A hydrogen electrode placed in a buffer solution of $C H_{3} C O O N a$ and $C H_{3} C O O H$ in the molar ratios x : y and y : x has oxidation potentials $E_{1}$ and $E_{2}$ volts respectively at 298 $K, (p_{H_{2}} = 1 a t m)$ . The $p K_{a}$ a value of acetic acid will be given by

$\frac{E_{1} - E_{2}}{0.118}$

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$\frac{E_{2} - E_{1}}{0.118}$

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$\frac{E_{1} + E_{2}}{0.118}$

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Solution

The correct option is C
$\frac{E_{1} + E_{2}}{0.118}$

$p H_{1} = p K_{a} + l o g \frac{x}{y} . . . . . . . . . . . . . . . . . . (1)$

$p H_{2} = p K_{a} + l o g \frac{y}{x} . . . . . . . . . . . . . . . . . . . (2)$

$O x i d a t i o n : H_{2} \to 2 H^{+} 2 e^{-}$

$E_{1} = E_{H_{2} / H^{+}}^{0} - \frac{0.059}{2} l o g [H^{+}]_{1}^{2} = 0.059 p H_{1}$

$E_{2} = 0.059 p H_{2}$

$\Rightarrow E_{1} + E_{2} = 0.059 (2 p K_{a})$

$\Rightarrow p K_{a} = \frac{E_{1} + E_{2}}{0.118}$

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A hydrogen electrode placed in a buffer solution of $C H_{3} C O O N a$ and $C H_{3} C O O H$ in the ratios $x : y$ and $y : x$ has electrode potential values $E_{1}$ volts and $E_{2}$ volts respectively at 25 $^{0}$ C . The $p K_{a}$ value of acetic acid is :