An electrochemical cell consists of a silver metal electrode im. mersed in a solution with \(\left[\mathrm{Ag}^{+}\right]=1.0 M\) separated by a porous disk from a copper metal electrode. If the copper electrode is placed in a solution of \(5.0 \mathrm{M} \mathrm{NH}_{3}\) that is also \(0.010 \mathrm{M}\) in \(\mathrm{Cu}\left(\mathrm{NH}_{3}\right)_{4}^{2+}\), what is the cell potential at \(25^{\circ} \mathrm{C} ?\) $$\begin{aligned}\mathrm{Cu}^{2+}(a q)+4 \mathrm{NH}_{3}(a q) \rightleftharpoons \mathrm{Cu}\left(\mathrm{NH}_{3}\right)_{4}{ }^{2+}(a q) & K=1.0 \times 10^{13}\end{aligned}$$

The equilibrium expression is given by: \[ K = \frac{[Cu(NH_{3})_4^{2+}]}{[Cu^{2+}] \times [NH_{3}]^4} \] Rearrange the equation to solve for \(Cu^{2+}\) concentration and substitute the given values: \[ [Cu^{2+}] = \frac{[Cu(NH_{3})_4^{2+}]}{K \times [NH_{3}]^4} = \frac{0.010}{1.0 \times 10^{13} \times (5.0)^4} \]

The Nernst equation can be used to calculate the cell potential: \(E_{cell} = E^\circ_{cell} - \frac{RT}{nF} \ln Q\) Where: \(E_{cell}\) = cell potential \(E^\circ_{cell}\) = standard cell potential R = gas constant (8.314 J/mol K) T = temperature in Kelvin (298 K, as 25°C = 298 K) n = number of electrons transferred (2, as both \(Cu\) and \(Ag\) have a valency of 2) F = Faraday constant (96485 C/mol) Q = reaction quotient Standard cell potentials (\(E^\circ_{cell}\)) for the respective half-reactions are: 1. \(Cu^{2+} + 2e^- \rightarrow Cu(s)\) : \(E^\circ_{Cu} = +0.34~V\) 2. \(Ag^+ + e^- \rightarrow Ag(s)\) : \(E^\circ_{Ag} = +0.80~V\) The overall standard cell potential (\(E^\circ_{cell}\)) is given by: \(E^\circ_{cell} = E^\circ_{Cu} - E^\circ_{Ag}\) Calculate Q using the given concentrations: \[ Q = \frac{[Ag^+][Cu^{2+}]}{[Cu(NH_{3})_4^{2+}]} \] Now, find the overall cell potential using the Nernst equation.

Calculate using the formula: \(E_{cell} = E^\circ_{cell} - \frac{RT}{nF} \ln Q\) Note that: n = 2 Temperature, T = 298 K R = 8.314 J/mol K F = 96485 C/mol Plug in the values and evaluate. This will give you the cell potential at 25°C.