Problem 116 - Chemistry The Central Science

Question

Cytochrome, a complicated molecule that we will represent as CyFe \(^{2+},\) reacts with the air we breathe to supply energy required to synthesize adenosine triphosphate (ATP). The body uses ATP as an energy source to drive other reactions (Section 19.7). At pH 7.0 the following reduction potentials pertain to this oxidation of \(\mathrm{CyFe}^{2+} :\) $$ \begin{aligned} \mathrm{O}_{2}(g)+4 \mathrm{H}^{+}(a q)+4 \mathrm{e}^{-} \longrightarrow 2 \mathrm{H}_{2} \mathrm{O}(l) & E_{\mathrm{red}}^{\circ}=+0.82 \mathrm{V} \\ \mathrm{CyFe}^{3+}(a q)+\mathrm{e}^{-} \longrightarrow \mathrm{CyFe}^{2+}(a q) & E_{\mathrm{red}}^{\circ}=+0.22 \mathrm{V} \end{aligned} $$ (a) What is \(\Delta G\) for the oxidation of CyFe \(^{2+}\) by air? (b) If the synthesis of 1.00 mol of ATP from adenosine diphosphate (ADP) requires a \(\Delta G\) of 37.7 \(\mathrm{kJ}\) , how many moles of ATP are synthesized per mole of \(\mathrm{O}_{2} ?\)

Step-by-Step Solution

Answer

The oxidation of CyFe²⁺ by air has a ΔG of -401.41 kJ/mol, and about 10.7 moles of ATP are synthesized per mole of O₂.

Step 1: Determine the overall cell potential

The overall cell potential can be found by subtracting the reduction potential of the oxidation half-reaction (CyFe²⁺) from the reduction potential of the reduction half-reaction (O₂): E°cell = E°red(reduction) - E°red(oxidation) For the given half-reactions, \(E°_{red}(O₂) = 0.82\) V \(E°_{red}(CyFe^{2+}) = +0.22\) V Now we can calculate the overall cell potential: E°cell = 0.82 V - (-0.22 V) = 1.04 V

Step 2: Calculate the Gibbs free energy change

We can now use the overall cell potential to calculate the Gibbs free energy change for the oxidation reaction. The formula for this is: ΔG = -nFE°cell where n is the number of moles of electrons transferred, F is the Faraday constant (96,485 C/mol), and E°cell is the overall cell potential. Since 4 moles of electrons are transferred in the reaction, we can now calculate the Gibbs free energy change: ΔG = -4 mol × 96,485 C/mol × 1.04 V ΔG = -401,406.72 J/mol ΔG = -401.41 kJ/mol

Step 3: Determine the moles of ATP synthesized per mole of O₂

Now that we know the Gibbs free energy change per mole of O₂ for the oxidation of CyFe²⁺, we can determine how many moles of ATP are synthesized per mole of O₂. Given that the synthesis of 1 mol of ATP requires ΔG of 37.7 kJ, we can use the following relationship to solve for the number of moles of ATP: moles of ATP = ΔG(oxidation of CyFe²⁺) / ΔG(synthesis of 1 mol ATP) moles of ATP = (-401.41 kJ/mol) / (37.7 kJ/mol) moles of ATP ≈ 10.7 Therefore, about 10.7 moles of ATP are synthesized per mole of O₂.

Key Concepts

Gibbs Free Energy

The concept of Gibbs free energy is central to understanding chemical reactions and processes such as ATP synthesis. It tells us whether a reaction can occur spontaneously under constant temperature and pressure. The Gibbs free energy, denoted as \( \Delta G \), is calculated using the following equation:

\[ \Delta G = \Delta H - T \Delta S \]
where \( \Delta H \) represents the change in enthalpy, \( T \) stands for absolute temperature in Kelvins, and \( \Delta S \) symbolizes the change in entropy. For the oxidation of \( \mathrm{CyFe}^{2+} \) by air, we calculated a \( \Delta G \) value using the cell potential, which is related to the work that can be done by the reaction. A negative \( \Delta G \) indicates a spontaneous process, which in this exercise, leads to the creation of energy that can be used for ATP synthesis.

Electrochemical Cell Potential

Electrochemical cell potential, also known as the standard reduction potential, is a measure of the tendency of a chemical species to be reduced, and by extension, the tendency of its conjugate form to lose electrons, or oxidize. Measured in volts \( (V) \), it is a critical factor in predicting the direction of electron flow in redox reactions and plays a key role in electrochemical cells.

In the exercise provided, the electrochemical cell potential was determined by combining the reduction potential of oxygen with the oxidation potential of \( \mathrm{CyFe}^{2+} \) to find the overall cell potential. This overall potential is indicative of the driving force behind the electrons during the reaction and is essential for calculating the Gibbs free energy change, which informs us about spontaneity and energetic feasibility of the oxidation process.

ATP Synthesis

ATP synthesis is a fundamental biological process where adenosine diphosphate (ADP) is converted to adenosine triphosphate (ATP) using energy from various sources. ATP acts as a universal energy currency in living organisms, and its synthesis is often coupled with energy-releasing processes like the oxidation of \( \mathrm{CyFe}^{2+} \) by air, as described in the exercise.

The energy required to synthesize one mole of ATP, illustrated by the \( \Delta G \) value, is a constant factor in deriving the efficiency and capacity of biochemical pathways that generate ATP. By comparing the \( \Delta G \) of the oxidation reaction to the energy required for ATP synthesis, we can determine how many moles of ATP can be produced from a given reaction, which is crucial for understanding metabolic energy yield.

Faraday's Constant

Faraday's constant, denoted by \( F \), is a fundamental physical constant used in the calculation of the Gibbs free energy change during an electrochemical reaction. Representing the charge of 1 mole of electrons, 96,485 coulombs per mole \( (C/mol) \), it directly links the amount of chemical change to the quantity of electricity involved in an electrochemical reaction.

When calculating the Gibbs free energy change using the formula \( \Delta G = -nFE^{\circ}_{cell} \), Faraday's constant ensures that the electrochemical cell potential is correctly translated into units of energy per mole. It's integral for bridging the gap between electrochemistry and thermodynamics, as it did in our exercise to estimate the change in free energy for the oxidation of \( \mathrm{CyFe}^{2+} \) and thereby predict the biological yield of ATP.

Reduction Potentials

Reduction potentials, often listed as standard reduction potentials in a half-cell notation, are intrinsic values that characterize the tendency of a substance to gain electrons. These potentials are presented in volts and provide a scale against which the relative strength of reductants and oxidants can be compared.

In the given exercise, reduction potentials for both \( \mathrm{O}_{2} \) and \( \mathrm{CyFe}^{3+} \) are provided, allowing us to calculate the associated change in free energy. Understanding how to manipulate and interpret these potentials is essential for solving problems related to electrochemistry, such as balancing redox reactions, predicting the feasibility of chemical processes, and assessing energetic changes, such as in the production of ATP.