Problem 7 - Chemistry: The Central Science in SI Units, Global Edition

Question

Consider a redox reaction for which \(E^{\circ}\) is a negative number. (a) What is the sign of \(\Delta G^{\circ}\) for the reaction? (b) Will the equilibrium constant for the reaction be larger or smaller than \(1 ?\) (c) Can an electrochemical cell based on this reaction accomplish work on its surroundings?

Step-by-Step Solution

Answer

(a) Since \(E^{\circ}\) is negative, \(\Delta G^{\circ}\) will be positive for the reaction. (b) The equilibrium constant (K) for the reaction will be smaller than 1. (c) An electrochemical cell based on this reaction cannot accomplish work on its surroundings.

Step 1: (Step 1: Relate Standard Cell Potential and Gibbs Free Energy)

We will use the above relationships to express the relationship between the standard potential and Gibbs free energy: \[\Delta G^{\circ} = -nFE^{\circ}\] where \(n\) is the number of moles of electrons transferred and \(F\) is Faraday's constant.

Step 2: (Step 2: Determine the Sign of Gibbs Free Energy)

Since \(E^{\circ}\) is given to be a negative number and both \(n\) and \(F\) are positive values, the multiplication of all three results in a positive number. Therefore, \(\Delta G^{\circ}\) is a positive number for this reaction as given: \[\Delta G^{\circ} > 0\]

Step 3: (Step 3: Relate Gibbs Free Energy and the Equilibrium Constant)

To answer (b), we will need to express the equilibrium constant (K) in terms of Gibbs free energy. We can do this using: \[\Delta G^{\circ} = -RT\ln{K}\] where \(R\) is the ideal gas constant and \(T\) is the temperature in Kelvin.

Step 4: (Step 4: Determine Whether the Equilibrium Constant is Greater or Smaller than 1)

Since we've determined in Step 2 that \(\Delta G^{\circ}>0\), we have: \[-RT\ln{K} > 0\] Divide both sides by \(-RT\): \[\ln{K} < 0\] Exponentiate both sides: \[K < 1\] This means that the equilibrium constant for this reaction will be smaller than 1.

Step 5: (Step 5: Determine the Electrochemical Cell's Ability to Accomplish Work)

As the standard cell potential (\(E^{\circ}\)) is negative, this implies that the electrochemical cell is not spontaneous. Thus, an electrochemical cell based on this reaction cannot accomplish work on its surroundings. To make the reaction spontaneous, an external voltage equal to or greater than the absolute of standard potential value must be applied. In summary: (a) The sign of \(\Delta G^{\circ}\) for the reaction is positive. (b) The equilibrium constant for the reaction will be smaller than 1. (c) An electrochemical cell based on this reaction cannot accomplish work on its surroundings.

Key Concepts

Redox Reaction

A redox reaction, short for reduction-oxidation reaction, is a chemical process in which electrons are transferred between two substances. This process involves two half-reactions: oxidation, where a substance loses electrons, and reduction, where another substance gains electrons.

Understanding redox reactions is crucial in electrochemistry, where these reactions facilitate the generation of electrical energy or driving chemical reactions via electric current. The movement of electrons occurs via an electrochemical cell, which can be a galvanic or voltaic cell in spontaneous reactions or an electrolytic cell in non-spontaneous reactions.

• Oxidation: Involves the loss of electrons from a species, increasing its oxidation state.
• Reduction: Involves the gain of electrons, resulting in a decrease in oxidation state.

The overall redox reaction is balanced regarding charge and mass, ensuring that electrons lost in the oxidation half are equal to those gained in the reduction half.

In this context, redox reactions are analyzed to determine various cell-related properties such as standard cell potential, Gibbs free energy, and the equilibrium constant.

Standard Cell Potential

The standard cell potential (\[ E^{\circ} \]) is a measure of the driving force behind a redox reaction and is indicative of the cell's ability to perform work. Measured in volts, this potential represents the difference in potential energy between the two half-cells of an electrochemical cell under standard conditions (1 M concentration, 1 atm pressure, and 25°C).

Calculation of the standard cell potential is achieved by subtracting the standard reduction potential of the anode from that of the cathode:\[ E^{\circ} = E^{\circ}_{\text{cathode}} - E^{\circ}_{\text{anode}} \]When the standard cell potential (\[ E^{\circ} \]) is positive, the reaction is spontaneous, capable of producing electrical work. Conversely, a negative\[ E^{\circ} \]indicates a non-spontaneous reaction, unable to perform work without external influence.

In electrochemistry, the value of \[ E^{\circ} \]relates directly to the reaction's thermodynamic feasibility and its capacity to release or absorb energy.

Gibbs Free Energy

Gibbs free energy (\[ \Delta G^{\circ} \]) is a thermodynamic property that provides insights into the spontaneity of chemical reactions. It is calculated using the formula:\[ \Delta G^{\circ} = -nFE^{\circ} \]where:

• \( n \): Number of moles of electrons transferred.
• \( F \): Faraday's constant (96485 C/mol).
• \( E^{\circ} \): Standard cell potential.

A negative \[ \Delta G^{\circ} \]indicates a spontaneous reaction, while a positive \[ \Delta G^{\circ} \]denotes non-spontaneity.

This concept is essential in electrochemistry because it also links to other fundamental properties such as the equilibrium constant. By understanding how Gibbs free energy relates to redox reactions, we can predict whether a reaction will occur under given conditions and how much useful work can be derived from it, reflecting either energy generation or consumption.

Equilibrium Constant

The equilibrium constant (\[ K \]) is a dimensionless value representing the ratio of the concentration of products to reactants at equilibrium. It can be expressed in terms of Gibbs free energy as:\[ \Delta G^{\circ} = -RT\ln K \]where:

• \( R \): Universal gas constant (8.314 J/mol·K).
• \( T \): Temperature in Kelvin.

When \[ \Delta G^{\circ} \]is greater than zero, the natural logarithm of \[ K \]is negative, indicating that \[ K \]is less than 1. This situation implies that the reactants are favored over products at equilibrium.

Understanding the equilibrium constant is crucial in predicting the position of equilibrium in redox reactions and assessing how changes in conditions might affect chemical balances. Such insights are vital for applications ranging from industrial chemical processes to biological systems, where equilibrium shifts can have significant outcomes.