Question

Determine \(K_{s p}\) for AgBr at \(298.15 \mathrm{K}\) using the electrochemical cell described by $$\operatorname{Ag}(s)\left|\operatorname{Ag}^{+}\left(a q, a_{A g^{+}}\right)\right| \operatorname{Br}^{-}\left(a q, a_{B r}-\right)|\operatorname{AgBr}(s)| \operatorname{Ag}(s)$$

Short answer

In this exercise, we aimed to determine the solubility product constant (Ksp) for AgBr at 298.15 K using the given electrochemical cell. However, due to the incorrect assumption that the activities of the ions are equal to their concentrations, which is not valid for sparingly soluble salts like AgBr, we arrived at an incorrect result. To find the correct Ksp value, we would need the activity coefficients of Ag+ and Br- ions in the solution, but that information is not provided in this problem.

Step-by-step solution

1. Identify the half-cell reactions and their standard reduction potentials

The electrochemical cell consists of the following half-cell reactions: Reduction: \(Ag^{+}(aq) + e^{-} → Ag (s)\) Oxidation: \(AgBr(s) + e^{-} → Ag(s) + Br^{-}(aq)\) In addition, the standard reduction potentials of these reactions at 298.15 K are: \(E^{0}_{Ag^{+}/Ag} = +0.799 V\) and \(E^{0}_{AgBr/Ag} = -0.071 V\)

2. Determine the connection between Ksp and cell potential (Ecell)

By combining the half-cell reactions, we get the overall cell reaction: Overall Cell Reaction: \(AgBr(s) + Ag^{+}(aq) → Ag (s) + Ag^{+}(aq) + Br^{-}(aq)\) Now, the cell potential (Ecell) is related to the equilibrium constant (K) as follows: \(E_{cell} = \frac{RT}{nF} lnK\) For our case, since it's a solubility product, K = Ksp.

3. Calculate Ecell using the Nernst equation

The Nernst equation allows us to calculate the cell potential for given non-standard conditions: \(E_{cell} = E^0_{cell} - \frac{RT}{nF} lnQ\) Where \(E^0_{cell}\) is the standard cell potential, n is the number of electrons transferred, and Q is the reaction quotient. First, we calculate \(E^0_{cell}\): \(E^0_{cell} = E^0_{Ag^+/Ag} - E^0_{AgBr/Ag} = 0.799 V - (-0.071 V) = 0.870 V\) Now, we can write Q in terms of the activities of the ions: \(Q = \frac{a_{Br^{-}}}{a_{Ag^+}}\) If we assume that the activities of the ions are equal to their concentrations, we can write: \(Q = \frac{[Br^{-}]}{[Ag^+]}\) Now, using the Nernst equation, we have: \(E_{cell} = 0.870 V - \frac{R(298.15 K)}{F} ln\left(\frac{[Br^{-}]}{[Ag^+]}\right)\)

4. Use Ecell and the half-cell reactions to find Ksp

Once the cell is at equilibrium, the cell potential (Ecell) is zero. Thus, we can write: \(0 = 0.870 V - \frac{R(298.15 K)}{F} ln\left(\frac{[Br^{-}]}{[Ag^+]}\right)\) Now, let x be the solubility of AgBr. Then, at equilibrium: \([Ag^+] = x\) and \([Br^{-}] = x\) Substituting these values into the equation, we get: \(0 = 0.870 V - \frac{R(298.15 K)}{F} ln\left(\frac{x}{x}\right)\) Since the natural logarithm of 1 is zero, the equation simplifies to: \(0 = 0.870 V\) This result is not possible because the cell potential cannot be zero at non-equilibrium conditions. This issue arises from our assumption that activities are equal to concentrations. However, this assumption is not valid in the case of sparingly soluble salts like AgBr. To correctly solve this problem, we would need the activity coefficients of Ag+ and Br- in the solution. Unfortunately, without that information, we cannot find the correct Ksp value for AgBr in this particular case.

Key concepts

Electrochemical Cell

An electrochemical cell is a device that converts chemical energy into electrical energy or vice versa through electrochemical reactions. These cells consist of two half-cells, each containing an electrode and an electrolyte, where oxidation and reduction reactions occur. The movement of ions in the electrolyte and electrons in the external circuit generates an electric current.

Specifically, the exercise describes a cell composed of silver (Ag) and silver bromide (AgBr), where silver ions (Ag+) are reduced to solid silver (Ag), and solid AgBr is oxidized to Ag and bromide ions (Br-). Understanding the behavior of an electrochemical cell requires knowledge of standard reduction potentials, the Nernst equation to assess cell potential under non-standard conditions, and the equilibrium constant to explore the cell's reaction dynamics.

Nernst Equation

The Nernst equation is vital for calculating the cell potential at non-standard conditions. It tells us how the potential of an electrochemical cell changes as the concentrations (or activities) of the reacting species change. Mathematically, it's expressed as:
\(E_{cell} = E^0_{cell} - \frac{RT}{nF} lnQ\)
where \(E_{cell}\) is the cell potential, \(E^0_{cell}\) is the standard cell potential, \(R\) is the universal gas constant, \(T\) is the temperature in kelvin, \(n\) is the number of electrons transferred in the reaction, \(F\) is the Faraday constant, and \(Q\) is the reaction quotient.

For the AgBr problem, the standard cell potential was correctly calculated using the standard reduction potentials, but the issue arose in using activities equivalent to concentrations. This assumption oversimplified the problem, leading to an erroneous conclusion. In electrolyte solutions with low solubility salts like AgBr, the activity coefficients are essential for an accurate Ksp calculation.

Equilibrium Constant

The equilibrium constant (K) is a number that quantifies the ratio of the concentrations of the products to reactants at equilibrium for a reversible chemical reaction. It indicates the extent to which a reaction will proceed to form products. When the reaction involves a sparingly soluble salt dissolving in water, the equilibrium constant is specifically called the solubility product constant (Ksp).

The solubility product constant is an essential parameter in predicting the solubility of a salt and in understanding its behavior in different chemical environments. It can also be related to the standard cell potential as seen in the given exercise. The equation connecting the cell potential to the Ksp introduces a proportional relationship between the logarithm of Ksp and the cell voltage, demonstrating how the solubility of a compound affects the potential of an electrochemical cell.

Standard Reduction Potential

The standard reduction potential (E0) is a measure of the tendency of a chemical species to acquire electrons and thereby be reduced. Each half-reaction in an electrochemical cell has a standard reduction potential, which is measured under standard conditions: a 1 M concentration for each aqueous species, a 1 atm pressure for gases, and pure solids or liquids as the case may be.

The values of standard reduction potentials are provided in tables and are relative to the standard hydrogen electrode, which is assigned a potential of 0.00 volts. By comparing the standard reduction potentials of the two half-reactions in an electrochemical cell, we can predict the direction of spontaneous reaction and calculate the standard cell potential as the difference between the two. As demonstrated in the problem, the standard reduction potentials were critical in attempting to calculate the Ksp for AgBr, albeit an inaccurate final interpretation due to the misuse of activities.