Problem 170 - A Complete Resource Book in Chemistry for JEE Main

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 \([\) lig \(2=0.3010, \log 3=0.4771\) and \(\log 3.4=0.5315]\)

Step-by-Step Solution

Answer

The value of \(x\) is 3.14.

Step 1: Understand the Nernst Equation

The Nernst equation is used to calculate the electrode potential at any given concentration. It can be written as: \[ E = E^0 - \left( \frac{0.0591}{n} \right) \log [ \text{concentration of ions}] \]where \(E\) is the electrode potential, \(E^0\) is the standard electrode potential, \(n\) is the number of electrons transferred, and the concentration is in molarity (M).

Step 2: Substitute Known Values into Nernst Equation

For the reduction reaction \( \text{Cu}^{2+} + 2e^- \rightarrow \text{Cu} \), the standard electrode potential \(E^0\) is given as 0.34 V. We want \(E\) to be zero. Therefore:\[ 0 = 0.34 - \left( \frac{0.0591}{2} \right) \log [ \text{Cu}^{2+}] \]

Step 3: Rearrange to Solve for Ion Concentration

Rearrange the equation to solve for \( \log [ \text{Cu}^{2+}] \):\[ \left( \frac{0.0591}{2} \right) \log [ \text{Cu}^{2+}] = 0.34 \]\[ \log [ \text{Cu}^{2+}] = \frac{0.34 \times 2}{0.0591} \]

Step 4: Calculate Logarithm Value

Calculate the logarithmic value:\[ \log [ \text{Cu}^{2+}] = \frac{0.68}{0.0591} \approx 11.501 \]

Step 5: Convert Logarithm to Concentration

Converting the logarithmic form to concentration form:\[ [ \text{Cu}^{2+}] = 10^{-11.501} \]This gives us the concentration as \(x \times 10^{-12}\).

Step 6: Determine the Value of x

Since \( 10^{-11.501} \text{ can be written as } 3.14 \times 10^{-12} \), the value of \(x\) that corresponds to the concentration \([ \text{Cu}^{2+}] = x \times 10^{-12} \text{M} \) is 3.14.

Key Concepts

Nernst equation

The Nernst equation is a fundamental concept in electrochemistry used to calculate the electrode potential of a cell when the ion concentration is not at standard conditions. This equation helps link the standard electrode potential to the actual conditions in a cell, such as ion concentration. It is expressed as:\[ E = E^0 - \left( \frac{0.0591}{n} \right) \log [ \text{ion concentration} ] \]**Key components of the Nernst equation include:**

• \(E\): The actual electrode potential at the given conditions.
• \(E^0\): The standard electrode potential measured under standard conditions (usually when all ions are at 1 M concentration).
• \(n\): The number of electrons transferred in the reaction.
• \([\text{ion concentration}]\): The concentration of the ions involved, which directly affects the electrode potential.

Breaking down this equation, the difference between \(E\) and \(E^0\) depends on the ion concentration, which is modified using the logarithm. The factor \(0.0591\) is specifically for temperature at 25°C and uses base–10 logarithms. Recognizing how subtly changes in ion concentration can impact potential is crucial in predicting and adjusting reactions in batteries and electrochemical cells.

Standard electrode potential

The standard electrode potential, symbolized as \(E^0\), is the measure of the energy gained or lost when a chemical species undergoes reduction or oxidation under standard conditions. It is a critical component in determining the driving force of an electrochemical reaction. Standard conditions imply that all solutes are at 1 M concentration, gases at 1 atm pressure, and the temperature is 25°C.**Importance of Standard Electrode Potential:**

• \(E^0\) values help predict the direction of electron flow in redox reactions. Positive \(E^0\) values typically indicate a tendency to gain electrons and become reduced.
• Comparing \(E^0\) values for different half-reactions can help in determining which reactions are more likely to proceed spontaneously.
• They are used to calculate the overall cell potential for galvanic cells, which is pivotal in evaluating the capability of electrochemical cells to do work.

In practice, standard electrode potentials provide a baseline from which real-world potentials can be adjusted using the Nernst equation, accommodating variations from standard conditions.

Concentration of ions

The concentration of ions within a solution is fundamental to the Nernst equation and impacts electrode potentials significantly. In electrochemistry, changing the concentration of ions can either enhance or diminish the driving force for electron movement, altering the electrochemical behavior of a system. **Influence of Ion Concentration:**

• Increasing ion concentration can drive a reaction towards completion by shifting the equilibrium, often resulting in a higher electrode potential.
• Decreasing ion concentration has the opposite effect, reducing the electrode potential as fewer ions are available to participate in redox reactions.
• Logarithms in the Nernst equation emphasize that even small changes in concentration can significantly impact the cell potential.

Manipulating ion concentration is a powerful tool in electrochemistry used in applications such as batteries, electroplating, and sensors, making it essential to understand and control in both industrial and laboratory settings.