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Consider a buffer solution containing an acid, HA and its salt \({ Na }^{ + }\) and \({ A }^{ - }\). Ionization of weak acid may be represented as, $$ HA + { H }_{ 2 }O \rightleftharpoons { H }_{ 3 }{ O }^{ + } + { A }^{ - }\qquad ...(1) $$ The ionization constant is given by, $$ { K }_{ a } = \frac { \left[ { H }_{ 3 }{ O }^{ + } \right] \left[ { A }^{ - } \right] }{ \left[ HA \right] } \qquad ...(2) $$ or $$ \left[ { H }_{ 3 }{ O }^{ + } \right] = { K }_{ a }\frac { \left[ HA \right] }{ \left[ { A }^{ - } \right] } $$ lets take negative log, $$ -log \left[ { H }_{ 3 }{ O }^{ + } \right] = -log { K }_{ a } - log\frac { \left[ HA \right] }{ \left[ { A }^{ - } \right] } $$ $$ pH = p{ K }_{ a } + log\frac { \left[ { A }^{ - } \right] }{ \left[ HA \right] } $$ $$ or pH = p{ K }_{ a } + log\frac { \left[ Salt \right] }{ \left[ Acid \right] } \qquad ...(3) $$ This is handersons equation. Similarly for bases we can write,$$ pH = p{ K }_{ b } + log\frac { \left[ Salt \right] }{ \left[ Base \right] } \qquad ...(4) $$
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