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 \(\mathrm{CyFe}^{2+}\) : $$ \begin{aligned} \mathrm{O}_{2}(g)+4 \mathrm{H}^{+}(a q)+4 \mathrm{e}^{-} \longrightarrow 2 \mathrm{H}_{2} \mathrm{O}(l) & E_{\text {red }}^{*}=+0.82 \mathrm{~V} \\ \mathrm{CyFe}^{3+}(a q)+\mathrm{e}^{-} \longrightarrow \mathrm{CyFe}^{2+}(a q) & E_{\text {red }}^{+}=+0.22 \mathrm{~V} \end{aligned} $$ (a) What is \(\Delta G\) for the oxidation of CyFe \({ }^{2+}\) by air? (b) If the synthesis of \(1.00 \mathrm{~mol}\) 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 \(\mathrm{O}_{2}\) ?

Short answer

(a) The ΔG for the oxidation of CyFe 2+ by air is -231.6 kJ/mol. (b) Approximately 6.14 moles of ATP are synthesized per mole of O2.

Step-by-step solution

Write the half-reactions and the overall reaction.

The given half-reactions are: Reduction half-reaction: \( \mathrm{O}_{2}(g)+4 \mathrm{H}^{+}(a q)+4 \mathrm{e}^{-} \longrightarrow 2 \mathrm{H}_{2} \mathrm{O}(l) , E_{\text {red }}^{*}=+0.82 \mathrm{~V} \) Oxidation half-reaction: \( \mathrm{CyFe}^{3+}(a q)+\mathrm{e}^{-} \longrightarrow \mathrm{CyFe}^{2+}(a q) , E_{\text {red }}^{+}=+0.22 \mathrm{~V} \) Now we need to balance electrons between the two half-reactions. For that purpose, we will multiply the Oxidation half-reaction by 4. Then, we will combine both half-reactions to obtain the overall reaction: Overall Reaction: \( \mathrm{O}_{2}(g)+4 \mathrm{H}^{+}(a q)+4 \mathrm{CyFe}^{2+}(a q) \longrightarrow 2 \mathrm{H}_{2} \mathrm{O}(l) +4 \mathrm{CyFe}^{3+}(a q) \)

Calculate the overall cell potential, E cell.

To find the overall cell potential, we will subtract the E° for the oxidation half-reaction from the E° for the reduction half-reaction: \( E_\text{cell} = E_\text{red}^* - E_\text{red}^+ = (0.82 \ \text{V}) - (0.22 \ \text{V}) = 0.60 \ \text{V} \)

Calculate the Gibbs free energy change, ΔG.

Now, we calculate the Gibbs free energy change, ΔG, using the following equation: \( ΔG = -nFE_\text{cell} \) where n is the number of electrons transferred, F is Faraday's constant (96485 C/mol) and E cell is the overall cell potential. Since 4 electrons are transferred during the overall reaction, we have: \( ΔG = -4 \times 96485 \ \text{C/mol} \times 0.60 \ \text{V} = -231,564 \ \text{J/mol} = -231.6 \ \text{kJ/mol} \) (a) The ΔG for the oxidation of CyFe 2+ by air is -231.6 kJ/mol.

Calculate the number of moles of ATP synthesized per mole of O2.

Given that the synthesis of 1.00 mol of ATP from adenosine diphosphate (ADP) requires a ΔG of 37.7 kJ, we can now find the number of moles of ATP synthesized per mole of O2 using the energy available from the oxidation of CyFe 2+: \( \frac{\text{moles of ATP}}{\text{mol O}_2} = \frac{-231.6 \;\text{kJ/mol}}{37.7\;\text{kJ/mol}} = 6.14 \) (b) Approximately 6.14 moles of ATP are synthesized per mole of O2.

Key concepts

Electrochemistry and Redox Reactions

Electrochemistry is the study of chemical processes that cause electrons to move. This movement of electrons is what we refer to as electricity. A core part of electrochemistry involves redox reactions. These are reactions where oxidation and reduction occur simultaneously. In oxidation, a substance loses electrons, while in reduction, a substance gains electrons.

In the provided exercise, we look at a redox reaction involving a cytochrome:

• The reduction half-reaction is the transformation of oxygen into water, which has a specific reduction potential of +0.82 V.
• The oxidation half-reaction involves CyFe and results in the transformation of CyFe²⁺ to CyFe³⁺, with a reduction potential of +0.22 V.

To understand the overall cell potential, we subtract the reduction potential of the oxidation reaction from that of the reduction reaction. This gives us the voltage that drives the reaction forward.

The key takeaway here is that redox reactions are fundamental to driving electrochemical processes, including those that generate cellular energy.

Gibbs Free Energy in Redox Reactions

Gibbs free energy (\( \Delta G \) ) represents the maximum amount of work a thermodynamic system can perform at constant temperature and pressure. In chemistry, it's crucial for understanding whether a given reaction can occur spontaneously. For a reaction to be spontaneous, \( \Delta G \) must be negative.

The relationship between cell potential and Gibbs free energy is given by the formula:\( \Delta G = -nFE_{cell} \). Here,

• \( n \) is the number of electrons transferred in the reaction.
• \( F \) is Faraday's constant, which is approximately 96485 C/mol.
• \( E_{cell} \) is the overall cell potential, which we previously calculated as 0.60 V.

In the exercise, 4 electrons are being transferred. Substituting these values into the equation gives us:\( \Delta G = -4 \times 96485 \times 0.60 = -231.6 \text{ kJ/mol} \).

This negative \( \Delta G \) value indicates that the reaction is indeed capable of proceeding without any outside interference, and the resulting energy can be used by the cell.

ATP Synthesis and its Relation to Energy

Adenosine triphosphate (ATP) is often referred to as the "energy currency" of cells. It stores and supplies the energy needed for many biochemical cellular processes. ATP synthesis primarily occurs through cellular respiration, where redox reactions play a pivotal role.

In this particular problem, the energy released from the oxidation of CyFe²⁺ is used for the synthesis of ATP. We know from the problem that synthesizing one mole of ATP from adenosine diphosphate (ADP) requires \( \Delta G = 37.7 \text{ kJ} \).

• This means that the energy we calculated from the redox reaction (−231.6 kJ/mol) can be harnessed to form multiple ATP molecules.
• When calculated, approximately 6.14 moles of ATP can be synthesized per mole of oxygen consumed from this energy.

Understanding ATP synthesis in relation to energy from redox reactions clarifies how essential these processes are in biological systems for efficient energy production and usage.