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Equilibrium

Question
CBSEENCH11006575

Show that for an aqueous solution of a salt of weak base and strong acid straight alpha space equals space square root of straight K subscript straight w over straight K subscript straight b. straight c end subscript end root where straight alpha is the degree of hydrolysis. 

Solution
Let the salt of weak base and strong acid be XY and hydrolysis reaction represented as:
straight X to the power of plus space plus space straight H subscript 2 straight O space space space rightwards harpoon over leftwards harpoon space space space space stack XOH space plus space space straight H to the power of plus with left square bracket Cationic space hydrolysis right square bracket below
Let c be the initial concentration of the salt and a be the degree of hydrolysis. At equilibrium, the concentration of various species is represented as:
                          X +   H2O        rightwards harpoon over leftwards harpoon       XOH   +     H+
Initial conc.                     c                        0              0
Equilibrium
conc.                            straight c left parenthesis 1 minus straight alpha right parenthesis               cα             cα
                   straight K subscript straight h space plus space fraction numerator cα space cross times space cα over denominator straight c left parenthesis 1 minus straight alpha right parenthesis end fraction space equals space fraction numerator cα squared over denominator left parenthesis 1 minus straight alpha right parenthesis end fraction
Since straight alpha is very small as compared to unity. 
   therefore space space space space space space space space 1 space minus space straight alpha space equals space space 1
therefore space space space space space space space space space space space space straight K subscript straight h space equals space cα squared
or space space space space space space space space space space space space space space straight alpha squared space equals space straight K subscript straight h over straight c
or space space space space space space space space space space space space space space straight alpha space equals space square root of straight K subscript straight h over straight c end root
But space space space straight K subscript straight h space space equals space straight K subscript straight w over straight K subscript straight b
therefore space space space space space space space space space space straight alpha space equals space square root of fraction numerator straight K subscript straight w over denominator straight K subscript straight b space cross times space straight c end fraction end root space space space