Problem 71 - Chemistry and Chemical Reactivity

Question

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

Step-by-Step Solution

Answer

The standard free energy change, \(\Delta G^\circ\), is \(-409.08\) kJ/mol.

Step 1: Recall the Formula for Standard Free Energy Change

The standard free energy change \(\Delta G^\circ\) is related to the standard electromotive force \(E^\circ\) by the formula \(\Delta G^\circ = -nF E^\circ\), where \(n\) is the number of moles of electrons transferred and \(F\) is the Faraday constant \(96,485 \) C/mol.

Step 2: Identify the Reaction and Electrons Transferred

The reaction involves zinc reacting with carbon. For this exercise, and given no additional information on the balanced equation, assume that zinc is oxidized to \(\text{Zn}^{2+}\) and carbon moves to a hypothetical state, but the reaction uses \(n = 2\) electrons, which is common for zinc standard reactions.

Step 3: Substitute Known Values into the Formula

Use the formula \(\Delta G^\circ = -nF E^\circ\) with \(n = 2\), \(F = 96,485 \) C/mol, and \(E^\circ = 2.12\) V. Substitute these values into the equation: \[ \Delta G^\circ = -(2)(96,485 \text{ C/mol})(2.12 \text{ V}) \].

Step 4: Calculate \(\Delta G^\circ\)

Perform the calculation: \[ \Delta G^\circ = -(2)(96,485)(2.12) \]. This results in \(\Delta G^\circ = -409,076.4\) J/mol. To convert joules to kilojoules, divide by 1,000: \(\Delta G^\circ = -409.08\) kJ/mol.

Key Concepts

Standard Potential

In electrochemistry, the _standard potential_ is denoted by the symbol \(E^{\circ}\). It is a measure of the tendency of a chemical species to be reduced, and it is measured in volts \(V\). A positive standard potential indicates that the reaction is spontaneous under standard conditions. Conversely, a negative \(E^{\circ}\) suggests that the reaction is non-spontaneous. In the context of the original exercise, the standard potential for the reaction of zinc with carbon is given as \(+2.12\) V. This positive \(E^{\circ}\) signifies that the reaction would proceed spontaneously under standard conditions. Understanding \(E^{\circ}\) is crucial as it helps predict the direction and feasibility of electrochemical reactions.

To find the standard potential:

• Perform half-reactions to identify potential values.
• Use a reference electrode, commonly the standard hydrogen electrode.
• Combine cell potentials of half-reactions accordingly.

Standard Free Energy Change

The _standard free energy change_, \(\Delta G^{\circ}\), helps to predict the spontaneity of a reaction. Linked with \(E^{\circ}\) through the equation \(\Delta G^{\circ} = -nFE^{\circ}\), it reflects the energy available to do work. In this formula:

• \(n\) represents the number of moles of electrons transferred.
• \(F\) is the Faraday constant \(96,485\) C/mol.
• \(E^{\circ}\) is the electromotive force.

For zinc's reaction with carbon, substituting the known values \(n = 2\), \(F = 96,485\) C/mol, and \(E^{\circ} = 2.12\) V yields \(\Delta G^{\circ} = -409.08\) kJ/mol.

A negative \(\Delta G^{\circ}\) indicates a spontaneous process, consistent with the positive \(E^{\circ}\). Converting joules to kilojoules is essential for practicality, reflecting meaningful energy changes in chemical contexts.

Faraday Constant

The _Faraday Constant_, \(F\), measures the electric charge carried by one mole of electrons. It is vital for understanding reactions in electrochemistry, where charge transfer occurs. The value of \(F\) is \(96,485\) C/mol. This constant bridges chemical reactions and electronic effects.

Why is it important?

• Links electron counts to measurable currents.
• Helps compute energy changes in electrochemical cells.
• Used in equations relating \(\Delta G^{\circ}\) and \(E^{\circ}\).

In calculations involving \(\Delta G^{\circ}\), \(F\) allows translation of mole-based reactions to an energy perspective. This ensures comprehensible predictions and assessments in electrochemical studies.

Electromotive Force

_Electromotive Force_ (EMF), also known as \(E^{\circ}\), quantifies the voltage or potential difference of an electrochemical cell. It reflects the cell's ability to drive an electric current. The term "force" can be misleading; it's more about potential energy per charge than a force in the physical sense.

Understanding EMF includes:

• Recognizing it as a predictor of reaction direction.
• Realizing it's the driving "force" behind electron flow.
• Calculating it as a measure of work done moving charge.

In the context of the exercise, the reaction of zinc and carbon yields an \(E^{\circ}\) of \(+2.12\) V, indicating the cell can spontaneously work towards equilibrium. EMF is crucial for assessing the net potential and efficiency of electrochemical cells.