Question

For a spontaneous reaction \(\mathrm{A}(a q)+\mathrm{B}(a q) \longrightarrow \mathrm{A}^{-}(a q)+\) \(\mathrm{B}^{+}(a q),\) answer the following questions: (a) If you made a voltaic cell out of this reaction, what halfreaction would be occurring at the cathode, and what half-reaction would be occurring at the anode? (b) Which half-reaction from (a) is higher in potential energy? (c) What is the sign of \(E_{\text {cell }}^{\circ}\) [ Section 20.3 ]

Short answer

(a) Cathode half-reaction: \(\mathrm{A}(aq) + e^{-} \longrightarrow \mathrm{A}^{-}(aq)\) and Anode half-reaction: \(\mathrm{B}(aq) \longrightarrow \mathrm{B}^{+}(aq) + e^{-}\) (b) The oxidation half-reaction (involving species B) has higher potential energy. (c) The sign of \(E_{\text{cell}}^{\circ}\) is positive because the reaction is spontaneous.

Step-by-step solution

Identify the half-reactions

The given redox reaction is: $$ \mathrm{A}(aq) + \mathrm{B}(aq) \longrightarrow \mathrm{A}^{-}(aq) + \mathrm{B}^{+}(aq) $$ We can see that species A is gaining an electron, and species B is losing an electron. So, the half-reactions are: $$ \mathrm{A}(aq) + e^{-} \longrightarrow \mathrm{A}^{-}(aq) $$ $$ \mathrm{B}(aq) \longrightarrow \mathrm{B}^{+}(aq) + e^{-} $$

Determine cathode and anode reactions

In a voltaic cell, the cathode is the site of reduction (gain of electrons), and the anode is the site of oxidation (loss of electrons). So, the half-reaction occurring at the cathode (reduction) is: $$ \mathrm{A}(aq) + e^{-} \longrightarrow \mathrm{A}^{-}(aq) $$ The half-reaction occurring at the anode (oxidation) is: $$ \mathrm{B}(aq) \longrightarrow \mathrm{B}^{+}(aq) + e^{-} $$

Determine the half-reaction with higher potential energy

The half-reaction with higher potential energy is the one that releases more energy when its electrons are transferred to the other half-reaction. In a voltaic cell, the half-reaction with higher potential energy occurs at the anode (where oxidation occurs). Thus, the half-reaction involving species B (oxidation half-reaction) is higher in potential energy.

Find the sign of \(E_{\text{cell}}^{\circ}\)

The standard cell potential, \(E_{\text{cell}}^{\circ}\), is the difference between the standard reduction potentials of the cathode and anode half-reactions. When a reaction is spontaneous, the cell potential is positive. In this case, we are given that the reaction is spontaneous. Therefore, the sign of \(E_{\text{cell}}^{\circ}\) is positive. #Conclusion# (a) In a voltaic cell, the reduction half-reaction (A gaining an electron) occurs at the cathode, and the oxidation half-reaction (B losing an electron) occurs at the anode. (b) The oxidation half-reaction (involving species B) has higher potential energy. (c) The sign of \(E_{\text{cell}}^{\circ}\) is positive because the reaction is spontaneous.

Key concepts

Spontaneous Redox Reactions

Spontaneous redox reactions are chemical processes that occur without the need for external energy. These reactions involve the transfer of electrons from one substance to another, and they are fundamental to the functioning of voltaic cells, which are also known as galvanic cells.

In a spontaneous redox reaction, one species (the reductant) loses electrons – this process is known as oxidation – while another species (the oxidant) gains electrons, known as reduction. These complementary processes always occur together because the electrons lost by one species must be gained by another.

In our textbook example, species A is reduced by gaining an electron, and species B is oxidized by losing an electron. The movement of electrons from the oxidation to the reduction half-reactions generates electrical energy in a voltaic cell, making spontaneous redox reactions the powerhouse of such electrochemical cells.

Cathode and Anode in Voltaic Cells

Understanding the roles of the cathode and anode in voltaic cells is crucial. The cathode is where reduction occurs, meaning it is the electrode at which electrons are gained by the chemical species. Conversely, the anode is where oxidation happens – electrons are lost by the chemical species here.

The textbook exercise provides a clear distinction between these two electrodes: species A, which undergoes reduction, will do so at the cathode, whereas species B will be oxidized at the anode. It's important to note that in a voltaic cell, the convention is that the cathode is positive and the anode is negative due to the direction of electron flow from anode to cathode.

Moreover, at the anode of a voltaic cell, the half-reaction with higher potential energy takes place since it is the source of the electrons that flow through the external circuit before reaching the cathode.

Standard Cell Potential

Standard cell potential (\(E_{\text{cell}}^{\text{o}}\)) is a measurement of the electromotive force of a voltaic cell under standard conditions – typically with solutes at 1 M concentration, gases at 1 atm pressure, and at a temperature of 25°C (298 K). It reflects the capability of the cell to do electrical work.

The standard cell potential is positive for spontaneous reactions, exemplified in our textbook case where the sign of \(E_{\text{cell}}^{\text{o}}\) is positive because the reaction is stated as spontaneous. This value is calculated by taking the difference between the standard reduction potentials of the cathode and anode; a higher difference indicates a higher voltage of the cell and, thereby, a greater capacity to do work.

This concept is essential not only in theoretical exercises but also in practical applications because it helps determine the energy efficiency and feasibility of electrochemical processes, such as those found in batteries and fuel cells.