Calculate the $p H$ of the following half-cells solutions:
(a) $P t H_{2} (1 a t m) | H^{+} (H C l); E = 0.25 v o l t$
(b) $P t H_{2} (1 a t m) | H^{+} (H_{2} S O_{4}); E = 0.3 v o l t$

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Solution

(a) $H_{2} \to 2 H^{+} + 2 e$
$E_{H_{2} / H^{+}} = E_{H_{2} / H^{+}}^{\circ} - \frac{0.0591}{2} log \frac{[H^{+}]^{2}}{P_{H_{2}}}$
$= 0.0 - 0.0591 log \frac{[H^{+}]}{1}$
$0.25 = 0.0591 p H$
$p H = \frac{0.25}{0.0591} = 4.23$
(b) $E_{H_{2} / H^{+}} = E_{H_{2} / H^{+}}^{\circ} - \frac{0.0591}{2} log \frac{[H^{+}]^{2}}{P_{H_{2}}}$
$= 0.0 - 0.591 log \frac{[H^{+}]}{1}$
$0.3 = 0.0591 p H$
$p H = \frac{0.3}{0.0591} = 5.076$ .

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