Problem 63 - Chemistry and Chemical Reactivity

Question

The standard potential, \(E^{\circ},\) for the reaction of \(\mathrm{Zn}(\mathrm{s})\) and \(\mathrm{Cl}_{2}(\mathrm{g})\) is \(+2.12 \mathrm{V}\). What is the standard free energy change, \(\Delta_{\mathrm{r}} G^{\circ},\) for the reaction?

Step-by-Step Solution

Answer

The standard free energy change, \(\Delta_{\mathrm{r}} G^{\circ}\), is \(-408.78 \, \text{kJ/mol}\).

Step 1: Understanding the Nernst Equation

The relationship between the standard potential \(E^{\circ}\) and the standard free energy change \(\Delta_{\mathrm{r}}G^{\circ}\) is given by the equation: \[\Delta_{\mathrm{r}} G^{\circ} = -nFE^{\circ}.\] In this equation, \(n\) is the number of moles of electrons exchanged in the reaction, and \(F\) is Faraday's constant \( (F = 96485 \, \text{C/mol}) \).

Step 2: Identify the Number of Electrons Transferred

Look at the balanced chemical reaction. For the reaction \(\mathrm{Zn}(\mathrm{s}) + \text{Cl}_{2}(\text{g}) \rightarrow \mathrm{ZnCl}_2(\mathrm{s})\), zinc metal \(\mathrm{Zn}(\mathrm{s})\) is oxidized by chlorine gas \(\text{Cl}_2\) to form \(\mathrm{ZnCl}_2\). In this reaction, each \(\mathrm{Zn}^{2+}\) ion requires two electrons: \(n = 2\).

Step 3: Calculate the Standard Free Energy Change, \(\Delta_{\mathrm{r}} G^{\circ}\)

Use the values \(n = 2\), \(F = 96485 \, \text{C/mol}\), and \(E^{\circ} = 2.12 \, \text{V}\) in the formula: \[\Delta_{\mathrm{r}} G^{\circ} = - (2)(96485)(2.12).\] Calculate: \[\Delta_{\mathrm{r}} G^{\circ} = -408780.4 \, \text{J/mol} = -408.78 \, \text{kJ/mol}.\]

Step 4: Interpret the Result

The negative sign of \(\Delta_{\mathrm{r}} G^{\circ}\) indicates that the reaction is spontaneous under standard conditions. The calculated change of \(-408.78 \, \text{kJ/mol}\) signifies the amount of free energy available to do work.

Key Concepts

Nernst Equation

In electrochemistry, the Nernst Equation is crucial for understanding how different factors influence the potential of a galvanic cell. It primarily describes the relationship between the electromotive force (EMF) of a cell under non-standard conditions and the concentrations of the reacting species. However, it's important to remember that this exercise focuses on the relationship between standard potential and standard free energy change, not concentration effects. The Nernst Equation can be simplified, under standard conditions, to:\[ \Delta_{\mathrm{r}} G^{\circ} = -nFE^{\circ}, \]where:

• \( \Delta_{\mathrm{r}} G^{\circ} \) is the standard free energy change,
• \( n \) is the number of moles of electrons transferred,
• \( F \) is Faraday's constant, and
• \( E^{\circ} \) is the standard potential of the cell.

For the reaction given in the exercise, this simplified Nernst Equation helps us relate the standard cell potential to the free energy change standard under equilibrium conditions.

Standard Potential

The standard potential, denoted as \( E^{\circ} \), refers to the voltage or electrical potential difference of a cell under standard conditions—typically 1 M concentration, 1 atm pressure, and 25 °C. It reflects the tendency of a chemical species to be reduced or oxidized. If the standard potential is positive, as is the case in this exercise with \( E^{\circ} = +2.12 \, \text{V} \), it indicates a spontaneous reaction; the higher the value, the greater the driving force for the reaction.
In the context of electrochemical cells, the standard potential difference is essentially the maximum potential difference when the cell is operating efficiently, without external influences. This value is critical when calculating the standard free energy change using the formula \( \Delta_{\mathrm{r}} G^{\circ} = -nFE^{\circ}. \)

Electrons Transferred

When it comes to electrochemical reactions, understanding how many electrons are being transferred is pivotal. In the balanced reaction provided, \( \mathrm{Zn}( ext{s}) + \mathrm{Cl}_2( ext{g}) \to \mathrm{ZnCl}_2(\text{s}) \), the zinc (Zn) loses two electrons to form \( \mathrm{Zn}^{2+} \). Consequently, two chlorine anions \( \text{Cl}^- \) are produced, each gaining one electron. Hence, the total number of electrons transferred, \( n \), is 2.

• Zinc undergoes oxidation: \( \mathrm{Zn} \to \mathrm{Zn}^{2+} + 2e^- \).
• Chlorine undergoes reduction: \( \mathrm{Cl}_2 + 2e^- \to 2\mathrm{Cl}^- \).

This transfer number \( n = 2 \) is essential in the calculation of the standard free energy change, connecting it directly to both the Nernst Equation and Faraday's constant.

Faraday's Constant

Faraday's constant is a fundamental constant used in electrochemistry, represented by the symbol \( F \). It signifies the charge of one mole of electrons, approximately \( 96485 \, \text{C/mol} \). This constant is vital for converting chemical moles into electric charge, crucial for calculations involving electron moles, such as calculating the \( \Delta_{\mathrm{r}} G^{\circ} \), or standard free energy change.

• It helps quantify the amount of electricity needed to drive an electrochemical reaction.
• By combining \( n \) (moles of electrons) with Faraday’s constant, you can determine the total charge transferred in a reaction.
• This is especially significant in practical applications like determining the efficiency of a battery or electrochemical cell.

Using Faraday’s constant alongside the standard potential allows you to compute the energy efficiency of reactions accurately, as demonstrated in the exercise's solution.