# Stepwise and Overall Stability Constants

## Stepwise and Overall Formation or Stability Constants

Coordination compounds are assumed to be formed from their constituents in aqueous solution by stepwise replacement of coordinated water molecules by ligand molecules or ions. The extent to which a metal cation combines with ligands to form a complex ion is expressed in terms of**stability constant or formation constant**.

According to Bjerrum, the formation of complex (i.e. ML

_{n}) proceeds by stepwise addition of the ligand to the metal ion. Each step is characterized by its individual equilibrium constant called stepwise formation constant or stepwise stability constant.

For the formation of ML

_{n}complex from an aquated metal cation [M(H

_{2})

_{n}]

^{+n}and monodentate ligand L, there will be 'n' consecutive steps and 'n' stepwise formation constants.

where- K

_{1}, K

_{2}, K

_{3}...K

_{n}are the stepwise formation constants or stepwise stability constants, and β

_{n}is the overall formation constant or overall stability constant in which reaction takes place in a single step.

The formation of complexes ML, ML

_{2}, ML

_{3}...ML

_{n}may be exprssed by the following steps and equilibrium constants β

_{1}, β

_{2}, β

_{3}...β

_{n}respectively.

where- β

_{1}, β

_{2}, β

_{3}...β

_{n}are the equilibrium constant called overall formation constant or overall stability constant.

*Higher the value of stability constant for a complex ion, the greater will be its stability*.

Inverse if the formation onstant (K

_{f}= 1/k) is called dissociation constant (K

_{d}) or instability constant of the complex.

Stepwise and overall (or cumulative) stability constants are also expressed as-

log

_{10}K

_{1}, log

_{10}K

_{2}, log

_{10}K

_{3}...log

_{10}K

_{n}and log

_{10}β

_{n}respectively.

## Relationship Between Stepwise and Overall Stability Constants

Cosider the expression for β_{3}-

Eq.-5 indicates that the overall constant (β) is equal to the product of the stepwise formation constants K

_{1}, K

_{2}, K

_{3}, ... K

_{n}. It also indicates that the formation of a complex takes place in various stepwise equilibria.

## Trends in K Value

The K values generally decrease gradually with increasing substitution of water by ligand L. This gradual decrease in K values is mainly due to three factors, namely, the statistical factor, steric factor and the electrostatic factor.**1. Statistical Factor**

As the coordinated water molecules are replaced by ligand molecules or ions (L), the number of water molecules in the complex formed decreases. Hence the probability of replacing water molecules also decreases. As a result K values decrease gradually.

**2. Steric Factor**

This arises only when the incoming ligands are bulkier in size than the coordinated water molecules. As the small sized water molecules are replaced by bulkier ligands (L), the steric crowding around the central metal ion increases. As a result of these steric repulsions the subsequent steps are retarded. Hence the K values gradually decrease.

**3. Electrostatic Factor**

In the first step of complex formation, one ligand (L) replaces one coordinated water molecule to give [M(H

_{2}O)

_{n-1L}]. In the second step, another ligand (L) of the same charge approaches the first stage product. Now there is an electrostatic repulsion between the incoming ligand and a similar ligand already present in the complex. As a result of this electrostatic repulsion between the ligands, the subsequent steps are retarded. Hence K values decrease gradually.

Therefore, the general trend in K values is

K

_{1}> K

_{2}> K

_{3}... > K

_{n}.

Any anomaly in the trend of K value suggests a major change in the structure.

Generally, aqua complexes are octahedral (six coordinated) whereas the halo complexes are tetrahedral(four coordinated).

Example-

In Cd

^{+2}/Br

^{−}system, the reaction of fourth Br

^{−}group and the complex with three Br

^{−}group is-

β[Cdbr

_{3}(H

_{2})

_{3}]

^{−}+ Br

^{−}→ [CdBr

_{4}]

^{−2}+ 3H

_{2}

Octahedral Tetrahedral

1 + 1 = 2 Particles 1 + 3 = 4 Particles

This reaction is accompanied by an increase in the number of particles and hence it is entropically favoured. This results in an increase in K

_{4}value.

## Factors Affecting the Stability of Complex Compounds

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