Write a balanced equation from each cell notation: (a) \(\operatorname{Mn}(s)\left|\mathrm{Mn}^{2+}(a q) \| \mathrm{Cd}^{2+}(a q)\right| \mathrm{Cd}(s)\) (b) \(\mathrm{Fe}(s)\left|\mathrm{Fe}^{2+}(a q) \| \mathrm{NO}_{3}^{-}(a q)\right| \mathrm{NO}(g) \mid \operatorname{Pt}(s)\)

The half-reaction method is a systematic approach used to balance redox (oxidation-reduction) reactions. This method involves splitting the overall reaction into two individual half-reactions: one that describes the oxidation process and another that outlines the reduction process. This approach simplifies the balancing of redox reactions and ensures that both mass and charge are conserved.
The first step is to write the unbalanced half-reactions by identifying the species being oxidized and reduced. Each half-reaction is then balanced separately for atoms and charge. Electrons are added to one side of the half-reaction to balance the charge difference.
After both half-reactions are balanced, they are combined. However, before combining, it's essential to ensure that the number of electrons lost in oxidation matches the number of electrons gained in reduction. This may require multiplying one or both half-reactions by appropriate factors. Finally, the balanced half-reactions are added together, and any redundant species (like electrons) are canceled out.
The half-reaction method helps maintain clarity and precision in balancing complex redox reactions.

Oxidation and reduction are two key processes in redox reactions. Oxidation involves the loss of electrons from a species, whereas reduction involves the gain of electrons.
To identify these processes, look for changes in oxidation states of the participating elements. During oxidation, the oxidation state of an element increases. Conversely, during reduction, the oxidation state decreases. For example, in the half-reaction \(\text{Mn}(s) \rightarrow \text{Mn}^{2+}(aq) + 2e^-\), manganese is oxidized as it loses electrons and its oxidation state increases from 0 to +2.
In contrast, in the half-reaction \(\text{Cd}^{2+}(aq) + 2e^- \rightarrow \text{Cd}(s)\), cadmium ions gain electrons, reducing their oxidation state from +2 to 0.
Oxidation and reduction always occur simultaneously in redox reactions because the lost electrons from the oxidized species must be gained by another species. Thus, every redox reaction consists of an oxidation half-reaction paired with a reduction half-reaction.

Electrochemical cells are devices that convert chemical energy into electrical energy (or vice versa) through redox reactions occurring in separate compartments known as half-cells.
Each half-cell consists of an electrode immersed in an electrolyte solution containing ions. The oxidation half-reaction occurs at the anode, while the reduction half-reaction occurs at the cathode.
In the given exercise, for example, the electrochemical cell notation \(\text{Mn}(s)\big|\text{Mn}^{2+}(aq) \big\big\text| \text{Cd}^{2+}(aq)\big|\text{Cd}(s)\big\) represents an anode where manganese oxidizes and a cathode where cadmium reduces.
The overall cell reaction is obtained by combining the balanced half-reactions, which creates a flow of electrons from the anode to the cathode through an external circuit. This electron flow generates an electric current, which can be harnessed to perform work. Additionally, ions migrate through a salt bridge or membrane to maintain electrical neutrality.
Understanding electrochemical cells is essential, as they are the basis of batteries, electroplating, corrosion, and many other industrial and everyday applications.