Problem 70 - Chemistry The Central Science

Question

A voltaic cell is constructed with two silver-silver chloride electrodes, each of which is based on the following half-reaction: $$ \operatorname{AgCl}(s)+\mathrm{e}^{-} \longrightarrow \mathrm{Ag}(s)+\mathrm{Cl}^{-}(a q) $$ The two half-cells have \(\left[\mathrm{Cl}^{-}\right]=0.0150 \mathrm{M}\) and \(\left[\mathrm{Cl}^{-}\right]=\) \(2.55 \mathrm{M},\) 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 \(\left[\mathrm{Cl}^{-}\right]\) will increase, decrease, or stay the same as the cell operates.

Step-by-Step Solution

Answer

In summary, for the given voltaic cell with silver-silver chloride electrodes: (a) the cathode has a higher chloride ion concentration of 2.55 M, (b) the standard emf of the cell is 0 V, (c) the cell emf for the given concentrations is approximately -0.0536 V, and (d) as the cell operates, the concentration of Cl- ions will decrease at the cathode and increase at the anode.

Step 1: Identify which electrode is the cathode

From the given information, it is clear that the cathode is the electrode with higher chloride ion concentration. This is because, as the voltaic cell operates, the electrode with higher concentration (the cathode) will receive electrons, leading to a decrease in its concentration, while the electrode with the lower chloride ion concentration (the anode) will lose electrons, leading to an increase in its concentration. In this case, the cathode has a chloride ion concentration of 2.55 M, while the anode has a chloride ion concentration of 0.0150 M.

Step 2: Calculate the standard emf of the cell

The standard emf of a cell, under standard conditions, can be calculated using the following equation: \[E_{cell}^0 = E_{cathode}^0 - E_{anode}^0\] Given that the half-reaction is identical for both electrodes, their standard reduction potential cancels out and the standard emf of the cell is 0 V.

Step 3: Calculate the cell emf for the given concentrations

The cell emf for given concentrations can be calculated using the Nernst equation: \[E_{cell} = E_{cell}^0 - \frac{RT}{nF} \ln{Q}\] Since we have a single electron transfer (1 mole of electrons), n = 1. In this case, we have: \(Q = \frac{[\mathrm{Cl}^{-}]_{cathode}}{[\mathrm{Cl}^{-}]_{anode}}\) Plugging the values into the equation, we get: \[E_{cell} = 0 - \frac{8.314 \times 298}{1 \times 96,485} \ln{\frac{2.55}{0.0150}}\] Calculating the value, we have: \[E_{cell} \approx -0.0536\: V\]

Step 4: Predict concentration changes of Cl- ions as the cell operates

As the cell operates, the cathode (the electrode with higher chloride ion concentration) will receive electrons, this will make its concentration of Cl- ions decrease. On the other hand, at the anode (the electrode with lower concentration), Cl- ions will increase in concentration as Ag+ ions lose electrons to become Ag(s) and releasing Cl- ions in the process. In conclusion: a) The cathode is the electrode with higher chloride ion concentration (2.55 M). b) The standard emf of the cell is 0 V. c) The cell emf for the given concentrations is approximately -0.0536 V. d) The concentration of Cl- ions will decrease at the cathode and increase at the anode as the cell operates.

Key Concepts

Understanding Electrochemistry in the Context of a Voltaic Cell

Electrochemistry is a branch of chemistry that deals with the relationship between electricity and chemical reactions, particularly the conversion of chemical energy into electrical energy and vice versa. Voltaic cells, also known as galvanic cells, are a classic example of electrochemical cells that generate electricity through spontaneous redox reactions. In a typical voltaic cell, two metals (or metal ions) are immersed in electrolyte solutions and connected by a salt bridge, with electrons flowing from the anode (oxidation) to the cathode (reduction).

Understanding the dynamics of a voltaic cell involves recognizing that the electrode with a higher concentration of reacting ions will serve as the cathode, as it provides the required reduction potential for the cell to function. For example, in an exercise involving silver-silver chloride electrodes, the electrode with a higher chloride ion concentration becomes the cathode, which is essential in the context of predicting the cell behavior and calculating the cell's electromotive force (emf).

Calculating Standard Emf of a Cell

The standard emf of a cell, denoted as \(E_{cell}^0\), is essentially the difference in potential between the cathode and anode under standard conditions, typically 1 M concentration for all reactants and products, a pressure of 1 atmosphere, and at a temperature of 25°C (298 K). The standard emf calculation is crucial, as it indicates the maximum potential difference between electrodes and the cell's ability to do work when no current flows.

In textbook exercises, calculating the standard emf may involve subtracting the standard reduction potential of the anode from that of the cathode. If both electrodes have the same half-reaction, such as in the case with two silver-silver chloride electrodes, their standard reduction potentials cancel each other out, resulting in a standard emf of 0 V. This scenario simplifies the problem and emphasizes the significance of solution conditions on the actual emf generated by the cell.

Applying the Nernst Equation to Calculate Actual Emf

The Nernst equation enables the calculation of the cell's emf based on non-standard conditions, taking into account the actual concentrations of reactants and products. It's given by the formula:
\[E_{cell} = E_{cell}^0 - \frac{RT}{nF} \ln{Q}\]
Where \(E_{cell}^0\) is the standard emf, \(R\) is the universal gas constant, \(T\) is the temperature in Kelvin, \(n\) is the number of moles of electrons transferred, \(F\) is the Faraday constant, and \(Q\) is the reaction quotient. In a textbook problem, such as the one involving silver-silver chloride electrodes, you would use the Nernst equation to determine the actual emf by inputting the concentrations of chloride ions at the cathode and the anode.

This equation also explains why even a cell with a standard emf of 0 V can have an actual emf under different concentration conditions. When answering such textbook exercises, it's imperative to understand and apply the Nernst equation correctly to predict the cell's behavior under a given set of circumstances.