Problem 64 - Chemistry The Central Science In Si Units

Question

A voltaic cell utilizes the following reaction: $$ \mathrm{Al}(s)+3 \mathrm{Ag}^{+}(a q) \longrightarrow \mathrm{Al}^{3+}(a q)+3 \mathrm{Ag}(s) $$ What is the effect on the cell emf of each of the following changes? (a) Water is added to the anode half-cell, diluting the solution. (b) The size of the aluminum electrode is increased. (c) A solution of \(\mathrm{AgNO}_{3}\) is added to the cathode half-cell, increasing the quantity of \(\mathrm{Ag}^{+}\) but not changing its concentration. (d) HCl is added to the \(\mathrm{AgNO}_{3}\) solution, precipitating some of the \(\mathrm{Ag}^{+}\) as \(\mathrm{AgCl}\).

Step-by-Step Solution

Answer

(a) Increases emf, (b) no change, (c) no change, (d) decreases emf.

Step 1: Identify Components of the Reaction

In the given voltaic cell reaction \( \mathrm{Al}(s)+3 \mathrm{Ag}^{+}(aq) \rightarrow \mathrm{Al}^{3+}(aq)+3 \mathrm{Ag}(s) \), aluminum (Al) acts as the anode and silver ions (Ag\(^+\)) act as the cathode. The cell involves the oxidation of Al to Al\(^{3+}\) and the reduction of Ag\(^+\) to Ag.

Step 2: Understanding Emf and the Nernst Equation

The electromotive force (emf) of a voltaic cell is given by the Nernst equation: \( E = E^\circ - \frac{RT}{nF} \ln{Q} \), where \( E^\circ \) is the standard emf, R is the gas constant, T is the temperature, n is the number of moles of electrons transferred, F is Faraday's constant, and Q is the reaction quotient. \( E \) is affected by changes in concentration of reactants and products.

Step 3: Evaluate the Effect of Adding Water to Anode Half-Cell

Adding water to the anode half-cell decreases the concentration of \( \mathrm{Al}^{3+} \), the product of the anodic reaction. According to the Nernst equation, a decrease in product concentration decreases Q, thus increasing the cell emf.

Step 4: Effect of Increasing the Size of the Aluminum Electrode

Increasing the size of the aluminum electrode does not affect the emf. The emf is dependent on the concentration of the ions in solution, not the size of the electrodes.

Step 5: Analyze the Effect of Adding AgNO₃ Solution to Cathode Half-Cell

Adding a solution of AgNO₃ increases the quantity of \( \mathrm{Ag}^{+} \) ions but if the concentration remains unchanged, there is no effect on Q, and thus no change in the cell emf.

Step 6: Impact of Adding HCl to AgNO₃ Solution

Adding HCl to the \( \mathrm{AgNO}_{3} \) solution precipitates Ag\(^+\) as AgCl. This reduces the concentration of \( \mathrm{Ag}^{+} \), increasing Q in the Nernst equation and thus decreasing the cell emf.

Key Concepts

Electromotive Force (emf)

The electromotive force (emf) in a voltaic cell is the potential difference between the anode and cathode. It essentially acts as the cell's battery, driving the flow of electrons from the anode to the cathode through an external circuit. The emf can be thought of as the energy provided by the cell's chemical reactions to move electrons.

Emf is usually influenced by the concentration of ions in the cell. When the conditions around the cell change, such as the concentration of reactants or products, the emf can also change. These conditions can be evaluated mathematically using the Nernst equation, which helps calculate the actual working emf of the voltaic cell under non-standard conditions. Therefore, understanding emf is essential in predicting how a voltaic cell will perform in various situations.

Nernst Equation

The Nernst Equation is a key tool used to understand how the electromotive force (emf) of a voltaic cell changes with the concentration of its chemical components. The formula is: \[E = E^\circ - \frac{RT}{nF} \ln Q\] Where:

• \(E\) is the cell's emf under non-standard conditions.
• \(E^\circ\) is the standard emf when all components are at standard conditions.
• \(R\) is the universal gas constant (8.314 J/mol·K).
• \(T\) is the temperature in Kelvin.
• \(n\) represents the number of moles of electrons transferred in the reaction.
• \(F\) is Faraday's constant (96,485 C/mol).
• \(Q\) is the reaction quotient.

The Nernst Equation shows us that as the concentration of reactants and products changes, so does the cell's emf. It helps in predicting the direction and magnitude of the emf change due to concentration shifts. This is particularly useful in practical applications where conditions are often deviating from the ideal.

Reaction Quotient (Q)

The Reaction Quotient, denoted as \(Q\), represents the ratio of the concentrations of the products to the reactants at any point in time for a chemical reaction. For the given reaction:\[\mathrm{Al}(s) + 3 \mathrm{Ag}^{+}(aq) \rightarrow \mathrm{Al}^{3+}(aq) + 3 \mathrm{Ag}(s)\]The reaction quotient \(Q\) is:\[Q = \frac{[\mathrm{Al}^{3+}]}{[\mathrm{Ag}^{+}]^3}\]Changes in \(Q\) indicate shifts in reaction dynamics. For instance, if the concentration of \(\mathrm{Al}^{3+}\) decreases or the concentration of \(\mathrm{Ag}^{+}\) increases, \(Q\) differs from its initial value. This change in \(Q\), as plugged into the Nernst equation, can increase or decrease the cell's emf. Monitoring \(Q\) can thus provide insights into how close a reaction is from reaching equilibrium and helps in adjusting concentrations accordingly to control the cell's emf.

Anode and Cathode

In any voltaic cell, the anode and cathode serve crucial roles. The anode is the electrode where oxidation occurs. In the provided reaction, aluminum (\(\mathrm{Al}\)) serves as the anode, as it loses electrons ( undergoing oxidation) to form \(\mathrm{Al}^{3+}\).

The cathode, on the other hand, is where reduction happens. Silver ions (\(\mathrm{Ag}^{+}\)) receive electrons at the cathode to form solid silver (\(\mathrm{Ag}\)). This flow of electrons from the anode to the cathode via an external circuit is what generates the cell’s power.
Both electrodes facilitate a fundamental redox reaction in the cell. Because their roles are defined by the type of chemical changes they undergo, altering conditions at either electrode can impact the overall cell emf. Hence, understanding their specific functions can help you predict changes in a cell’s performance effectively.

Redox Reaction

Redox reactions are the sum of two complementary processes: reduction and oxidation. In the context of voltaic cells, they are essential because they facilitate the conversion of chemical energy into electrical energy.

In the given reaction:

• Oxidation occurs at the anode, where aluminum (\(\mathrm{Al}\)) loses electrons:
\[\mathrm{Al}(s) \rightarrow \mathrm{Al}^{3+}(aq) + 3e^{-}\]
• Reduction occurs at the cathode, where silver ions (\(\mathrm{Ag}^{+}\)) gain electrons:
\[\mathrm{Ag}^{+}(aq) + e^{-} \rightarrow \mathrm{Ag}(s)\]

Overall, the redox reaction drives the electron flow within the cell. Alterations, such as changes in concentration or electrode size, can disrupt the electron balance, thereby affecting the reaction rate and consequently the cell's emf. Understanding redox reactions will help you grasp how energy flow is controlled and how it can be maximized for efficient cell function.