Question

Calculate Ecell expression

$Pt_{(s)}|H_{2(g)}|ACOOH_{(aq)}||ACOOH_{(aq)}|H_{2(g)}|Pt_{(s)}$ 1 atm $C_{1} K_{a_{1}}$ $C_{2} K_{a_{2}}$ 1 atm

$ACOOH \rightleftharpoons H^+ + ACO\Theta$

$C_{1}$ $C_{1}-C_{1}\alpha_{1}$ $C_{1}\alpha$ $C_{1}\alpha_{1}$

Anode: $(H_{2})_{a(g)} \rightarrow 2H_{a}^{\oplus} + 2e^{\ominus}$ Cathode: $2H_{c}^{\oplus} + 2e^{\ominus} \rightarrow (H_{2})_{c(g)}$

Answer 1

E_{cell} = 0.0591 \log_{10} \left( \frac{-K_{a2} + \sqrt{K_{a2}^2 + 4K_{a2} C_2}}{-K_{a1} + \sqrt{K_{a1}^2 + 4K_{a1} C_1}} \right)

Answer 2

The cell is a concentration cell with identical hydrogen electrodes but different electrolyte concentrations. $E_{cell}^\circ = 0$. The Nernst equation for this cell simplifies to $E_{cell} = 0.0591 \log_{10} \left( \frac{[H^+_{cathode}]}{[H^+_{anode}]} \right)$. For a weak acid $ACOOH$, the hydrogen ion concentration $[H^+]$ is determined by solving the equilibrium expression $K_a = \frac{[H^+]^2}{C-[H^+]}$, which yields $[H^+] = \frac{-K_a + \sqrt{K_a^2 + 4K_a C}}{2}$. Substituting these exact expressions for $[H^+]$ in the anode ($C_1, K_{a1}$) and cathode ($C_2, K_{a2}$) compartments into the Nernst equation gives the final Ecell expression.