$$ \begin{aligned} &\text { The e.m.f. of the cell } M \mid M^{n+}(0.02 M) \| \mathrm{H}^{+}(1 M) \mathrm{H}_{2(\mathrm{~g})}(1 \mathrm{~atm}) \mathrm{Pt} \text { at }\\\ &25^{\circ} \mathrm{C} \text { is } 0.81 \mathrm{~V} \text { . Calculate the valence of metal if } E^{\circ}{ }_{\ldots 1 / 2}^{n+}=0.76 \mathrm{~V} \end{aligned} $$

Cell potential calculation is central to understanding the workings of electrochemical cells. The overall voltage, or electromotive force (e.m.f.), of a cell determines its ability to drive an electric current through a circuit.
To calculate the cell potential, we must consider both the standard electrode potential and the actual conditions under which the cell operates. The Nernst equation allows us to calculate the cell potential under non-standard conditions by incorporating the concentrations of the reactants and products involved in the cell's electrochemical reaction.
When using the Nernst equation, as we've seen in the solution, we take the standard electrode potential and adjust it based on the ratio of product and reactant concentrations. This reflects the real-world scenario where concentrations often differ from the standard state. To ensure a solid understanding of the cell potential calculation, remember that the cell potential diminishes as the reaction proceeds and the concentration of the reactants decreases.

Electrochemistry is the branch of chemistry that studies the interrelation of electrical currents and chemical reactions. It involves the conversion between chemical energy and electrical energy, which is the basis for batteries, fuel cells, and electrolysis.
Key concepts that we encounter in electrochemistry include oxidation and reduction processes, which occur at the electrodes during the operation of an electrochemical cell. The flow of electrons from one substance to another is what generates an electric current outside the cell and a transfer of charge through ions within the cell. Electrochemistry is not only about generating electrical energy but also encompasses the use of electrical energy to drive chemical reactions that are not spontaneous, a process known as electrolysis.
In the context of our exercise, we dive into the practical application of electrochemistry through understanding cell potential and its calculation, which directly relates to the efficiency and functionality of electrochemical cells.

The standard electrode potential (denoted as E°) is a measure of the propensity of a substance to lose or gain electrons in an electrochemical reaction, also known as a redox reaction.
It is a half-cell potential measured under standard conditions: a solute concentration of 1 M, a pressure of 1 atm for gases, and a specified temperature, usually 25°C (298 K). By convention, the standard electrode potential for hydrogen is set at 0 V, and other potentials are determined relative to it.
This concept is crucial because it allows chemists to predict the direction of redox reactions and to construct electrochemical cells with a desired e.m.f. Understanding the standard electrode potential is also instrumental in figuring out which metals can displace others from solutions, and in determining the spontaneity of redox reactions. It's also a valuable tool for troubleshooting exercises such as the one given, wherein we apply the standard electrode potential to find an unknown metal valence.