrons, form ligands, to form coordination bonds. For example,

## Stability Constants of Metal Complexes

The complex formation between a metal ion and a ligand is formed through a coordination bond and is considered to be a substitution reaction.

A reaction between metal and the complex is given as :

M + nL $\to$ MLn

The stability constant for the above metal complexes is given by

$\beta= \frac{[ML]}{[M][L]}$

Where [ML], [M] and [L] are the concentration of metal - ligand complex, metal and ligand respectively.

This is for a single metal and a single ligand.

For more than 1 metal and 1 ligand, the equation for the stability constant is given as:

pM + qL $\to$ MpLq

$\beta_{pq...}=\frac{M_pL_q...}{[M]^p[L]^q...}$

There are different methods to find out the stability constants, they are stepwise and cumulative constants, hydrolysis products and acid base complexes.

### Stepwise and cumulative constants

M + 2L $\to$ ML2

the stability constant is given as

ß= $\frac{[ML_2]}{[M][L]^2}$

For the above reaction, the step-wise stability constant can also be written, i.e., for the reaction between a single metal and a single ligand,

$M+L\rightleftharpoons ML; K_1=\frac{[ML]}{[M][L]}$

For the reaction between a metal and 2 ligands

$M+L\rightleftharpoons ML_2; K_2=\frac{[ML]}{[M][L]}$

From this we can conclude that ß = K1K2

Stability constant for hydrolysis products

It is an example for hydrolysis reaction and the stability constant for hydrolysis product is given by:

M+OH $\to$ M(OH)

K=$\frac{[M(OH)]}{[M][OH]}$

but $K_w=[H^+][OH^-] so [OH^-]=\frac{k_w}{H^-}$

After Simplification

K=$\frac{M(OH)}{[M]K_w[H]^-1}$

$\beta^*_{1-1}=\frac{K}{K_w}=\frac{M(OH)}{[M][H]^-1}$

### Stability constant for Acid-base complexes:

A + B $\to$ AB so K = $\frac{[AB]}{[A][B]}$

## Stability of Metal Complexes

• The chelate effect

• The macro cyclic effect

• Geometrical factors

• ## Stability Constant

A complex ion does not always dissociate into its individual components in aqueous solution. However the extent of dissociation is very small and it depends on the strength of the metal–ligand bond. For a strong metal – ligand the extent of dissociation will be very less and the complex ion will be stable. Hence stronger metal –ligand bond corresponds to more stable complex ion.

The stability of a complex can be measured by the amount of resistance shown to the process of replacement of one ligand by the other. The dissociation leads to the equilibrium concentration of undissociate and dissociate complex hence the stability of a complex can be expressed in terms of equilibrium constant of the dissociation equilibrium.

This equilibrium constant is called as instability constant or dissociation constant.
For example: Cationic complex ion [Cu(NH3)4]2+ in aqueous solution dissociates in following way,

[Cu(NH3)4]2+ $\rightleftharpoons$ Cu2+ + 4NH3

Dissociation constant or Instability constant, Kf = [Cu2+ ] [NH3]4
[Cu(NH3)4]2+

Since dissociation of a complex is a reversible reaction, hence the equilibrium constant for the backward reaction will be constant for the formation of a complex ion from their components.

Cu2+ + 4NH3$\rightleftharpoons$ [Cu(NH3)4]2+

Equilibrium constant for the formation of a complex ion from its components is called as stability constant. Stability constant is represented by ß , which is equal to

ß = $\frac{[Cu(NH_{3})_{4}]^{2+}} {[Cu^{2+}][NH_{3}]^{4}}$

Hence stability constant is reciprocal of the instability constant. Another term used for stability constant is formation constant or binding constant, as it is an equilibrium constant for the formation of a complex ion from its component. A general reaction of metal ion and ligand to form a complex ion will be,

Ma+ + nLx- $\rightleftharpoons$ MLnb+

Stability constant for a given reaction

ß = $\frac{[ML_{n}^{b+}]} {[M^{a+}] [ nL^{x-}]}$

Greater the magnitude of stability constant, more stable is the complex. For stronger ligands, the bond strength will be high and the magnitude of stability constant will be more.
For the formation of complex ion, ligand bonds with metal ion in a stepwise manner. There is a stepwise stability constant for each step. For example, formation of ML4 complex can be completed in given steps;
M + L $\to$ ML K1 = $\frac{ [ML]}{ [M] [L]}$

ML + L $\to$ ML2 K2 = $\frac{[ML2]}{[ML] [L] }$

ML2 + L $\to$ ML3 K3 = $\frac{[ML3]}{[ML2] [L] }$

ML3 + L $\to$ ML4 K4 = $\frac{[ML4]}{[ML3] [L] }$

where K1, K2, etc., are referred to as "stepwise stability constants".
Hence "Overall Stability Constant" will be

M + 4L $\to$ ML4 ß4 = $\frac{[ML_{4}]} {[M] [L]^{4}}$
The relation between the stepwise and overall stability constant is as follows:

ß4 =K1.K2.K3.K4
or
ßn =K1.K2.K3.K4--------------Kn

For example, the formation of Ag(NH3)2+, the reactions are:

Ag+(aq)+NH3(aq) $\Leftrightarrow$ Ag(NH3)+(aq) K1=2.1$\times$103
Ag(NH3)+(aq)+NH3(aq) $\Leftrightarrow$ Ag(NH3)+2 (aq) K2=8.1$\times$103
-----------------------------------------------------------------------------------------------------------------------
Ag+(aq)+2NH3(aq) $\Leftrightarrow$ Ag(NH3)+2 (aq) Kf=K1*K2=1.7$\times$107

Stability constant is affected by
• Stability constant is affected by the charge on the central metal ion. Greater the charge on the metal ion, greater is the stability of the complex ion.
• It is also affected by the nature of the metal atom or ion. Group 1, 2 and earlier members of the transition metals form stable complexes with those ligands which have N, O, and F as a donor atom. While transition metals form stable complex with those ligands which have a higher member of the N, O and F family.
• More basic nature of the ligand makes a stable complex.
• Formation of stable ch-elate ring increases the stability of complex as well as the value of the stability constant.

## Formation Constant Table

 Complex ion Stability Constant Value (mol-1 dm3) Log (Stability constant) [Cu(NH3)(H2O)5]2+ K1 1.78 x 104 4.25 [Cu(NH3)2(H2O)4]2+ K2 4.07 x 103 3.61 [Cu(NH3)3(H2O)3]2+ K3 9.55 x 102 2.98 [Cu(NH3)4(H2O)2]2+ K4 1.74 x 102 2.24 [Cu(NH3)4]2+ β 4.5 x 1011 11.653 [Cu(CN)4]2- β 2.0 x 1027 27.301 [Ag(NH3)2]+ β 1.6 x 107 7.204 [Co(NH3)6]3+ β 5.0 x 1033 33.6989 [Cu(en)2]2+ β 1.0 x 1020 20 [Ni(H2O)4(en)]2+ β 3.3 x 107 7.518 [Ni(en)3]2+ β 1.1 x 1018 18.041

## Conditional Formation Constant

In predicting the completeness of any reaction like complexation or precipitation, we need to account for all equilibrium involving M and L. Under specific conditions for a given system, the formation constant is called conditional Formation Constants (K').

The titration between metal ion like Ca2+ and EDTA (ethylenediammine) which is a hexadentate ligand, the formation constant at a certain pH value is termed as conditional formation constant.

The formation constant for the formation of Metal-EDTA complex will be;

Kf = $\frac{[MEDTA]}{[M][EDTA]}$

Let’s take metal ion as M+n and EDTA4- as Y4- .

If K’ =$\frac{[My^{n-4} ]}{[M^{+n}][Y^{4-} ]}$

Since [Y4- ] = αy-4 F(EDTA) -4 ; Where α is fraction of Y4-

Then we can combine these to get the equation

Kf = $\frac{[My^{ y-4} ]}{[M^{+n} ] α_{y-4} F(EDTA)^{-4}}$

Then rearrange to get

K f αy-4 = $\frac{[My^{ y-4} ]}{[M^{+n} ] F(EDTA)^{-4}}$

Thus we can calculate the conditional formation constant for complex formation of metal ion and hexadentate ligand, EDTA at any pH value.

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