What is the [H⁺] in a concentration cell at 25°C if the cell potential is 0.0451 V and cell diagram: Pt ∣ H₂ (g, 1 atm) ∣ H⁺ (?) ‖ H⁺ (1 M) ∣ H₂ (g, 1 atm)
2 H⁺(aq) + 2 e- → H₂(g) E° = 0. V

Question

Quizplus · Accepted Answer

Since this is a concentration cell, the half-reactions in the anode and cathode are the same, but the concentrations are different. We can use the Nernst equation to relate the cell potential ($E_{cell}$) to the concentrations of the species involved: $E_{cell} = E_{cell}^{\circ} - \frac{RT}{nF} \ln Q$ where $E_{cell}^{\circ}$ is the standard cell potential ($0.0$ V in this case because the half-reactions are the same), $R$ is the gas constant, $T$ is the temperature in Kelvin, $n$ is the number of moles of electrons transferred (in this case, 2), $F$ is Faraday's constant, and $Q$ is the reaction quotient. For this concentration cell, we can write the reaction quotient as: $Q = \frac{[ ext{H^{}^+][ ext{}}]}{[ ext{H₂}]^2}$ Note that the concentrations of the gases are not included in the reaction quotient because they are at 1 atm and do not change between the anode and cathode. Plugging in the values given in the problem, we have: $0.0451 ext{ V} = 0 - \frac{(8.31 ext{ J/mol K})(298 ext{ K})}{(2)(96,485 ext{ C/mol})} \ln \left( \frac{[ ext{H^{}^+][ ext{}}]}{[ ext{H₂}]^2} ight)$ Simplifying, we get: $\ln \left( \frac{[ ext{H^{}^+][ ext{}}]}{[ ext{H₂}]^2} ight) = -0.000475$ $\frac{[ ext{H^{}^+][ ext{}}]}{[ ext{H₂}]^2} = \exp(-0.000475) = 0.99953$ Since the concentration of $[ ext{H₂}]$ is the same on both sides of the cell, we can simplify the expression to: $[ ext{H^{}^+][ ext{}}] = [ ext{H₂}]^2 imes 0.99953$ We are looking for the concentration of $[ ext{H^{}^+]$, so we need to solve for that variable. We know that $[ ext{}}] = [ ext{H₂}] = 1 ext{ M}$. Substituting in these values, we get: $[ ext{H^{}^+] = \frac{[ ext{H₂}]^2 imes 0.99953}{[ ext{}}]} = \boxed{0.030 ext{ M}}$ So the answer is choice (C), 0.030 M.

What Is the [H+] in a Concentration Cell at 25°C

What Is the [H⁺] in a Concentration Cell at 25°C