Problem 116 - Chemistry The Central Science In Si Units

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

Step-by-Step Solution

Answer

\( \Delta G = -231.564 \mathrm{~kJ/mol} \); about 6.14 moles of ATP are made per mole of \( \mathrm{O}_2 \).

Step 1: Identify Half-Reactions

First, identify the two half-reactions from the provided data:1. Reduction of oxygen: \( \mathrm{O}_2(g) + 4 \mathrm{H}^+(aq) + 4 \mathrm{e}^- \rightarrow 2 \mathrm{H}_2 \mathrm{O}(l) \) with \( E_{\mathrm{red}}^{\circ} = +0.82 \mathrm{~V} \).2. Reduction of \(\mathrm{CyFe}^{3+}\): \( \mathrm{CyFe}^{3+}(aq) + \mathrm{e}^- \rightarrow \mathrm{CyFe}^{2+}(aq) \) with \( E_{\mathrm{red}}^{\circ} = +0.22 \mathrm{~V} \).

Step 2: Calculate Cell Potential

Calculate the standard cell potential \(E_{\mathrm{cell}}^{\circ}\) as: \[ E_{\mathrm{cell}}^{\circ} = E_{\mathrm{red}}^{\circ}(\mathrm{O}_2) - E_{\mathrm{red}}^{\circ}(\mathrm{CyFe}^{3+}) = +0.82 \mathrm{~V} - 0.22 \mathrm{~V} = +0.60 \mathrm{~V} \]

Step 3: Calculate Gibbs Free Energy Change

Use the formula \( \Delta G^{\circ} = -nFE_{\mathrm{cell}}^{\circ} \), where:\( n = 4 \) is the number of electrons,\( F = 96485 \) C/mol is the Faraday constant,and \( E_{\mathrm{cell}}^{\circ} = +0.60 \) V. Calculate \( \Delta G^{\circ} \): \[ \Delta G^{\circ} = -(4 \times 96485 \mathrm{~C/mol}) \times 0.60 \mathrm{~V} = -231564 \mathrm{~J/mol} \]Convert to kJ/mol: \( \Delta G^{\circ} = -231.564 \mathrm{~kJ/mol} \).

Step 4: Calculate ATP Production

Given \( \Delta G = 37.7 \mathrm{~kJ} \) per mole of ATP, calculate moles of ATP synthesized: \[ \text{Moles of ATP} = \frac{|\Delta G^{\circ}(\mathrm{O}_2)|}{\Delta G(\mathrm{ATP})} = \frac{231.564 \mathrm{~kJ/mol}}{37.7 \mathrm{~kJ/mol}} \approx 6.14 \]

Step 5: Conclusion

Thus, approximately 6.14 moles of ATP are synthesized per mole of \( \mathrm{O}_2 \).

Key Concepts

Reduction Potentials

Reduction potentials are vital in understanding how oxidation-reduction (redox) reactions generate energy. In these reactions, electrons are transferred between molecules, which can result in energy release if the reaction is spontaneous. The standard reduction potential, denoted as \( E_{\mathrm{red}}^{\circ} \), indicates how likely a chemical species is to gain electrons, essentially acting as an oxidizing agent. A positive \( E_{\mathrm{red}}^{\circ} \) suggests a strong tendency to gain electrons, while a negative value implies the opposite.

• A higher reduction potential means a greater likelihood of reduction occurring under standard conditions.
• Reduction potentials are measured in volts, with more positive values signifying more favorable reductions.

To calculate the overall cell potential \( E_{\mathrm{cell}}^{\circ} \) for a reaction, we use the formula:\[E_{\mathrm{cell}}^{\circ} = E_{\mathrm{red}}^{\circ}(\text{cathode}) - E_{\mathrm{red}}^{\circ}(\text{anode})\]This calculation helps in determining whether a redox reaction will proceed spontaneously. Together with the number of electrons transferred, it allows us to find out the Gibbs Free Energy change, \( \Delta G \). A positive cell potential indicates that the reaction can proceed spontaneously, often producing energy that can drive other processes like ATP synthesis.

Half-Reactions

In the context of redox chemistry, reactions are often split into two parts called half-reactions. Each half-reaction shows either the oxidation process (loss of electrons) or the reduction process (gain of electrons). By studying half-reactions, we can better understand how electrons move during a chemical reaction and how this movement generates energy.

• Every redox reaction consists of two half-reactions: one oxidation and one reduction.
• The sum of these half-reactions provides the full balanced redox equation.
• Half-reactions are useful because they highlight the electron transfer processes.

For example, in the provided exercise:- The reduction of oxygen is represented as: \( \mathrm{O}_2(g) + 4 \mathrm{H}^+(aq) + 4 \mathrm{e}^- \rightarrow 2 \mathrm{H}_2 \mathrm{O}(l) \) with \( E_{\mathrm{red}}^{\circ} = +0.82 \text{ V} \).- The reduction of \( \mathrm{CyFe}^{3+} \) is shown by: \( \mathrm{CyFe}^{3+}(aq) + \mathrm{e}^- \rightarrow \mathrm{CyFe}^{2+}(aq) \) with \( E_{\mathrm{red}}^{\circ} = +0.22 \text{ V} \).Understanding these half-reactions allows us to calculate the overall reaction's potential energy change, which in turn is critical for assessing whether the reaction can drive processes such as ATP synthesis effectively.

ATP Synthesis

ATP, or adenosine triphosphate, is the energy currency of the cell. It's essential for a vast number of cellular processes, including muscle contraction, neurotransmission, and chemical synthesis. Energy is stored in ATP in the form of high-energy phosphate bonds, releasing energy when those bonds are broken to form ADP, or adenosine diphosphate.
The synthesis of ATP from ADP and inorganic phosphate requires energy, which is often derived from redox reactions in biological systems. One common source of this energy is the electron transport chain, where molecules like oxygen are reduced, releasing energy that drives ATP production:

• Each mole of ATP requires a \( \Delta G \) of about 37.7 kJ.
• Molecules like \( \mathrm{CyFe}^{2+} \) play a role in transferring electrons, allowing efficient ATP synthesis.

In the biological context, ATP synthesis is facilitated by proteins, such as ATP synthase, which catalyze the conversion of ADP to ATP using the energy released from electron transfers. The efficiency of ATP production can be calculated using Gibbs Free Energy changes from redox reactions, such as those involving oxygen and other reactants in the transport chain.
Thus, understanding the concept of ATP synthesis is crucial not only in biology but also in bioenergetics and medicine, as it provides the foundational energy mechanism upon which countless cellular activities rely.