Question

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 \(25^{\circ} \mathrm{C} ?\) $$\begin{aligned}\mathrm{Cu}^{2+}(a q)+4 \mathrm{NH}_{3}(a q) \rightleftharpoons \mathrm{Cu}\left(\mathrm{NH}_{3}\right)_{4}{ }^{2+}(a q) & K=1.0 \times 10^{13}\end{aligned}$$

Short answer

The cell potential at \(25^{\circ}C\) can be calculated using the Nernst equation. First, calculate the concentration of \(Cu^{2+}\) using the equilibrium constant expression: \([Cu^{2+}] = \frac{0.010}{1.0 \times 10^{13} \times (5.0)^4}\). The standard cell potential (\(E^\circ_{cell}\)) can be calculated as \(E^\circ_{cell} = E^\circ_{Cu} - E^\circ_{Ag}\). Then, calculate the reaction quotient (Q) using the given concentrations. Finally, find the overall cell potential using the Nernst equation: \(E_{cell} = E^\circ_{cell} - \frac{RT}{nF} \ln Q\). Plug in the values and evaluate to obtain the cell potential at \(25^{\circ}C\).

Step-by-step solution

Calculate the concentration of \(Cu^{2+}\) using the equilibrium constant expression for the formation of \(Cu(NH_3)_4^{2+}\)

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: \[ [Cu^{2+}] = \frac{[Cu(NH_{3})_4^{2+}]}{K \times [NH_{3}]^4} = \frac{0.010}{1.0 \times 10^{13} \times (5.0)^4} \]

Calculate the cell potential using the Nernst Equation

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 the cell potential

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.

Key concepts

Nernst Equation

The Nernst equation is a relation used in electrochemistry to determine the electrode potential of a cell under non-standard conditions. It relates the reduction potential of an electrochemical reaction to the standard electrode potential, temperature, activity, and concentration of the reactants and products involved.

When applying the Nernst equation to calculate the cell potential, as seen in the exercise, you must identify the standard cell potential (\(E^\text{o}_{\text{cell}}\)), temperature (\(T\)), the number of electrons transferred in the half-reactions (\(n\)), and the reaction quotient (\(Q\)). In the given step by step solution, the Nernst equation is rearranged and the values are substituted to find the cell potential for the electrochemical cell at 25°C.

Equilibrium Constant

An equilibrium constant (\(K\)) quantifies the ratio of concentrations of products to reactants at equilibrium in a chemical reaction. In the context of this exercise, the equilibrium concept applies to the chemical reaction where copper ions (\(Cu^{2+}\)) react with ammonia (\(NH_3\)) to form tetraamminecopper(II) complex ions (\(Cu(NH_3)_4^{2+}\)).

Understanding the equilibrium constant is crucial for calculating the concentrations of reactants or products at equilibrium. It's important to note that a high value of the equilibrium constant (\(K=1.0 \times 10^{13}\)) suggests the reaction strongly favors the formation of the complex ion, which is key information for solving the problem.

Electrode Reactions

Electrode reactions involve the transfer of electrons between a chemical species and an electrode. These reactions are the fundamental processes in electrochemical cells that generate an electromotive force (EMF). In the exercise provided, the electrode reactions would be the reduction of copper ions at the cathode and the oxidation of silver at the anode.

It's also essential to use the correct stoichiometry and understand the nature of the electrode reactions to determine the number of moles of electrons transferred (\(n\)), which directly feeds into the Nernst equation for the calculation of cell potential.

Standard Cell Potential

The standard cell potential (\(E^\text{o}_{\text{cell}}\)) is the potential difference between two half-cells at standard conditions, which typically include all solutes at 1M concentration, a pressure of 1 atm for gases, and a temperature of 25°C. It's the sum of the potentials for the half-reactions occurring at the anode and cathode.

In the provided problem, the standard cell potential is calculated by taking the difference between the standard reduction potentials for the cathode and anode reactions. It represents the driving force of the cell reaction and determines whether a cell will undergo spontaneous reaction under standard conditions. This potential is a reference point for using the Nernst equation to determine the actual cell potential under non-standard conditions.