An electrochemical cell is constructed such that on one side a pure nickel electrode is in contact with a solution containing \(\mathrm{Ni}^{2+}\) ions at a concentration of \(3 \times 10^{-3} M\). The other cell half consists of a pure Fe electrode that is immersed in a solution of \(\mathrm{Fe}^{2+}\) ions having a concentration of \(0.1 M\). At what temperature will the potential between the two electrodes be \(+0.140 \mathrm{~V} ?\)

Understanding the Nernst equation is crucial for analyzing the behavior of electrochemical cells under non-standard conditions. While the standard cell potential gives us the potential difference under ideal conditions, real-world scenarios require an adjustment for the actual concentrations of the reacting species.

The Nernst equation provides this adjustment by incorporating the effect of ion concentrations on the cell potential. Given by the formula:
\[ E = E^{0} - \frac{RT}{nF}\ln{Q} \]
where \( E \) is the cell potential, \( E^{0} \) is the standard cell potential, \( R \) is the universal gas constant, \( T \) is the temperature in Kelvin, \( n \) is the number of moles of electrons transferred, \( F \) is Faraday's constant, and \( Q \) is the reaction quotient.

This equation is vital when considering the effects of concentration, as it allows us to calculate how the potential of a cell will change when the amounts of the reactants and products vary from their standard conditions. For example, if the concentration of the cathode increases, the potential will decrease, according to the relationship given by the Nernst equation. This is essential for calculating the exact cell potential needed to predict the direction of electron flow and the feasibility of a chemical reaction in the cell.

The standard cell potential, denoted as \( E^{0} \), is a measure of how much voltage, or electric potential, a cell can produce under standard conditions. These conditions generally include all solutes at a concentration of 1 M and all gases at a pressure of 1 atm, with a temperature usually set at 298 K (25 °C).

The standard cell potential is calculated from the difference in standard electrode potentials of the two half-cells, which are based on the reduction potential of each half-reaction at the electrode. In the given exercise, the standard cell potential is:
\[ E^{0} = E^0_{\mathrm{Ni}} - E^0_{\mathrm{Fe}} = 0.25\, \mathrm{V} - (-0.44\, \mathrm{V}) = 0.69\, \mathrm{V} \]
This value represents the maximum potential difference that can be achieved by the cell when operating under standard conditions and is crucial for predicting the direction of spontaneous reaction and for calculating non-standard cell potentials using the Nernst equation.

The relationship between electrode potential and temperature is a reflection of the fundamental thermodynamic principles governing electrochemical cells. According to the Nernst equation, temperature directly affects the voltage of an electrochemical cell.

When the temperature increases, the potential typically decreases if the reaction quotient \( Q \) is greater than 1, and vice versa. This implies that the temperature can influence the direction and extent of chemical reactions taking place within the cell.

In our exercise, the temperature at which the potential is +0.140 V was found by rearranging the Nernst equation to solve for \( T \). The increase or decrease of temperature affects the potential energy of the reactants and products, which in turn affects the overall cell potential. The exercise demonstrates how the Nernst equation can be used to determine temperatures at which specific cell potentials will occur, highlighting the direct and quantifiable connection between temperature and electrochemical potential.