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 \(\mathrm{CyFe}^{2+}\) : $$ \begin{aligned} \mathrm{O}_{2}(\mathrm{~g})+4 \mathrm{H}^{+}(a q)+4 \mathrm{e}^{-}--\rightarrow 2 \mathrm{H}_{2} \mathrm{O}(l) & E_{\mathrm{red}}^{\circ}=+0.82 \mathrm{~V} \\ \mathrm{CyFe}^{3+}(a q)+\mathrm{e}^{-}--\rightarrow \mathrm{CyFe}^{2+}(a q) & E_{\mathrm{red}}^{\mathrm{o}}=+0.22 \mathrm{~V} \end{aligned} $$ (a) What is \(\Delta G\) for the oxidation of \(C y F e^{2+}\) 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 \(\mathrm{O}_{2}\) ?

Short answer

The oxidation of \(\mathrm{CyFe}^{2+}\) by air results in a Gibbs free energy change of \(\Delta G = -231{,}564\,\text{J/mol}\). Approximately 6 moles of ATP are synthesized per mole of \(\mathrm{O}_{2}\).

Step-by-step solution

Calculate the overall reaction

In order to find the Gibbs free energy change, we need to determine the overall reaction that takes place when \(\mathrm{CyFe}^{2+}\) is oxidized by air. To do this, we will combine the two half-cell reactions given in the exercise: \[ \begin{aligned} \mathrm{O}_2(\mathrm{g}) + 4\mathrm{H}^{+}(aq) + 4\mathrm{e}^{-} \rightarrow 2\mathrm{H}_2\mathrm{O}(l) \\ \mathrm{CyFe}^{3+}(aq) + \mathrm{e}^{-} \rightarrow \mathrm{CyFe}^{2+}(aq) \end{aligned} \] Since 4 electrons are involved for the reduction of \(\mathrm{O}_2\) to water, while 1 electron is involved for the oxidation of \(\mathrm{CyFe}^{2+}\), we have to multiply the second half-cell reaction by 4 to balance the number of electrons: \[ 4(\mathrm{CyFe}^{3+}(aq) + \mathrm{e}^{-} \rightarrow \mathrm{CyFe}^{2+}(aq)) \] Now, add the reactions together and balance the equation to obtain the overall reaction: \[ \mathrm{O}_2(\mathrm{g}) + 4\mathrm{H}^{+}(aq) + 4\mathrm{CyFe}^{2+}(aq) \rightarrow 2\mathrm{H}_2\mathrm{O}(l) + 4\mathrm{CyFe}^{3+}(aq) \]

Calculate the cell's standard potential

Using the Nernst equation, we can calculate the cell's standard potential, \(E_\text{cell}^\circ\). The Nernst equation is: \[ E_\text{cell}^\circ = E_\text{red}^\circ(\text{cathode}) - E_\text{red}^\circ(\text{anode}) \] As \(\mathrm{O}_2\) is reduced, it will be the cathode; as \(\mathrm{CyFe}^{2+}\) is oxidized, it will be the anode. Therefore, the equation is: \[ E_\text{cell}^\circ = (+0.82\,\text{V}) - (+0.22\,\text{V}) = 0.60\,\text{V} \]

Calculate the Gibbs free energy change

Now that we have the cell's standard potential, we can calculate the Gibbs free energy change, \(\Delta G\), using the following formula: \[ \Delta G = -nFE_\text{cell}^\circ \] where \(n\) is the number of electrons transferred in the reaction and \(F\) is the Faraday constant, approximately 96,485 C/mol. In this case, \(n=4\) because 4 electrons were transferred in the balanced equation. \[ \Delta G = -(4\,\text{mol})(96{,}485\,\text{C/mol})(0.60\,\text{V}) = -231{,}564\,\text{C·V} = -231{,}564\,\text{J/mol} \]

Calculate the moles of ATP synthesized per mole of \(\mathrm{O}_{2}\)

In part (b), we are given that the synthesis of 1 mol of ATP requires a \(\Delta G\) of 37.7 kJ. We can use the ratio of \(\Delta G\) values to find the number of moles of ATP synthesized per mole of \(\mathrm{O}_{2}\): \[ \frac{\Delta G_\text{reduction}}{\Delta G_\text{synthesis}} = \frac{-231{,}564\,\text{J/mol}}{37{,}700\,\text{J/mol}} = -6.14 \] Rounding the result to the nearest whole number, we find that approximately 6 moles of ATP are synthesized per mole of \(\mathrm{O}_{2}\).

Key concepts

Cytochrome Function

Cytochromes are vital proteins found within cells that function as essential components in the electron transport chain, a key part of cellular respiration. These molecules contain heme groups, which include an iron (Fe) atom capable of transitioning between different oxidation states. This ability allows cytochromes to accept and transfer electrons. When cytochromes receive electrons, they move through different stages of oxidation and reduction, facilitating the transfer of electrons along the chain. This transfer is crucial for the production of ATP, an essential energy currency in cells. By acting as a link between different enzymes in the chain, cytochromes help ensure efficient electron flow, which ultimately supports ATP synthesis needed for various cellular processes. Without cytochromes, the chain would fail to operate effectively, impeding energy production.

Oxidation-Reduction Reactions

Oxidation-reduction reactions, often abbreviated as redox reactions, involve the transfer of electrons between chemical species. These reactions are fundamental in many biological processes, including metabolism and respiration.
In a redox reaction, one species undergoes oxidation, meaning it loses electrons, while another species undergoes reduction by gaining electrons. These reactions are generally paired, as the loss of electrons by one molecule must be matched by the gain of those electrons by another. In cellular respiration, the electron transport chain is a series of these paired reactions occurring within mitochondria or cell membranes. The chain starts with the oxidation of electron donors such as NADH and FADH2, which release electrons. These electrons pass through complexes and cofactors, such as cytochromes, in a series of controlled steps. As electrons travel through the chain, their energy is gradually released, allowing the cell to efficiently harvest energy. In this process, oxygen often serves as the final electron acceptor, being reduced to form water.

ATP Synthesis

ATP synthesis is the process by which cells produce adenosine triphosphate (ATP), a molecule that serves as an immediate energy source for various biological functions. The synthesis predominantly occurs in the mitochondria during cellular respiration. The energy for ATP production comes from the electron transport chain, particularly through a process known as oxidative phosphorylation. This involves using energy derived from electrons moving through the protein complexes in the electron transport chain to pump hydrogen ions across the mitochondrial membrane. This creates a proton gradient, often referred to as a chemiosmotic potential. The stored energy in this gradient is released as the protons flow back into the mitochondrial matrix through ATP synthase, an enzyme that catalyzes the formation of ATP from adenosine diphosphate (ADP) and inorganic phosphate. Each complete cycle of the chain results in the creation of several ATP molecules, converting the energy stored in food molecules into a usable form of cellular power.