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Equilibrium

Question
CBSEENCH11006578

Derive the relations:
straight K subscript straight h space equals space straight K subscript straight w over straight K subscript straight a
for a solution of a salt of weak acid and strong base.

Solution
Consider the hydrolysis of a simple salt of a weak acid with a strong base i.e. BA. The hydrolysis of BA may be represented as:
space space space space space space space space space BA with Salt below space plus space straight H subscript 2 straight O space space space leftwards harpoon over rightwards harpoon space space space space space BOH with Strong below space space plus space HA
or space space space space space straight B to the power of plus space plus straight A to the power of minus space plus space straight H subscript 2 straight O space space leftwards harpoon over rightwards harpoon space space space straight B to the power of plus space plus space OH to the power of minus space plus space HA
space space space space space Cancelling space the space common space ion comma
space space space space space space space space space straight A to the power of minus space plus space straight H subscript 2 straight O space space leftwards harpoon over rightwards harpoon space space space space stack space OH to the power of minus space plus space HA with left square bracket Anionic space hyrolysis right square bracket below
The hydrolysis constant Kh is given by
                              straight K subscript straight h space equals space fraction numerator open square brackets OH to the power of minus close square brackets space open square brackets OH close square brackets over denominator open square brackets straight A to the power of minus close square brackets end fraction space space space space space space space... left parenthesis 1 right parenthesis
For the weak acid HA, the dissociation equilibrium is
                                 HA space rightwards harpoon over leftwards harpoon space space space straight H to the power of plus space plus space straight A to the power of minus
therefore space space space space space space space straight K subscript straight a space equals space fraction numerator open square brackets straight H to the power of plus close square brackets space open square brackets straight H to the power of minus close square brackets over denominator open square brackets HA close square brackets end fraction space space space space space... left parenthesis 2 right parenthesis
where space straight K subscript straight a space is space the space dissociation space constant space of space HA.
Also space ionic space product space of space water space straight K subscript straight w space is space given space by
space space space space space straight K subscript straight w space equals space open square brackets straight H to the power of plus close square brackets space open square brackets OH to the power of minus close square brackets space space space space space space space space space space space space space... left parenthesis 3 right parenthesis
Dividing space left parenthesis 3 right parenthesis space by space left parenthesis 2 right parenthesis comma space we space get comma space
space
          straight K subscript straight w over straight K subscript straight a space equals space fraction numerator open square brackets OH to the power of minus close square brackets open square brackets HA close square brackets over denominator open square brackets straight A to the power of minus close square brackets end fraction space equals space straight K subscript straight h
therefore space space space space space space straight K subscript straight h space equals space straight K subscript straight w over straight K subscript straight a
It space is space clear space from space equation space left parenthesis 4 right parenthesis comma space that
space space space space space space straight K subscript straight h space space proportional to space space space 1 over straight K subscript straight a
i.e. Hydrolysis constant is inversely proportional to the dissociation constant Ka of the weak acid. Weaker the acid, the greater is the hydrolysis constant of the salt.