Question

The standard electrode potential of \(\mathrm{Cu}^{2+} / \mathrm{Cu}=0.34 \mathrm{~V}\). The electrode potential will be zero, when the conc. of \(\mathrm{Cu}^{2+}\) is as \(\mathrm{x} \times 10^{-12} \mathrm{M}\). the value of \(\mathrm{x}\) is ___ .

Short answer

To solve this problem, we need to use the Nernst equation, which relates the concentration of ions to the electrode potential. The Nernst equation is given by:\[ E = E^0 - \frac{RT}{nF} \ln \left( \frac{1}{[ ext{Cu}^{2+}]} \right) \]Where:- \( E \) is the electrode potential.- \( E^0 \) is the standard electrode potential.- \( R \) is the universal gas constant \((8.314 \text{ J/mol K})\).- \( T \) is the temperature in Kelvin.- \( n \) is the number of moles of electrons exchanged in the reaction (for Cu, \( n = 2 \)).- \( F \) is the Faraday's constant \((96485 \text{ C/mol})\).

Step-by-step solution

Understanding the Nernst Equation

To solve this problem, we need to use the Nernst equation, which relates the concentration of ions to the electrode potential. The Nernst equation is given by:\[ E = E^0 - \frac{RT}{nF} \ln \left( \frac{1}{[ ext{Cu}^{2+}]} \right) \]Where:- \( E \) is the electrode potential.- \( E^0 \) is the standard electrode potential.- \( R \) is the universal gas constant \((8.314 \text{ J/mol K})\).- \( T \) is the temperature in Kelvin.- \( n \) is the number of moles of electrons exchanged in the reaction (for Cu, \( n = 2 \)).- \( F \) is the Faraday's constant \((96485 \text{ C/mol})\).

Key concepts

Electrode Potential

The electrode potential (E) of an electrochemical cell is the voltage difference between its electrodes. It tells us how strong the drive for electrons to move through the system is. When you think of electrode potential, envision it as the push that gets electrons moving from one place to another. This movement is not only important for powering batteries but also for various chemical reactions.
It's crucial to know that the value of electrode potential depends on several factors, such as:

• The identity of the metal and its ion involved.
• The concentration of the ions in the solution, which we will explore more.
• The temperature of the solution.

The solution sought in the exercise is an instance where the electrode potential becomes zero, meaning there's no net drive for electrons to move because the concentration of ions matches a particular threshold.

Standard Electrode Potential

The standard electrode potential (E^0) represents the electrode potential under standard conditions. It is measured when the ion concentration is 1 M, the temperature is 25^C (or 298 K), and the pressure is 1 atm. Given this set of fixed conditions, E^0 is a consistent measure, providing a benchmark to compare the tendencies of different cells to lose or gain electrons. Think of it as a way to rank electrochemical reactions by their potential power.
For instance, in the exercise, the standard electrode potential for Cu^{2+}/Cu is 0.34 V. This is the potential measured when copper ions in a solution are 1 M. A positive E^0 value, such as 0.34 V, suggests a tendency to gain electrons and undergo reduction.

Concentration of Ions

The concentration of ions plays a pivotal role in determining the electrode potential using the Nernst equation. As the ion concentration changes, so does the electrode potential. When ions in a solution are less than 1 M, the system will have a different potential than when ions are at standard state conditions.
Ions in a lower concentration than their standard state raise the electrode potential, and vice versa. To comprehend this through the Nernst equation \[ E = E^0 - \frac{RT}{nF} \ln \left( \frac{1}{[\text{Cu}^{2+}]} \right) \]we observe that a reduction in copper ions' concentration will impact the electrode potential, potentially leading it to zero, as discussed in the problem. This is because the fewer ions there are to undergo reduction, the less the driving force for the electron transfer, aligning the potential towards neutrality.