Problem 96 - Chemistry Principles and Practice

Question

The standard free energy change at \(25^{\circ} \mathrm{C}, \Delta G^{\circ},\) is equal to \(-34.3 \mathrm{~kJ}\) for $$ 2 \mathrm{Fe}(\mathrm{CN})_{6}^{3-}(\mathrm{aq})+2 \mathrm{I}^{-}(\mathrm{aq}) \rightarrow 2 \mathrm{Fe}(\mathrm{CN})_{6}^{4-}(\mathrm{aq})+\mathrm{I}_{2}(\mathrm{~s}) $$ Calculate the standard potential for this reaction.

Step-by-Step Solution

Answer

The standard potential \( E^{\circ} \) is approximately 0.178 V.

Step 1: Understand the Relation between ΔG° and E°

The standard free energy change (ΔG°) and the standard electrode potential (E°) are related through the equation: \[ \Delta G^{\circ} = -nFE^{\circ} \] where \( n \) is the number of moles of electrons transferred in the balanced equation, \( F \) is the Faraday constant (about 96485 C/mol), and \( E^{\circ} \) is the standard cell potential.

Step 2: Identify the Number of Electrons Transferred

The given reaction is: \[ 2\mathrm{Fe} (\mathrm{CN})_{6}^{3-} + 2 \mathrm{I}^{-} \rightarrow 2 \mathrm{Fe}(\mathrm{CN})_{6}^{4-} + \mathrm{I}_{2} \] In this reaction, each \(\mathrm{Fe} (\mathrm{CN})_{6}^{3-}\) is reduced to \( \mathrm{Fe}(\mathrm{CN})_{6}^{4-}\) by gaining one electron, and two \( \mathrm{I}^{-} \) ions together lose two electrons to form \( \mathrm{I}_2 \). Thus, \( n = 2 \).

Step 3: Substitute Values into the Equation

Now, substitute the known values into the equation: \[ \Delta G^{\circ} = -nFE^{\circ} \] which becomes: \[ -34.3 \times 10^3 \mathrm{~J/mol} = -(2)(96485 \mathrm{~C/mol}) E^{\circ} \]

Step 4: Solve for E°

Rearrange the equation to solve for \( E^{\circ} \): \[ E^{\circ} = \frac{-34.3 \times 10^3 \mathrm{~J/mol}}{-(2)(96485 \mathrm{~C/mol})}\] Calculate this to find: \[ E^{\circ} = \frac{34.3 \times 10^3}{192970} \approx 0.178 \mathrm{~V} \]

Key Concepts

Standard Free Energy Change

In electrochemistry, the standard free energy change, represented as ΔG°, is a vital concept that helps us understand the spontaneity of a chemical reaction. The key idea here is that free energy change is a measure of work a system can perform.
- **Negative ΔG°:** If ΔG° is negative, the reaction proceeds spontaneously under standard conditions.- **Positive ΔG°:** Conversely, a positive value implies that the reaction is non-spontaneous.
In the exercise, we see that ΔG° is (-34.3 \mathrm{~kJ/mol}), showing that the reaction proceeds spontaneously towards product formation. This value is linked with the standard cell potential, E°, via the formula: \[ \Delta G^{\circ} = -nFE^{\circ} \]
where \( n \) is the number of moles of electrons transferred, and \( F \) is the Faraday constant. Understanding this equation is key as it ties together energy and electrochemical potential, providing insights into how chemical energy is converted into electrical energy during reactions.

Standard Cell Potential

The standard cell potential, E°, is a crucial concept in understanding how electrochemical cells work. It can be thought of as the 'electromotive force' of a cell, indicating the voltage produced when the cell operates under standard conditions.
E° is determined using the formula:\[ E^{\circ} = \frac{\Delta G^{\circ}}{-nF} \]This formula combines concepts of energy change (ΔG°) and electron transfer, allowing for the calculation of the E° value in volts.
- **Impact of Positive E°:** A positive E° indicates a reaction that releases energy, suggesting that it's spontaneous.- **Impact of Negative E°:** A negative E° thus implies that energy is required for the reaction to occur.
In the exercise, we calculated E° as approximately 0.178 V, which means that the reaction can produce electrical energy spontaneously. This is crucial for applications such as battery design, where the goal is to maximize energy extraction efficiently.

Faraday Constant

The Faraday constant, represented as \( F \), links electrical charge to moles of electrons. It's a fundamental constant in electrochemistry with a value of approximately 96485 C/mol (coulombs per mole). It enables us to calculate the charge transferred in a reaction when given the amount of substance.
- **Understanding its Role:** The Faraday constant forms a bridge between the macroscopic world of electrical currents and the microscopic world of chemistry.- **Applications:** By knowing \( F \), we can convert between energy changes in a chemical system and the corresponding electrical potential or current produced.
In the given exercise, \( F \) is essential for calculating the standard cell potential (E°) from the known ΔG° value. Its use confirms the deep connection between the movement of electrons and the free energy changes of reactions. Keeping \( F \) in mind helps us appreciate the power of electrochemical reactions in practical applications like batteries and fuel cells.