Problem 96 - Chemistry: Principles and Reactions

Question

In biological systems, acetate ion is converted to ethyl alcohol in a two- step process: \(\mathrm{CH}_{3} \mathrm{COO}^{-}(a q)+3 \mathrm{H}^{+}(a q)+2 e^{-} \longrightarrow \mathrm{CH}_{3} \mathrm{CHO}(a q)+\mathrm{H}_{2} \mathrm{O}\) \(E^{o \prime}=-0.581 \mathrm{~V}\) \(\mathrm{CH}_{3} \mathrm{CHO}(a q)+2 \mathrm{H}^{+}(a q)+2 e^{-} \longrightarrow \mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}(a q) \quad E^{o \prime}=-0.197 \mathrm{~V}\) \(\left(E^{\circ \prime}\right.\) is the standard reduction voltage at \(25^{\circ} \mathrm{C}\) and \(\mathrm{pH}\) of \(7.00 .\) ) (a) Calculate \(\Delta G^{\circ \prime}\) for each step and for the overall conversion. (b) Calculate \(E^{\circ \prime}\) for the overall conversion.

Step-by-Step Solution

Answer

Question: Calculate the standard Gibbs free energy change, ΔG°', and the standard reduction voltage, E°', for the overall conversion of acetate ion to ethyl alcohol in the given biological system. Answer: For the overall conversion of acetate ion to ethyl alcohol, the standard Gibbs free energy change, ΔG°', is 150049.2 J/mol, and the standard reduction voltage, E°', is -0.389 V.

Step 1: Calculating \(\Delta G^{\circ \prime}\) for each step

To calculate the standard Gibbs free energy change, use the Nernst equation: \(\Delta G^{\circ \prime} = -nFE^{\circ \prime}\) Where: \(ΔG^{\circ \prime}\) - standard Gibbs free energy change \(n\) - number of electrons transferred in the reaction \(F\) - Faraday's constant (96485 C/mol) \(E^{\circ \prime}\) - standard reduction voltage For step 1: \(n_1 = 2 \, e^{-}\) \(E^{\circ \prime}_1 = -0.581 \, V\) \(\Delta G^{\circ \prime}_1 = -2 \times 96485 \times (-0.581)\) \(\Delta G^{\circ \prime}_1 = 112069 \, J/mol\) For step 2: \(n_2 = 2 \, e^{-}\) \(E^{\circ \prime}_2 = -0.197 \, V\) \(\Delta G^{\circ \prime}_2 = -2 \times 96485 \times (-0.197)\) \(\Delta G^{\circ \prime}_2 = 37980.2 \, J/mol\)

Step 2: Calculating the overall reaction

Add the two half-reactions to get the balanced overall reaction: $\mathrm{CH}_{3} \mathrm{COO}^{-}(a q)+3 \mathrm{H}^{+}(a q)+2 e^{-} \longrightarrow \mathrm{CH}_{3} \mathrm{CHO}(a q)+\mathrm{H}_{2} \mathrm{O}$ $+\; \;\mathrm{CH}_{3} \mathrm{CHO}(a q)+2 \mathrm{H}^{+}(a q)+2 e^{-} \longrightarrow \mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}(a q)$ \(--------------------------------------------------------------\) $\mathrm{CH}_{3} \mathrm{COO}^{-}(a q)+5 \mathrm{H}^{+}(a q)+4 e^{-} \longrightarrow \mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}(a q)+\mathrm{H}_{2} \mathrm{O}$

Step 3: Calculating \(\Delta G^{\circ \prime}\) for the overall conversion

Add the \(\Delta G^{\circ \prime}\) values for each step to calculate the overall standard Gibbs free energy change: \(\Delta G^{\circ \prime}_{\text{overall}} = \Delta G^{\circ \prime}_1 + \Delta G^{\circ \prime}_2\) \(\Delta G^{\circ \prime}_{\text{overall}} = 112069 \, J/mol + 37980.2 \, J/mol\) \(\Delta G^{\circ \prime}_{\text{overall}} = 150049.2\, J/mol\)

Step 4: Calculating \(E^{\circ \prime}\) for the overall conversion

Since we have the total Gibbs free energy change and the number of electrons transferred (4) for the overall reaction, we can calculate the standard reduction voltage for the overall conversion using the Nernst equation: \(\Delta G^{\circ \prime}_{\text{overall}} = -n_{\text{overall}}FE^{\circ \prime}_{\text{overall}}\) \(150049.2 = -4 \times 96485 \times E^{\circ \prime}_{\text{overall}}\) \(E^{\circ \prime}_{\text{overall}} = -\frac{150049.2}{4 \times 96485}\) \(E^{\circ \prime}_{\text{overall}} = -0.389 \, V\)

Key Concepts

Standard Reduction Potential

Standard reduction potential, denoted as \( E^{ullet '}(E^0') \), is crucial in determining the tendency of a chemical species to acquire electrons and be reduced. This potential is measured in volts. A lower (more negative) standard reduction potential suggests that the species is less likely to gain electrons and be reduced under standard conditions, typically at 25°C and \( pH \) 7.0 for biochemical reactions. In our exercise, the two reactions have \( E^{\circ \prime} \) values of -0.581 V and -0.197 V, respectively. Here, the second reaction (acetaldehyde to ethanol) has a less negative \( E^{\circ \prime} \), indicating a stronger tendency to be reduced compared to the first step.

Electrochemistry

Electrochemistry is the study of chemical changes that produce electrical current and vice versa. It involves the transfer of electrons from one compound to another. In our reaction process, electrochemistry helps explain how acetate ions are converted into ethanol through electron transfer.

• The two-step conversion process in the exercise is a set of redox reactions, which are reactions where oxidation and reduction occur.
• Each step involves the exchange of electrons, quantified as "n" in reactions, which is critical in calculating \( \Delta G^{\circ \prime} \) (Gibbs free energy change).
• Electrochemistry also highlights the role of cell potential and free energy changes in predicting the directionality and spontaneity of the reaction.

Understanding electrochemistry allows us to comprehend how electrical energy can drive chemical reactions, which is essential in contexts like batteries and biological energy transformations.

Nernst Equation

The Nernst equation provides a quantitative relationship between the reaction quotient and the cell potential. It is useful for calculating cell potentials under non-standard conditions and directly relates to the Gibbs free energy change. In standard conditions, the equation simplifies to consider the number of electrons exchanged, \( n \) and standard potentials.

• It provides a mathematical framework to calculate the free energy change \( \Delta G^{\circ \prime} \) by using \( \Delta G^{\circ \prime} = -nFE^{\circ \prime} \).
• The Nernst equation becomes indispensable for determining the equilibrium state in electrochemical cells and understanding the links between chemical energy and electrical potential.
• In this exercise, it guides the transition from theoretical cell voltages to real-world usable energy values.

Applying the Nernst equation allows scientists and students to predict how a change in reactant concentration, temperature, or pressure affects electrochemical cell behavior, making it an invaluable tool in both laboratory and industrial settings.