A) \[pH=1/2\left[ p{{K}_{w}}+p{{K}_{a}}+\log C \right]\]
B) \[pH=1/2\left[ p{{K}_{w}}+p{{K}_{a}}-\log C \right]\]
C) \[pH=1/2\left[ p{{K}_{w}}+p{{K}_{b}}-\log C \right]\]
D) None of these
Correct Answer: A
Solution :
[a] \[C{{H}_{3}}CO{{O}^{-}}+{{H}_{2}}O\rightleftharpoons C{{H}_{3}}COOH+O{{H}^{-}}\] \[\therefore [O{{H}^{-}}]=C.h=C\sqrt{\frac{{{K}_{H}}}{C}}=\sqrt{{{K}_{H}}.C}=\sqrt{\frac{{{K}_{w}}}{{{K}_{a}}}C}\] Or \[-\log OH=-\frac{1}{2}[log{{K}_{w}}+logC-log{{K}_{a}}]\] Or \[pOH=\frac{1}{2}[p{{K}_{w}}-logC-p{{K}_{a}}]\] Now \[pH+pOH=p{{K}_{w}}\] \[\therefore pH=\frac{1}{2}[p{{K}_{w}}+logC+p{{K}_{a}}]\]
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