Explanation

(a)
Anode reaction ${\ H}_{2}\left( p_{1} \right) \rightarrow 2H^{+}$
Cathode reaction $\ 2H^{+} \rightarrow H_{2}\left( p_{2} \right)$
$E_{cathode} = - \frac{RT}{2F}\ ln\frac{p_{2}}{\left\lbrack H^{+} \right\rbrack^{2}}$
$E_{anode} = - \frac{RT}{2F}\ ln\frac{\left\lbrack H^{+} \right\rbrack^{2}}{p_{1}}$
$E_{cell} = E_{anode} + E_{cathode}$
$= - \frac{RT}{2F}\ln\frac{\left\lbrack H^{+} \right\rbrack^{2}}{p_{1}} - \frac{RT}{2F}\ln\frac{p_{2}}{\left\lbrack H^{+} \right\rbrack^{2}}$
$= - \frac{RT}{2F}\ln\frac{p_{2}}{p_{1}} = \frac{RT}{2F}\ln\frac{p_{1}}{p_{2}}$