Question

Cytochrome, a complicated molecule that we will represent as ${\mathrm{CyFe}}^{2+}$, 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 $\mathrm{pH}7.0$ the following reduction potentials pertain to this oxidation of ${\mathrm{CyFe}}^{2+}$ : $$\begin{array}{rl}{\mathrm{O}}_2(\mathrm{g})+4{\mathrm{H}}^{+}(aq)+4{\mathrm{e}}^{-}--\to 2{\mathrm{H}}_2\mathrm{O}(l)& E_{\mathrm{red}}^{\mathrm{o}}=+0.82\mathrm{V}\\ {\mathrm{CyFe}}^{3+}(aq)+{\mathrm{e}}^{-}--\to {\mathrm{CyFe}}^{2+}(aq)& E_{\mathrm{red}}^{\circ }=+0.22\mathrm{V}\end{array}$$ (a) What is $\mathrm{\Delta }G$ for the oxidation of ${\mathrm{CyFe}}^{2+}$ by air? (b) If the synthesis of $1.00$ mol of ATP from adenosine diphosphate (ADP) requires a $\mathrm{\Delta }G$ of $37.7\mathrm{kJ}$, how many moles of ATP are synthesized per mole of ${\mathrm{O}}_2$ ?

Short answer

(a) The ΔG for the oxidation of CyFe²⁺ by air is -232.76 kJ/mol. (b) Approximately 6.17 moles of ATP are synthesized per mole of O₂.

Step-by-step solution

Obtain the balanced redox equation

Combine the two given half-reactions by multiplying each by a factor that will make the number of electrons equal. Then add the two half-reactions to get the overall balanced equation. Oxidation half-reaction: $CyF{e}^{2+}(aq)\to CyF{e}^{3+}(aq)+e^{-}$ Reduction half-reaction: $O₂(g)+4H⁺(aq)+4e^{-}\to 2H₂O(l)$ Oxidation half-reaction: * 4 $$4CyF{e}^{2+}(aq)\to 4CyF{e}^{3+}(aq)+4e^{-}$$ Complete redox reaction: $$4CyF{e}^{2+}(aq)+O₂(g)+4H⁺(aq)\to 4CyF{e}^{3+}(aq)+2H₂O(l)$$

Calculate the cell potential

Using the given standard reduction potentials, find the overall cell potential ($E_{cell}$) for the balanced redox reaction. Reduction potential for O₂: $E{°}_{red(O₂)}=+0.82V$ Reduction potential for CyFe: $E{°}_{red(CyFe)}=+0.22V$ E° for oxidation and reduction reactions are additive when combined. Thus, we need to convert $E{°}_{red(CyFe)}$ to an oxidation potential and add it to $E{°}_{red(O₂)}$. To convert a reduction potential to an oxidation potential, reverse the sign: $E{°}_{ox(CyFe)}=-E{°}_{red(CyFe)}=-0.22V$ Now add the oxidation potential for CyFe and the reduction potential for O₂: $E_{cell}=E{°}_{ox(CyFe)}+E{°}_{red(O₂)}=(-0.22V)+(+0.82V)=+0.60V$

Calculate the Gibbs free energy change

Using the cell potential, calculate the Gibbs free energy change (ΔG) for the reaction using the formula ΔG = -nFE₀, where n is the number of moles of electrons transferred (4 in this case), F is Faraday's constant (96485 C/mol), and E₀ is the overall cell potential. ΔG = -nFE₀ ΔG = -(4 mol e⁻)(96485 C/mol e⁻)(0.60 V) ΔG = -232764 C·V/mol (1 C·V = 1 J/mol) ΔG = -232764 J/mol To convert joules to kJ, divide by 1000: ΔG = -232.76 kJ/mol

Calculate the moles of ATP synthesized

Calculate the number of moles of ATP synthesized per mole of O₂ using the Gibbs free energy change of ATP synthesis (37.7 kJ/mol) and the Gibbs free energy change of the reaction. moles of ATP synthesized per mole of O₂ = $\frac{\mathrm{\Delta }{G}_{reaction}}{\mathrm{\Delta }{G}_{ATPsynthesis}}$ moles of ATP synthesized per mole of O₂ = $\frac{-232.76kJ/mol}{37.7kJ/mol}$ moles of ATP synthesized per mole of O₂ ≈ 6.17 Since 1 mole of O₂ is required for the reaction, approximately 6.17 moles of ATP are synthesized per mole of O₂. Answer: (a) ΔG for the oxidation of CyFe2+ by air is -232.76 kJ/mol. (b) Approximately 6.17 moles of ATP are synthesized per mole of O₂.

Key concepts

Cytochrome and ATP Synthesis

Cytochromes are proteins found within the cells that play a crucial role in the electron transport chain, a process that ultimately leads to ATP synthesis. ATP, or adenosine triphosphate, is considered the energy currency of the cell, used to power various biological activities. During this process, cytochromes undergo oxidation and reduction as they transfer electrons through the respiratory chain in mitochondria.

This transfer of electrons is coupled with the pumping of protons across the mitochondrial membrane, leading to a proton gradient. The energy from this gradient is used by ATP synthase to convert adenosine diphosphate (ADP) into ATP. The efficiency of ATP production is directly related to the redox reactions occurring within these cytochrome complexes, making their role significant in bioenergetics.

Gibbs Free Energy

Gibbs free energy, denoted as $\mathrm{\Delta }G$, is a thermodynamic quantity that measures the amount of usable energy that can be extracted from a chemical reaction to do work at constant temperature and pressure. It is a powerful indicator of the spontaneity of a process. A negative $\mathrm{\Delta }G$ value implies that a reaction is energetically favorable and can occur spontaneously, while a positive value indicates that the reaction requires an input of energy to proceed. In biochemistry, $\mathrm{\Delta }G$ is essential for understanding the energy changes that accompany metabolic reactions, including the synthesis of ATP, which is a primary form of energy storage and transfer within cells.

Standard Reduction Potential

The standard reduction potential is a measure that indicates how likely a chemical species is to gain electrons and be reduced during a redox reaction. It is represented in volts (V) and is determined under standard conditions, which comprise a 1M concentration of the ion, a pressure of 1 atmosphere for gases, and a temperature of 25°C. In biochemistry, the standard reduction potential helps assess the flow of electrons in biological systems. Higher or more positive reduction potentials indicate a greater tendency to accept electrons. When dealing with complex molecules such as cytochromes, knowing their reduction potential helps to map out how electrons are transferred in the process of ATP synthesis.

Bioenergetics

Bioenergetics is the study of the flow and transformation of energy within living organisms, a critical aspect of cellular and molecular biology. It includes examining processes like ATP synthesis, where energy is stored, and its release to fuel other cellular activities. The principles of bioenergetics incorporate concepts like Gibbs free energy, standard reduction potentials, and the role of enzymes and chemical reactions in energy metabolism. Understanding bioenergetics is fundamental in exploring how organisms obtain energy from their environment and how energy is efficiently converted to support life processes.

Redox Reactions

Redox reactions, or oxidation-reduction reactions, are processes in which one substance transfers electrons to another, signifying a change in oxidation states. In biochemistry, these reactions are central to the metabolism of cells, occurring in sequences that release or store energy. The interplay of these reactions drives many biological processes, including cellular respiration and photosynthesis. Through specific enzyme-assisted pathways, redox reactions facilitate the breakdown or assembly of biomolecules, contributing to the complex energy management system of living organisms. The balanced equation from the exercise provides a clear example of a redox reaction where cytochrome is oxidized while oxygen is reduced, fundamental for comprehending energy generation in cells.