For the cell: \(Ni (s)| Ni^{2+}(0.1M)||Cu^{2+}(0.1M)|Cu(s)\) the cell potential will increase when

\(Cu^{2+}\) ion concentration is increased

The correct answer is option 3. \(Cu^{2+}\) ion concentration is increased.

Let us look into why increasing the concentration of \(Cu^{2+}\) ions in this specific cell increases the cell potential.

Understanding the Nernst Equation:

The Nernst equation is a powerful tool in electrochemistry that relates the cell potential \((E)\) under non-standard conditions to various factors, including the standard potential \((E^o)\), temperature \((T)\), number of electrons transferred \((n)\), Faraday's constant \((F)\), and the reaction quotient \((Q)\).

The reaction quotient \((Q)\) represents the ratio of the product concentrations to the reactant concentrations, raised to their respective stoichiometric coefficients. In simpler terms, it reflects the relative amounts of reactants and products present at any given moment.

In the given cell:

\(Ni (s)| Ni^{2+}(0.1M)||Cu^{2+}(0.1M)|Cu(s)\)

\(Ni^{2+}\) acts as the reactant in the oxidation half-cell (anode).

\(Cu^{2+}\) acts as the product in the reduction half-cell (cathode).

According to the Nernst equation, a higher concentration of the product \((Cu^{2+})\) in the reduction half-cell corresponds to a larger value of \(Q\). A larger \(Q\) in the equation translates to a more positive cell potential \((E)\).

Imagine the cell reaction is ongoing. Initially, both \(Cu^{2+}\) and \(Ni^{2+}\) have concentrations of \(0.1 M\). As the reaction proceeds, \(Ni^{2+}\) gets oxidized, and \(Cu^{2+}\) gets reduced. This creates a tendency for the reaction to reach equilibrium, where the forward and backward rates become equal.

By increasing the external concentration of \(Cu^{2+}\), we essentially push the reaction back towards the reactants' side. The cell tries to counteract this change by further oxidizing Ni and reducing \(Cu^{2+}\) to maintain the equilibrium balance. This increased activity leads to a higher cell potential as the system tries to restore equilibrium.

Think of the cell potential as the voltage difference between two points. A higher concentration of \(Cu^{2+}\) acts like adding more "electrical pressure" on the reduction half-cell, forcing the reaction to occur and raising the overall voltage (cell potential).

By increasing the \(Cu^{2+}\) concentration, we manipulate the reaction quotient \((Q)\) in the Nernst equation, leading to a more positive cell potential as the system strives to maintain equilibrium.