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

Step-by-Step Solution

Answer

The value of \( x \) is approximately 3.4.

Step 1: Understanding the Nernst Equation

The Nernst equation relates the electrode potential to the concentration of ions in solution. It is given by the formula: \( E = E^0 - \frac{RT}{nF} \ln Q \), where \( E^0 \) is the standard electrode potential, \( R \) is the gas constant, \( T \) is the temperature in Kelvin, \( n \) is the number of moles of electrons exchanged, \( F \) is Faraday's constant, and \( Q \) is the reaction quotient. For \ \mathrm{Cu^{2+} + 2e^- \rightarrow Cu}, n = 2.

Step 2: Simplifying the Nernst Equation

The equation can be simplified using common constants and conditions (assuming \( T = 298 \) K). Then the equation simplifies to \( E = E^0 - \frac{0.059}{n} \log[Cu^{2+}] \). Setting \( E = 0 \) (since we're looking for the concentration at which the electrode potential is zero), we get \( 0 = 0.34 - \frac{0.059}{2} \log[Cu^{2+}] \).

Step 3: Solving for the Logarithm

Rearranging the equation, we get \( \log[Cu^{2+}] = \frac{0.34 \times 2}{0.059} \) after moving terms around. This becomes \( \log[Cu^{2+}] = \frac{0.68}{0.059} \). Calculate this value.

Step 4: Calculating the Logarithmic Value

Compute \( \frac{0.68}{0.059} \) which equals approximately 11.525. Therefore, \( \log[Cu^{2+}] = -11.525 \) after considering the negative sign since we are moving it across the equation.

Step 5: Determine the Concentration

From \( \log[Cu^{2+}] = -11.525 \), rewrite it in terms of \([Cu^{2+}]\): \([Cu^{2+}] = 10^{-11.525} \). This is equivalent to \([Cu^{2+}] = x \times 10^{-12} \), where \( x \approx 3.4 \), based on the given logs like \( \log 3.4 = 0.5315 \).

Key Concepts

Nernst Equation

The Nernst Equation is a fundamental equation in electrochemistry that describes how the electrochemical potential of a half-cell depends on the concentration of ions. When we delve into electrochemical cells, understanding this equation is crucial. It essentially shows how potential, or voltage, changes with ion concentration, allowing us to predict how a cell functions in different conditions.
For the reaction involving copper ions,
\[ \mathrm{Cu^{2+} + 2e^- \rightarrow Cu} \],
the Nernst Equation is given by:
\[ E = E^0 - \frac{RT}{nF} \ln Q \],
where:

• \( E \) is the electrode potential.
• \( E^0 \) is the standard electrode 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.

In our specific example where the temperature is 298 K, these constants allow the equation to simplify, making it easier to calculate the potential without complex mathematics. This simplicity enables students to focus on understanding core concepts rather than complicated computations in basic exercises.

Standard Electrode Potential

Standard Electrode Potential, often denoted as \( E^0 \), is a critical component when using the Nernst Equation. It represents the voltage associated with a reduction reaction at an electrode when concentrations are at standard conditions (1 M concentration).
The concept of standard conditions helps compare different half-cell reactions by providing a uniform reference point. For copper, as given in our problem:

• The standard electrode potential \( \mathrm{Cu^{2+}/Cu} \) is 0.34 V.

Understanding \( E^0 \) is essential because it forms the baseline potential of the electrode. The change from this potential when concentration varies is what the Nernst Equation calculates. Without this reference, determining the real-time potential is impossible.
Always make sure to remember that the standard conditions apply only as a basis for calculation—a real scenario will need adjustment as per the actual concentration and conditions.

Logarithmic Calculations

Logarithmic calculations play a vital role in using the Nernst Equation, especially when dealing with concentrations on a large scale. Instead of handling tiny or massive numbers, we use logarithms to simplify the math involved.
When you rearrange the simplified Nernst equation to solve for concentration in this exercise, you reach the step:

• \( \log[Cu^{2+}] = \frac{0.34 \times 2}{0.059} \)

This form allows you to solve for \([Cu^{2+}]\) without needing to perform cumbersome multiplication or division on large exponentiations. By converting a complex division into a single logarithm computation, the process is greatly streamlined.
In our example, solving produces a number like
\[ \log[Cu^{2+}] = -11.525 \],
implying
\([Cu^{2+}] = 10^{-11.525} \),
which translates into human-friendly terms through the use of given logarithm values for simplification.
As you become familiar with intuition on approximations and values, these calculations make for a powerful tool in chemistry!