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Equilibrium

Question
CBSEENCH11006583

Show that degree of hydrolysis of a salt of weak acid and weak base is independent of concentration of the solution.
Or
Derive the relation,  straight alpha space equals square root of fraction numerator straight K subscript straight w over denominator straight K subscript straight a. space straight K subscript straight b space end fraction end root in case of hydrolysis of a salt of weak acid and weak base.

Solution
Let us consider the example of ammonium acetate.
Let c be the initial concentration of ammonium acetate in moles per litres and a be the degree of hydrolysis. The equilibrium concentration of various species are:
                         CH subscript 3 COO to the power of minus space plus space NH subscript 4 superscript plus space plus space straight H subscript 2 straight O space space space rightwards harpoon over leftwards harpoon space space space space space CH subscript 3 COOH space plus space space NH subscript 4 OH
Initial conc.               c             c                                  0                    0
Equilibriumstraight c left parenthesis 1 minus straight alpha right parenthesis space straight c left parenthesis 1 minus straight alpha right parenthesis                                         cα                  cα
conc. 
                     straight K subscript straight h space equals space fraction numerator cα space cross times space cα over denominator straight c left parenthesis 1 minus straight alpha right parenthesis space cross times space straight c left parenthesis 1 minus straight alpha right parenthesis end fraction space equals space fraction numerator straight alpha squared over denominator left parenthesis 1 minus straight alpha squared right parenthesis end fraction
Since straight alpha is very small as compared to unity,
therefore space space space 1 minus space straight alpha space equals space 1
therefore space space space space space straight K subscript straight h space equals space straight alpha squared
or space space space space space straight alpha space equals space square root of straight K subscript straight h end root
AS space space straight K subscript straight h space equals space fraction numerator straight K subscript straight w over denominator straight K subscript straight a space cross times space straight K subscript straight b end fraction
therefore space space space space straight alpha space equals space space square root of fraction numerator straight K subscript straight w over denominator straight K subscript straight a space cross times space straight K subscript straight b end fraction end root
Hence degree of hydrolysis (R) is independent of the concentration of the solution.