Question

A voltaic cell is constructed with two silver-silver chloride electrodes, each of which is based on the following half-reaction: $$ \mathrm{AgCl}(s)+\mathrm{e}^{-}--\rightarrow \mathrm{Ag}(s)+\mathrm{Cl}^{-}(a q) $$ The two cell compartments have \(\left[\mathrm{Cl}^{-} \mathrm{J}=0.0150 \mathrm{M}\right.\) and \(\left[\mathrm{Cl}^{-}\right]=2.55 M\), respectively. (a) Which electrode is the cathode of the cell? (b) What is the standard emf of the cell? (c) What is the cell emf for the concentrations given? (d) For each electrode, predict whether [Cl \(^{-}\) ] will increase, decrease, or stay the same as the cell operates.

Short answer

(a) The cathode is the electrode with [Cl⁻] = 2.55 M, and the anode is the electrode with [Cl⁻] = 0.0150 M. (b) The standard emf of the cell is 0 V. (c) The cell emf for the given concentrations is 0.0589 V. (d) [Cl⁻] will decrease in both the anode and the cathode compartments as the cell operates.

Step-by-step solution

In a voltaic cell, the half-cell with higher potential will act as the cathode (reduction half-reaction), and the half-cell with lower potential will act as the anode (oxidation half-reaction). Here, since the electrode reactions are the same, we can conclude that the potential difference is due to the difference in chloride ion concentrations. By examining the half-reaction, we can conclude that a higher concentration of chloride ions will favor the reduction process (movement of electrons towards Cl-ions). Therefore, the electrode with the higher [Cl-] (2.55 M) will be the cathode, and the electrode with the lower [Cl-] (0.0150 M) will be the anode. #Step 2: Calculate the standard emf of the cell#

In this case, both the anode and the cathode share the same half-reaction. Hence, the standard emfs of the half-reactions cancel each other out, and the standard emf of the cell (E°cell) is 0 V. #Step 3: Calculate the cell emf for the given concentrations using the Nernst equation#

The Nernst equation is: $$ E_\text{cell} = E^\circ_\text{cell} - \frac{RT}{nF} \cdot \log Q_c $$ Where: - E°cell is the standard emf of the cell (0 V), - R is the gas constant (8.314 J∙K^(-1)∙mol^(-1)), - T is the temperature in Kelvin (Assume 298 K), - n is the number of electrons transferred (1), - F is Faraday's constant (96485 C∙mol^(-1)), - Qc is the reaction quotient, given by: $$ Q_c = \frac{[\mathrm{Cl}^-]_\text{anode}}{[\mathrm{Cl}^-]_\text{cathode}} $$ Plug in the given values: $$ E_\text{cell} = 0 - \frac{(8.314)(298)}{(1)(96485)} \cdot \log \frac{0.0150}{2.55} $$ Calculate E_cell: $$ E_\text{cell} \approx 0.0589 \, \text{V} $$ #Step 4: Predict the change in [Cl-] for each electrode as the cell operates#

As the cell operates, the anodic half-reaction will strip away Cl- ions from the anode compartment, turning AgCl to Ag and Cl- ions. This process will decrease [Cl-] at the anode. Conversely, at the cathode, Cl- ions will be consumed along with electrons to form AgCl. This process will decrease [Cl-] at the cathode. In summary: a) The cathode is the electrode with [Cl-] = 2.55 M, and the anode is the electrode with [Cl-] = 0.0150 M. b) The standard emf of the cell is 0 V. c) The cell emf for the given concentrations is 0.0589 V. d) [Cl-] will decrease in both the anode and the cathode compartments as the cell operates.

Key concepts

Nernst Equation

The Nernst equation is a fundamental principle in electrochemistry that allows us to calculate the electromotive force (emf) of a cell under non-standard conditions. It is particularly useful for understanding how cell potential is affected by ion concentration. The equation is expressed as:

\[ E_\text{cell} = E^\circ_\text{cell} - \frac{RT}{nF} \cdot \ln \frac{[\text{products}]}{[\text{reactants}]} \]

where:

• \(E_\text{cell}\) is the cell potential under non-standard conditions,
• \(E^\circ_\text{cell}\) is the standard emf of the cell,
• \(R\) is the universal gas constant,
• \(T\) is the temperature in Kelvin,
• \(n\) is the number of moles of electrons transferred in the reaction,
• \(F\) is the Faraday constant,
• \(\ln\) is the natural logarithm (logarithms to the base e).

In the case of the silver-silver chloride electrode problem, we utilize the Nernst equation to calculate the emf of a cell based on the different chloride ion concentrations at the anode and cathode. By substituting the given values into the Nernst equation, the emf of the cell is found to be approximately 0.0589 V.

Standard Emf

Standard emf (or standard electrode potential) is a measure of the intrinsic voltage generating capacity of an electrochemical cell under standard conditions, which means at a temperature of 298 K, 1 atm pressure, and concentration of 1 M for all solutions. It's expressed as \(E^\circ_\text{cell}\) and is crucial for determining the direction of spontaneous reactions in electrochemical cells.

In the textbook example, we determine the standard emf by considering the standard emfs of the half-reactions. Since the half-reactions are identical for both electrodes, their standard emfs cancel each other out, leading to a net standard emf for the cell of 0 V. It is essential to note that the standard emf is a starting reference point for calculating actual emf using the Nernst equation when concentrations differ from 1 M.

Electrochemical Cells

Electrochemical cells are the foundation of batteries and are devices constructed to convert chemical energy into electrical energy through redox reactions. There are two main types of electrochemical cells: voltaic or galvanic cells, which generate electrical energy from spontaneous chemical reactions, and electrolytic cells, which consume electrical energy to drive non-spontaneous chemical reactions.

The voltaic cell discussed in our exercise is formed with two half-cells, each consisting of a silver-silver chloride electrode, where a redox reaction occurs. The difference in concentration of ions creates a potential difference, and when connected through a conductive material, electrons flow from the anode to the cathode, generating an electric current.

Understanding the components and reactions within an electrochemical cell is paramount for predicting how the cell will behave under different conditions, such as changes in ion concentration and pressure, and is useful for practical applications such as battery technology and metal plating.

Concentration Cells

Concentration cells are a class of electrochemical cells where the electrodes are identical but immersed in solutions of different concentrations. The emf generated by these cells is due to the concentration gradient between the solutions. The natural tendency of ions to move from higher to lower concentrations (to achieve equilibrium) drives the potential difference in the cell.

In our textbook example, the cell is a concentration cell with two silver-silver chloride electrodes. The difference in chloride ion concentration between the cathode and anode creates the cell's emf. This potential prompts the reaction to proceed until the equilibrium is reached, meaning the concentration of \(\text{Cl}^-\) ions will eventually become uniform in both half-cells, provided the cell continues to operate and is not recharged.

Concentration cells are not only theoretical constructs but have practical implications. They serve as tools for measuring ion concentrations and are the principle behind certain types of potentiometric sensors, such as pH meters.