You have a concentration cell with Cu electrodes and [Cu^{2+} ] \(=1.00 M\) (right side) and \(1.0 \times 10^{-4} M\) (left side). a. Calculate the potential for this cell at \(25^{\circ} \mathrm{C}\). b. The \(\mathrm{Cu}^{2+}\) ion reacts with \(\mathrm{NH}_{3}\) to form \(\mathrm{Cu}\left(\mathrm{NH}_{3}\right)_{4}{ }^{2+}\) by the following equation: \(\mathrm{Cu}^{2+}(a q)+4 \mathrm{NH}_{3}(a q) \rightleftharpoons \mathrm{Cu}\left(\mathrm{NH}_{3}\right)_{4}^{2+}(a q)\) \(K=1.0 \times 10^{13}\) Calculate the new cell potential after enough \(\mathrm{NH}_{3}\) is added to the left cell compartment such that at equilibrium \(\left[\mathrm{NH}_{3}\right]=2.0 \mathrm{M}\)

The Nernst equation is fundamental in electrochemistry, as it relates the potential of an electrochemical cell to the concentration of the chemicals involved in the reaction. At its core, the Nernst equation allows us to predict how much voltage, or electrical potential, will be produced or required by a chemical reaction under non-standard conditions.

For an electrochemical cell composed of one half-cell with a concentration of 1.00 M of copper ions and another with a concentration of 1.0 x 10^-4 M, the Nernst equation can determine the cell potential. It takes into account the temperature (converted to Kelvin), the number of moles of electrons transferred in the reaction ( = 2), and the reaction quotient (Q), which is a ratio of the ionic concentrations. When applied to a concentration cell, the standard cell potential (E°) is zero, simplifying the equation further. The potential difference derived is directly related to the difference between the ion concentrations across the electrochemical cell.

Electrochemical concentration gradients are a driving force for electrons to move in a cell, creating electrical potential. Whenever there's a difference in the concentration of ions across a cell membrane, or in the case of a concentration cell, a potential is generated. This gradient prompts the movement of ions from an area of high concentration to an area of low to even out the distribution, similar to how water flows downhill.

In the exercise, the initial cell potential measured is a direct consequence of the existing concentration gradient between the copper ion solutions on each side of the cell. Over time, the system aims for equilibrium where the concentrations would become equal, thereby diminishing the potential. The process is analogous to diffusion in physical systems, just with charges and electrical potential involved. The Nernst equation allows us to quantify this potential, giving us a glimpse into the unseen force exerted by these invisible particles.

The chemical equilibrium constant (K) plays a critical role in determining the extent of chemical reactions. It reflects the particular ratio at which the reactants and products exist in equilibrium. A large equilibrium constant, such as the one provided in the exercise (K = 1.0 x 10¹³), indicates a reaction that strongly favors the formation of products.

In the context of the cell potential, by adding ammonium (NH₃) to react with the copper ions and form a complex ion, the equilibrium is shifted. This change in concentration, according to Le Chatelier's principle, affects the cell potential. Once again, the Nernst equation comes into play, enabling the calculation of the new potential, considering the near exhaustion of copper ions in the left compartment after reaction with NH₃. The significantly increased concentration gradient leads to an increased cell potential, as indicated by the negative value derived from the calculation.