Question

An electrochemical cell consists of a silver metal electrode immersed in a solution with [Ag \(^{+} ]=1.0 M\) separated by a porous disk from a copper metal electrode. If the copper electrode is placed in a solution of 5.0\(M \mathrm{NH}_{3}\) 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 \(25^{\circ} \mathrm{C}\) for the given electrochemical cell is 1.45 V. This is calculated using the Nernst equation, which involves determining the standard cell potential, calculating the reaction quotient (Q), and considering the temperature and number of electrons transferred in the reaction.

Step-by-step solution

Identify the half-reactions

We need to determine the half-reactions that occur at the two electrodes. For the silver electrode, we have: $$\mathrm{Ag^{+}(aq) + e^{-} \leftrightarrows Ag(s)}$$ For the copper electrode, we have: $$\mathrm{Cu^{2+}(aq) + 2e^{-} \leftrightarrows Cu(s)}$$

Determine the standard cell potential

In order to determine the cell potential for the given cell, we first need to determine the standard cell potential. The standard cell potential for the given cell can be calculated as follows: $$E_{cell}^{0} = E_{cathode}^{0} - E_{anode}^{0}$$ For silver half-cell reaction, $$E_{Ag}^{0} = 0.80 V$$ For copper half-cell reaction, $$E_{Cu}^{0} = 0.34 V$$ So, the standard cell potential is: $$E_{cell}^{0} = E_{Ag}^{0} - E_{Cu}^{0} = 0.80 V - 0.34 V = 0.46 V$$

Calculate the reaction quotient (Q)

Now, we need to calculate the reaction quotient (Q) as it is required for the Nernst equation. The reaction quotient for the given cell can be calculated using the concentrations of the species involved in the reaction. For the overall cell reaction, we have: $$\mathrm{Cu^{2+}(aq) + 2Ag^{+}(aq) \leftrightarrows Cu(s) + 2Ag(s)}$$ According to the balanced equation, the reaction quotient Q can be calculated as: $$Q = \frac{[\mathrm{Cu^{2+}}]}{[\mathrm{Ag^{+}}]^2}$$ Given, \([\mathrm{Ag^{+}}] = 1.0 M\) and \([\mathrm{Cu}\left(\mathrm{NH}_{3}\right)_{4}^{2+}] = 0.010 M\) The concentration of free \(\mathrm{Cu^{2+}}\) ions can be determined using the equilibrium constant: $$K = \frac{[\mathrm{Cu}\left(\mathrm{NH}_{3}\right)_{4}^{2+}]}{[\mathrm{Cu^{2+}}] [\mathrm{NH_3}]^4}$$ $$[\mathrm{Cu^{2+}}] = \frac{[\mathrm{Cu}\left(\mathrm{NH}_{3}\right)_{4}^{2+}]}{K [\mathrm{NH_3}]^4} = \frac{0.010 M}{1.0 \times 10^{13} (5.0 M)^4}$$ Now plug in these concentrations to calculate Q: $$Q = \frac{0.010/1.0\times10^{13}(5.0)^4}{(1.0)^2}=2.56\times10^{-16}$$

Apply the Nernst equation

Finally, we can use the Nernst equation to calculate the cell potential. The Nernst equation is given by: $$E_{cell} = E_{cell}^{0} - \frac{0.0592}{n}\log Q$$ Here, n = 2 (number of electrons transferred) and the temperature, T=25°C (or 298 K) Applying the values, we get $$E_{cell} = 0.46 - \frac{0.0592}{2}\log{2.56\times10^{-16}}$$ $$E_{cell} = 0.46 + 0.0296\times33.34 = 1.45 V$$ Therefore, the cell potential at \(25^{\circ} \mathrm{C}\) is 1.45 V.

Key concepts

Cell Potential

The cell potential, often referred to as the electromotive force (EMF) of an electrochemical cell, represents the ability of the cell to produce an electrical current. It is essentially the difference in potential energy between the two electrodes in a cell. When different metals, such as silver and copper, are used, they create a potential difference based on their tendency to lose or gain electrons.
In an electrochemical cell, the measured cell potential is influenced by several factors, including the concentration of ions, temperature, and the nature of electrodes. The potential helps to predict if a reaction will occur spontaneously. If the cell potential is positive, the reaction will proceed spontaneously as written. On the other hand, if it is negative, the reverse reaction is favored. Calculating this value accurately helps us understand and predict the cell's performance in practical applications.

Nernst Equation

The Nernst equation provides a link between the standard cell potential and the cell potential under non-standard conditions. It allows us to calculate the actual cell potential when concentrations are not at standard states. The equation is written as:
\[ E_{cell} = E_{cell}^{0} - \frac{0.0592}{n}\log Q \]

• Where \( E_{cell} \) is the cell potential at non-standard conditions.
• \( E_{cell}^{0} \) is the standard cell potential.
• \( n \) represents the number of electrons transferred in the reaction.
• \( Q \) is the reaction quotient, representing the ratio of the concentrations of the products to the reactants.

Using the Nernst equation, we can see how changes in concentration or temperature affect the cell's effectiveness. This equation is crucial when attempting to estimate the feasibility or direction of the electrochemical transformations in realistic conditions.

Standard Cell Potential

The standard cell potential, denoted as \( E_{cell}^{0} \), is measured under standardized conditions: 1 M ion concentrations, a pressure of 1 atm for gasses, and a temperature of 25°C (298 K).
For our electrochemical cell, which involves silver \( \mathrm{(Ag^+/Ag)} \) and copper \( \mathrm{(Cu^{2+}/Cu)} \) electrodes, their standard reduction potentials are \(0.80 \text{ V} \) and \(0.34 \text{ V} \), respectively. By applying the equation:
\[ E_{cell}^{0} = E_{cathode}^{0} - E_{anode}^{0} \] We find that the standard cell potential is calculated as \(0.46 \text{ V} \). This value is important because it serves as the benchmark against which other cell potentials are compared, indicating whether a cell can generate an electromotive force under specific conditions.

Reaction Quotient

The reaction quotient \( (Q) \) is a dimensionless number that provides a snapshot of the relative concentrations of reactants and products at any point in the reaction. Like equilibrium constants, it provides vital insight into reaction dynamics but under current, rather than equilibrium, conditions.
For the given cell, the reaction quotient \( Q \) is derived from the balanced redox reaction:

• \( \mathrm{Cu^{2+}(aq) + 2Ag^{+}(aq) \leftrightarrows Cu(s) + 2Ag(s)} \)

\[ Q = \frac{[\mathrm{Cu^{2+}}]}{[\mathrm{Ag^{+}}]^2} \]
Given the concentrations within the cell, especially \( [Ag^+] = 1.0 \text{ M} \) and \( [\text{Cu(NH}_3\text{)}_{4}^{2+}] = 0.010 \text{ M} \), the calculation of \( Q \) helps predict how far from equilibrium the reaction is and influences the cell potential when used in the Nernst equation. Understanding \( Q \) is crucial for interpreting real-time cell performance.