Question

Consider the following galvanic cell: A 15.0 -mole sample of \(\mathrm{NH}_{3}\) is added to the Ag compartment (assume \(1.00 \mathrm{L}\) of total solution after the addition). The silver ion reacts with ammonia to form complex ions as shown: $$\begin{aligned} \mathrm{Ag}^{+}(a q)+\mathrm{NH}_{3}(a q) \rightleftharpoons \mathrm{AgNH}_{3}^{+}(a q) \\\& K_{1}=2.1 \times 10^{3} \\ \mathrm{AgNH}_{3}^{+}(a q)+\mathrm{NH}_{3}(a q) \rightleftharpoons \mathrm{Ag}\left(\mathrm{NH}_{3}\right)_{2}^{+}(a q) \\\\\ & K_{2}=8.2 \times 10^{3} \end{aligned}$$ Calculate the cell potential after the addition of 15.0 moles of \(\mathrm{NH}_{3}\)

Short answer

In summary, to calculate the cell potential after the addition of 15.0 moles of NH3 to a galvanic cell with the given reactions and equilibrium constants, we followed these steps: 1. Determine the initial concentrations of species. 2. Solve the equilibrium expressions for complex ion concentrations in terms of K1, K2, and initial concentrations of Ag+ and NH3. 3. Calculate the final concentrations of species using mass balance equations. 4. Use the Nernst equation with the obtained concentrations to find the cell potential after the addition of NH3.

Step-by-step solution

Calculate initial concentrations of species

: First, we need to determine the initial concentrations of Ag+, NH3, AgNH3+, and Ag(NH3)2+. Since we have 15.0 moles of NH3 added to a 1.00 L solution, the initial concentration of NH3 can be calculated as follows: Initial concentration of \(\mathrm{NH}_{3}=15.0 \ \mathrm{moles}/1.00\ \mathrm{L}=15.0 \ \mathrm{M}\) We need more data for other species concentrations. We can make some reasonable assumptions. At the beginning of the reaction, the concentration of complex ions (AgNH3+ and Ag(NH3)2+) should be 0, so they do not contribute to the cell potential. But we will need the initial concentration of Ag+ ions to be able to proceed.

Solve the equilibrium expressions for complex ion concentrations

: Next, we will use the equilibrium expressions (K1, K2) to find the concentrations of AgNH3+ and Ag(NH3)2+ after reaching equilibrium. Let's denote the equilibrium concentration of AgNH3+ as [AgNH3+] and Ag(NH3)2+ as [Ag(NH3)2+]. From the problem, we have K1 = 2.1 × 10³ and K2 = 8.2 × 10³. We can express them as follows: $$K_{1}=\frac{[\mathrm{AgNH}_{3}^{+}]}{[\mathrm{Ag}^{+}][\mathrm{NH}_{3}]}$$ $$K_{2}=\frac{[\mathrm{Ag}\left(\mathrm{NH}_{3}\right)_{2}^{+}]}{[\mathrm{AgNH}_{3}^{+}][\mathrm{NH}_{3}]}$$ Now, we can solve these equations for [AgNH3+] and [Ag(NH3)2+] in terms of K1, K2 and the initial concentrations of Ag+ and NH3.

Calculate the final concentrations of species

: Now that we have the equilibrium concentrations of AgNH3+ and Ag(NH3)2+, we can use mass action principle to find new concentrations [Ag+] and [NH3]. To achieve that, we will write two mass balance equations for Ag and NH3: $$[\mathrm{Ag}+] + [\mathrm{AgNH}_3^+] + [\mathrm{Ag}\left(\mathrm{NH}_3\right)_2^+] = \mathrm{initial} \ [\mathrm{Ag}+]$$ $$[\mathrm{NH}_3] - [\mathrm{AgNH}_3^+] - 2 [\mathrm{Ag}\left(\mathrm{NH}_3\right)_2^+] = 15.0$$ Now, we have all 2 equations needed to find the final concentrations of all species.

Calculate the cell potential using the Nernst equation

: Now that we have the concentrations of all species after the reaction, we can use the Nernst equation to calculate the cell potential. The Nernst equation for this problem can be expressed as: $$E_{cell} = E^0_{cell} - \frac{RT}{nF} \ln \left(\frac{[\mathrm{Ag}\left(\mathrm{NH}_{3}\right)_2^+]}{[\mathrm{Ag^{+}}]}\right)$$ Where \(E_{cell}\) is the cell potential after adding 15.0 moles of NH3, \(E^0_{cell}\) is the standard cell potential, R is the gas constant (8.314 J/mol K), T is the temperature in Kelvin, n is the number of moles of electrons transferred in the reaction (in this case, n=1 for the formation of AgNH3+ and Ag(NH3)2+), and F is the Faraday constant (96,485 C/mol). We plug the concentrations obtained from steps 2 and 3 into this Nernst equation and solve for the cell potential after the addition of NH3.

Key concepts

Equilibrium Constant

Understanding the equilibrium constant, denoted as K, is crucial for anyone studying chemical reactions. It's a numerical value that represents the ratio of the concentrations of products to reactants at equilibrium, each raised to the power of their respective stoichiometric coefficients. In the context of galvanic cell potential calculations, these constants give us insights into the extent of complex ion formation. Larger values of K indicate more product formation, which, in turn, affects the cell potential.

In the exercise, we're given equilibrium constants for the stepwise formation of silver-ammonia complex ions: K1 for \( \mathrm{Ag}^{+}(aq) + \mathrm{NH}_{3}(aq) \rightleftharpoons \mathrm{AgNH}_{3}^{+}(aq) \) and K2 for \( \mathrm{AgNH}_{3}^{+}(aq) + \mathrm{NH}_{3}(aq) \rightleftharpoons \mathrm{Ag}(\mathrm{NH}_{3})_{2}^{+}(aq) \). These constants are used to calculate the equilibrium concentrations of complex ions, which are vital for the subsequent steps involving the Nernst equation to determine cell potential.

Complex Ion Formation

Complex ions are species formed when central metal ions bond with molecules or ions called ligands; ammonia (\(\mathrm{NH}_3\)) is a common example of a ligand. In our exercise, silver ions (\(\mathrm{Ag}^+\)) react with ammonia to form silver-ammonia complex ions. The formation of complex ions is significant because it alters the concentrations of species in solution, influencing the cell's electromotive force (EMF).

Complex ion formation is often a stepwise process, as observed in this exercise with the formation of \(\mathrm{AgNH}_3^+\) and then \(\mathrm{Ag}(\mathrm{NH}_{3})_{2}^{+}\), each with its own equilibrium constant. By understanding these stepwise reactions, students can systematically calculate the effects on equilibrium and thus on the cell potential.

Nernst Equation

The Nernst equation is a fundamental tool in electrochemistry for correlating the concentrations of reactants and products to the potential of an electrochemical cell. It modifies the standard cell potential (\(E^0_{cell}\)) by accounting for non-standard conditions, thereby enabling us to determine the actual cell potential (\(E_{cell}\)).

The equation accounts for temperature (T), the gas constant (R), the Faraday constant (F), the number of moles of electrons transferred in the reaction (n), and the concentrations or partial pressures of the reactants and products. In galvanic cells, the Nernst equation helps predict how the potential will change with varying reactant and product concentrations—essential for solving the given exercise. It's an indispensable equation for students to master as it bridges the gap between theoretical potential and actual observable potential under given conditions.

Concentration Calculation

Calculating concentrations is a key skill in not only chemistry but also in understanding electrochemical reactions. Accurate concentration values are needed to apply the Nernst equation effectively. In the sample problem, after the addition of ammonia to the silver ion solution, we need to know the new concentrations of the ions in question to calculate the cell potential.

For that, we use the initial molar amount of added substances, the volume of the solution, and the concept of conservation of mass to establish the mass balance equations. This calculation is a two-fold process involving both the initial concentration input and the changes brought about by reaction equilibria. Concentration calculations require careful accounting for all species in the reaction, including those forming complex ions, and the understanding that the total amount of each species remains constant, although they may be distributed differently between various forms at equilibrium.