Question

Consider the reaction $$ 2 \mathrm{Cu}^{2+}(a q)+\mathrm{Sn}^{2+}(a q) \longrightarrow \mathrm{Sn}^{4+}(a q)+2 \mathrm{Cu}^{+}(a q) $$ At what concentration of \(\mathrm{Cu}^{2+}\) is the voltage zero, if all other species are at \(0.200 \mathrm{M?}\)

Short answer

\(2\mathrm{Cu}^{2+}(aq) + \mathrm{Sn}^{2+}(aq) \longrightarrow 2\mathrm{Cu}^{+}(aq) + \mathrm{Sn}^{4+}(aq)\) Answer: The concentration of Cu²⁺ required for the cell voltage to be zero is 0.200 M.

Step-by-step solution

Identify the half-reactions

We can split the given redox reaction into two half-reactions: Oxidation half-reaction: \(\displaystyle \mathrm{Sn}^{2+}( aq) \longrightarrow \mathrm{Sn}^{4+}( aq) +2 e^{-}\) Reduction half-reaction: \(\displaystyle 2 \mathrm{Cu}^{2+}( aq) +2 e^{-}\longrightarrow 2 \mathrm{Cu}^{+}( aq)\)

Determine the standard reduction potentials

For the half-reactions, we can lookup the standard reduction potentials in the table of standard electrode potentials at 25°C: E° (Sn⁴⁺/Sn²⁺) = +0.15 V E° (Cu²⁺/Cu⁺) = +0.16 V

Calculate the standard cell potential

By subtracting the more positive value from the less positive one, we can obtain the standard cell potential (E° Cell) for the overall reaction: E° Cell = E° (Cu²⁺/Cu⁺) - E° (Sn⁴⁺/Sn²⁺) = 0.16 - 0.15 = 0.01V

Use the Nernst equation

We'll use the Nernst equation to find the concentration of Cu²⁺ at which the cell voltage is zero. The Nernst equation is given by: E = E° - \dfrac{0.0592}{n}logQ where E is the cell voltage, n is the number of electrons transferred (in this case, n=2), and Q is the reaction quotient. We are looking for the situation where E = 0, so we can set up the equation like this: 0 = E° - \dfrac{0.0592}{2}logQ

Set Q and find the concentration

During the reaction, the concentration of Cu²⁺ is unknown (x), while the concentration of Sn²⁺ is 0.200M. We can set the reaction quotient: Q = \dfrac{[\mathrm{Sn}^{4+}][\mathrm{Cu}^{+}]^2}{[\mathrm{Cu}^{2+}]^2[\mathrm{Sn}^{2+}]}=\dfrac{x(0.2)^2}{(0.2)(x)^2} Now, we need to plug the value of Q and E° into the Nernst equation and solve for x: 0 = 0.01 - \dfrac{0.0592}{2}log \dfrac{x(0.2)^2}{(0.2)(x)^2} Solving for x, we find that the concentration of Cu²⁺ is 0.200M for the cell voltage to be zero.