For each of the following redox reactions, (i) write the half-reactions; (ii) write a balanced equation for the whole reaction, (iii) determine in which direction the reaction will proceed spontaneously under standard-state conditions: (a) \(\mathrm{H}_{2}(g)+\mathrm{Ni}^{2+}(a q) \longrightarrow \mathrm{H}^{+}(a q)+\mathrm{Ni}(s)\) (b) \(\mathrm{MnO}_{4}^{-}(a q)+\mathrm{Cl}^{-}(a q) \longrightarrow\) \(\mathrm{Mn}^{2+}(a q)+\mathrm{Cl}_{2}(g)\) (in acid solution) (c) \(\mathrm{Cr}(s)+\mathrm{Zn}^{2+}(a q) \longrightarrow \mathrm{Cr}^{3+}(a q)+\mathrm{Zn}(s)\)

Understanding the half-reaction formulation is essential in grasping redox processes. A redox reaction involves both reduction (gain of electrons) and oxidation (loss of electrons). To break it down, we separate the overall reaction into two half-reactions: one for reduction and one for oxidation. This separation allows us to balance the reaction in complex environments, such as acidic or basic solutions.

For example, in reaction (a) from the exercise, we split the original reaction into two parts: the reduction of nickel ions to metallic nickel and the oxidation of hydrogen gas into hydrogen ions. Each half-reaction is balanced individually by ensuring the number of atoms of each element and the charges are equal on both sides. When we combine these half-reactions, we get a clearer insight into the electron transfer occurring in the overall redox process.

After we have outlined the half-reactions, the next step is to balance the overall redox equation. Balancing redox equations is a deliberate process that entails equating the number of electrons lost in the oxidation half-reaction to the number of electrons gained in the reduction half-reaction. This ensures that the electrons - which are the essence of redox reactions - are conserved.

In the exercise solution step 2, electron balance was achieved by multiplying the half-reactions by appropriate factors so that the electrons cancel out when the half-reactions are added together. This is a critical puzzle—it is only when the electrons balance that we can say the equation itself is balanced. Besides the electron count, we also need to check that atoms are balanced, which may require adding water molecules, hydrogen ions, or hydroxide ions, depending on whether the reaction occurs in an acidic or basic solution.

The standard reduction potentials (SRP) are a series of values that provide us a way to predict the behavior of redox reactions. These potentials, usually referenced to the standard hydrogen electrode, define the tendency of a species to gain electrons and thus be reduced.

In simple terms, the more positive the SRP value, the greater the species' affinity for electrons. We use these values to determine which substance in the reaction will act as the oxidizing agent (will be reduced) and which will act as the reducing agent (will be oxidized). When comparing two half-reactions, as we see in the exercise, the one with the higher SRP will predictably undergo reduction and the other oxidation. These standard potentials are not only useful for predicting the outcome of reactions but also for understanding the driving force behind the reaction's direction.

How do we know if a redox reaction will proceed on its own, without the input of external energy? This is where the idea of spontaneity comes in, which can be deduced from the standard reduction potentials we discussed earlier. A reaction is considered spontaneous if it leads to a decrease in free energy.

Using the rules mentioned in step 3 of the solution, we determine the spontaneity of a redox reaction by comparing the standard reduction potentials: if the SRP of the reactant being reduced is more positive than that of the reactant being oxidized, the reaction is likely spontaneous under standard conditions. This principle helps us predict whether a redox reaction can occur as written or if we need to apply energy to drive it. In the exercise, this is applied to conclude that reactions (a) and (c) are spontaneous, while reaction (b) is not, under standard-state conditions.