Question

$Derive space straight K subscript straight h space equals space fraction numerator straight K subscript straight w over denominator straight K subscript straight a space straight K subscript straight b end fraction for space straight a space solution space of space salt space of space weak space acid space and space weak space base.$

Solution

Suppose the salt of weak acid and weak base is BA and its hydrolysis is represented as:

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 BOH with weak below space plus space HA with weak below

or space space straight B to the power of plus space space plus straight A to the power of minus space plus space straight H subscript 2 straight O space space space rightwards harpoon over leftwards harpoon space space space space BOH space plus space HA

It involves both cationic as well as ionic hydrolysis.

Now space space space space space space space space straight K subscript straight h space equals space fraction numerator open square brackets BOH close square brackets space open square brackets HA close square brackets over denominator open square brackets straight B to the power of plus close square brackets space open square brackets straight A to the power of minus close square brackets end fraction space space space space space space space space space... left parenthesis 1 right parenthesis

Also for weak acid,

space space space 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 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 A 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

For weak base,

BOH space space rightwards harpoon over leftwards harpoon space space space space straight B to the power of plus space plus space OH to the power of minus straight K subscript straight b space equals space fraction numerator open square brackets straight B to the power of plus close square brackets space open square brackets OH to the power of minus close square brackets over denominator open square brackets BOH close square brackets end fraction space space space space space space space space space space space space space space space space... left parenthesis 3 right parenthesis

Also space space straight K subscript straight W space equals space open square brackets straight H close square brackets to the power of plus space open square brackets OH to the power of minus close square brackets space space space space space... left parenthesis 4 right parenthesis Multiplying space equations space left parenthesis 1 right parenthesis comma space left parenthesis 2 right parenthesis space and space left parenthesis 3 right parenthesis comma straight K subscript straight h space. space straight K subscript straight a space. space straight K subscript straight b 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 plus close square brackets space space space space space space space space space space space space space space space space space space equals space straight K subscript straight w space space space space space space space space space space space space space left square bracket From space eq. space left parenthesis 4 right parenthesis right square bracket space space therefore space space space space space 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 straight K subscript straight b end fraction space space space space space space space space space space space space space space space space space space space space

For Daily Free Study Material Join wiredfaculty