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) \(\mathrm{HCl}\) is added to the \(\mathrm{AgNO}_{3}\) solution, precipitating some of the \(\mathrm{Ag}^{+}\) as \(\mathrm{AgCl}\).

Short answer

(a) When the anode half-cell is diluted, the emf (E) of the cell will increase due to the decrease in the concentration of Al^3+ ions. (b) Increasing the size of the aluminum electrode does not affect the cell emf. (c) Adding AgNO3 solution to the cathode half-cell without changing the concentration of Ag+ ions will not affect the cell emf. (d) Precipitating some Ag+ as AgCl will decrease the cell emf as the concentration of Ag+ ions in the solution decreases.

Step-by-step solution

(a) Diluting the Anode Half-Cell

When water is added to the anode half-cell, the concentration of Al^3+ ions in the solution will decrease. According to the Nernst equation: \[ E = E^0 - \frac{0.0592}{n} \log\frac{[\mathrm{Al}^{3+}]}{[\mathrm{Ag}^+]^3} \] Decreasing the concentration of Al^3+ ions will cause the logarithmic term to decrease as well, and thus, the emf (E) will increase.

(b) Increasing the Size of the Aluminum Electrode

The size of the electrode does not affect the concentrations of the ionic species in the cell and therefore will not directly affect the cell emf. The emf is solely dependent on the concentrations of species involved in the redox reaction, according to the Nernst equation. So the emf of the cell remains unchanged in this situation.

(c) Adding AgNO3 Solution to the Cathode Half-Cell

By adding a solution of AgNO3 to the cathode half-cell, the quantity of Ag+ ions increases, but their concentration remains the same. Since the emf of the cell is determined by the concentrations of the involved species and not their absolute quantity, there will be no effect on the cell emf due to this addition.

(d) Precipitating Ag+ as AgCl

When HCl is added to the AgNO3 solution, it forms a precipitate with some of the Ag+ ions, producing AgCl. This process will reduce the concentration of Ag+ ions in the solution. Again, referring to the Nernst equation: \[ E = E^0 - \frac{0.0592}{n} \log\frac{[\mathrm{Al}^{3+}]}{[\mathrm{Ag}^+]^3} \] With the decrease in the concentration of Ag+ ions, the logarithmic term will increase, causing the emf (E) to decrease.

Key concepts

Nernst Equation

The Nernst Equation is a key tool used to calculate the electromotive force (EMF) of a voltaic cell, which arises from redox reactions. It relates the cell voltage to the concentrations of the reactants and products. When you have a reaction, like the one in your exercise, the Nernst Equation looks like this:\[E = E^0 - \frac{0.0592}{n} \log Q\]where:

• \(E\) is the cell potential under non-standard conditions.
• \(E^0\) is the standard cell potential.
• \(n\) is the number of moles of electrons transferred in the redox reaction.
• \(Q\) is the reaction quotient, a fraction representing the ratio of concentrations of products to reactants.

The Nernst Equation shows how cell voltage depends on concentration. For example, if you dilute one side of the cell with water, as in part (a) of your exercise, it changes the concentration of ions. Put simply, decreasing the concentration of ions can increase the cell EMF by reducing the value of the logarithmic term \(\log Q\) when applied to the reaction quotient.

Cell EMF

The electromotive force (EMF) or cell potential is essentially the measure of the voltage generated by a voltaic cell. It indicates the cell's ability to push electrons through a circuit, thus determining the cell's performance.**Key Points about EMF:**

• EMF is affected by the concentration of the reactants and products as suggested by the Nernst Equation.
• The standard EMF, \(E^0\), is when all reactants and products are in their standard states, usually at 1 M concentration.
• Changing the size of electrodes or adding a solution without changing concentration does not directly affect the EMF.

For instance, despite adding more silver ions to your cathode, if concentration stays constant, EMF remains unchanged. Understanding EMF is crucial in predicting the behavior of cell reactions under different conditions, making it easier to analyze and manipulate voltaic cells like those in your exercise.

Redox Reactions

Redox reactions form the backbone of voltaic cells. They involve the transfer of electrons between species in the reaction. In a typical redox reaction like the one in your exercise, aluminum gets oxidized while silver ions get reduced.**Redox Defined:**

• Oxidation refers to the loss of electrons.
• Reduction signifies the gain of electrons.
• Anode is where oxidation occurs; cathode is where reduction takes place.

This is significant because the cell potential (or EMF) is derived from the standard electrode potentials of these half-reactions. At the anode, aluminum transforms into \(\mathrm{Al}^{3+}\), releasing electrons, whereas at the cathode, \(\mathrm{Ag}^{+}\) ions gain electrons to become solid silver. Understanding which species gets oxidized or reduced helps decipher how a voltaic cell will operate and what conditions can shift its efficiency.