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 AgNO \(_{3}\) is added to the cathode half-cell, increasing the quantity of Ag' but not changing its concentration. (d) HCl is added to the AgNO\(_{3}\) solution precipitating some of the Ag' as AgCl.

Short answer

(a) Adding water to the anode half-cell increases the cell emf, as it dilutes the solution and decreases the concentration of aluminum ions. (b) Increasing the size of the aluminum electrode does not affect the cell emf, as it doesn't change ion concentrations. (c) Adding AgNO₃ to the cathode half-cell does not change the cell emf, as it maintains the concentration of Ag⁺ ions. (d) Adding HCl to the AgNO₃ solution increases the cell emf, as it reduces the concentration of Ag⁺ ions by forming AgCl precipitate.

Step-by-step solution

Effect of adding water to the anode half-cell

When water is added to the anode half-cell, it dilutes the solution, which reduces the concentration of aluminum ions in the half-cell. According to the Nernst equation, a decrease in ion concentration leads to an increase in cell emf. Therefore, adding water to the anode half-cell increases the cell emf.

Effect of increasing the size of the aluminum electrode

Increasing the size of the aluminum electrode does not directly affect the concentration of the ions in the solution. Therefore, the cell emf remains unchanged.

Effect of adding AgNO₃ to the cathode half-cell

Adding AgNO₃ to the cathode half-cell increases the quantity of Ag⁺, but its concentration remains unchanged. As the concentration of Ag⁺ is not changing, the cell emf remains constant.

Effect of adding HCl to the AgNO₃ solution

When HCl is added to the AgNO₃ solution, it reacts with Ag⁺ ions to form AgCl precipitate: \[ \mathrm{Ag^{+}}(a q)+\mathrm{Cl^{-}}(a q) \longrightarrow \mathrm{AgCl}(s) \] This reaction reduces the concentration of Ag⁺ ions in the solution. According to the Nernst equation, a decrease in ion concentration leads to an increase in cell emf. Thus, adding HCl to the AgNO₃ solution increases the cell emf.

Key concepts

Nernst Equation

The Nernst equation provides a quantitative description of the relationship between the electromotive force (emf) of a galvanic or voltaic cell and the concentrations of the ions involved in the cell's redox reaction. It is expressed as:
\[ E = E^{\circ} - \frac{RT}{nF} \ln Q \]
where \( E \) is the cell emf, \( E^{\circ} \) 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 exchanged, \( F \) is the Faraday's constant, and \( Q \) is the reaction quotient, which represents the ratio of the concentrations of the products over the reactants, each raised to the power of their respective stoichiometric coefficients.
The equation shows that as the concentration of the reactants increases or the concentration of the products decreases, the reaction quotient \( Q \) decreases, leading to an increase in the cell emf. Conversely, a lower concentration of reactants or higher concentration of products results in a higher \( Q \), hence a lower cell emf. This relationship is fundamental in understanding the effect of changes in ion concentration on cell emf.

Electrode Concentration Effect

The electrode concentration effect pertains to how changes in the concentration of ions at the electrode interface in a voltaic cell influence the overall cell voltage or emf. When the concentration of ions at one of the electrodes changes, it can shift the electrochemical equilibrium, affecting the cell's performance.
For instance, diluting the solution surrounding the anode, as in adding water, will reduce the concentration of the metal cations involved in the oxidation half-reaction. According to Le Chatelier's principle, the reaction will adjust to counter this change, leading to an increase in emf.
Furthermore, changes in electrode surface area can alter the reaction rates but do not inherently affect the concentration of ions. This is important when considering changes such as increasing the size of the aluminum electrode in a cell; while the reaction may proceed at a different rate due to increased surface area, if ion concentrations remain constant, the emf does not change.

Half-cell Reactions

Half-cell reactions are the individual oxidation and reduction reactions that occur at the two separate electrodes in a voltaic cell. These reactions are essential components in the cell's overall operation. Each half-cell contains an electrode and an electrolyte wherein either oxidation or reduction takes place.

• The anode half-cell undergoes oxidation, where electrons are lost by the substance.
• The cathode half-cell undergoes reduction whereby electrons are gained by another substance.

The net cell reaction is the sum of these half-cell reactions. The movements of electrons from the anode to the cathode through an external circuit and the migration of ions within the electrolytes maintain electrical neutrality and allow the cell to do work.
Understanding half-cell reactions can aid in determining the direction of electron flow and predicting how variations in reaction conditions, such as precipitating Ag+ as AgCl with the addition of HCl, can influence the overall cell emf. In this particular instance, precipitating Ag+ ions effectively removes them from the electrolyte, altering the cathode half-reaction and increasing the cell emf, per the Nernst equation.