Relation Between Ka And Kb, Common Ion Effect
Weak acid is related is related to its conjugate base through equilibrium constant expression ka
HA +H2O ↔ H3O+ +A-

Ka [ H 3 O + ][ A - ] [HA]                                                                                                                         

Weak base is related is related to its conjugate acid through equilibrium constant expression kb
BOH + H2O ↔ B+ + OH-

Kb =   [B + ][ OH - ] [BOH]   

As the conjugated base pairs are related through the equilibrium constant we can assume that the equilibrium constant for an acid-base pair also related to each other.
Consider the dissociation of weak acid in water
HA +H2O ↔ H3O+ +A-

 Ka [ H 3 O + ][ A - ] [HA]     ...........(1)

HA being a weak acid it has strong conjugate base A-
The dissociation equilibrium of strong conjugate base in water is
A-  + H2O ↔ HA +OH-

 Kb = [HA][ A - ] [ A - ]       ...........(2)

Multiplying equation 1 & 2

Ka * Kb  =  [ H 3 O + ][ A - ] [HA] * [HA][ A - ] [ A - ]  
             = [H3O+ ][ OH-] = k w
 k w  = Ionic product of water
∴  ka * kb = k w

Common ion effect:  The common ion effect is defined as the shift in the position of an ionic equilibrium caused by the addition of a solute that provides an ion which is the part of equilibrium.
Consider the dissociation of acetic acid in water
The equilibrium is represented as 
CH3COOH (aq) ↔ CH3COO- +H3O+
To this solution add a small amount of sodium acetate.
dissociation of sodium acetate is represented as
CH3COONa → CH3COO- +Na+
Since the concentration of CH3COO-  ions increased in the solution ,according to Le Chatelier,s principle the equilibrium will be shifted to left.
Due to the presence of common ions CH3COO- the equilibrium will be shifted to the left side. And the dissociation of  CH3COOH is suppressed.

Summary

Weak acid is related is related to its conjugate base through equilibrium constant expression ka
HA +H2O ↔ H3O+ +A-

Ka [ H 3 O + ][ A - ] [HA]                                                                                                                         

Weak base is related is related to its conjugate acid through equilibrium constant expression kb
BOH + H2O ↔ B+ + OH-

Kb =   [B + ][ OH - ] [BOH]   

As the conjugated base pairs are related through the equilibrium constant we can assume that the equilibrium constant for an acid-base pair also related to each other.
Consider the dissociation of weak acid in water
HA +H2O ↔ H3O+ +A-

 Ka [ H 3 O + ][ A - ] [HA]     ...........(1)

HA being a weak acid it has strong conjugate base A-
The dissociation equilibrium of strong conjugate base in water is
A-  + H2O ↔ HA +OH-

 Kb = [HA][ A - ] [ A - ]       ...........(2)

Multiplying equation 1 & 2

Ka * Kb  =  [ H 3 O + ][ A - ] [HA] * [HA][ A - ] [ A - ]  
             = [H3O+ ][ OH-] = k w
 k w  = Ionic product of water
∴  ka * kb = k w

Common ion effect:  The common ion effect is defined as the shift in the position of an ionic equilibrium caused by the addition of a solute that provides an ion which is the part of equilibrium.
Consider the dissociation of acetic acid in water
The equilibrium is represented as 
CH3COOH (aq) ↔ CH3COO- +H3O+
To this solution add a small amount of sodium acetate.
dissociation of sodium acetate is represented as
CH3COONa → CH3COO- +Na+
Since the concentration of CH3COO-  ions increased in the solution ,according to Le Chatelier,s principle the equilibrium will be shifted to left.
Due to the presence of common ions CH3COO- the equilibrium will be shifted to the left side. And the dissociation of  CH3COOH is suppressed.

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