Question

Consider the following reactions at \(25^{\circ} \mathrm{C}\) : $$ \begin{array}{r} \mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}(a q)+6 \mathrm{O}_{2}(g) \longrightarrow 6 \mathrm{CO}_{2}(g)+6 \mathrm{H}_{2} \mathrm{O} \\ \Delta G^{\circ}=-2870 \mathrm{~kJ} \\ \mathrm{ADP}(a q)+\mathrm{HPO}_{4}^{2-}(a q)+2 \mathrm{H}^{+}(a q) \longrightarrow \mathrm{ATP}(a q)+\mathrm{H}_{2} \mathrm{O} \\ \Delta G^{\circ}=31 \mathrm{~kJ} \end{array} $$ Write an equation for a coupled reaction between glucose, \(\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6},\) and \(\mathrm{ADP}\) in which \(\Delta G^{\circ}=-390 \mathrm{~kJ}\).

Short answer

Answer: The stoichiometry of the coupled reaction is: $$ C_6H_{12}O_6(aq) + 6O_2(g) + 80[ADP(aq) + HPO_4^{2-}(aq) + 2H^{+}(aq)] \rightarrow 6CO_2(g) + 6H_2O + 80[ATP(aq) + H_2O] $$

Step-by-step solution

Write down the given reactions and their Gibbs free energy changes

We are given the following reactions and their corresponding \(\Delta G^{\circ}\) values: 1) Glucose breakdown: \(C_6H_{12}O_6(aq) + 6O_2(g) \rightarrow 6CO_2(g) + 6H_2O\) \(\Delta G_1^{\circ} = -2870~kJ\) 2) ADP to ATP conversion: \(ADP(aq) + HPO_4^{2-}(aq) + 2H^{+}(aq) \rightarrow ATP(aq) + H_2O\) \(\Delta G_2^{\circ} = 31~kJ\)

Determine the stoichiometry of the coupled reaction

To find the stoichiometry of the coupled reaction, we need to find a linear combination of the two given reactions such that their combined \(\Delta G^{\circ}\) equals -390 kJ. Let's represent the stoichiometry of the two reactions as \(m\) and \(n\), respectively, where \(m\) and \(n\) are the stoichiometric coefficients. We need to find the values of \(m\) and \(n\) that satisfy the following equation: $$ m \Delta G_1^{\circ} + n \Delta G_2^{\circ} = -390~kJ $$

Calculate the stoichiometric coefficients of the coupled reaction

To find the values of \(m\) and \(n\), we can use the given \(\Delta G^{\circ}\) values: $$ m(-2870~kJ) + n(31~kJ) = -390~kJ $$ Now, we can see a relationship between the stoichiometric coefficients: $$ n = \frac{-390~kJ + 2870~kJ m}{31~kJ} $$ We need to find integer values for \(m\) and \(n\) that satisfy this relationship. After testing different integer values for \(m\), we find that \(m = 1\) and \(n = 80\) satisfy the above equation. Thus, the stoichiometry of the coupled reaction is: $$ C_6H_{12}O_6(aq) + 6O_2(g) + 80[ADP(aq) + HPO_4^{2-}(aq) + 2H^{+}(aq)] \rightarrow 6CO_2(g) + 6H_2O + 80[ATP(aq) + H_2O] $$ with a combined \(\Delta G^{\circ}\) of -390 kJ.

Key concepts

Gibbs Free Energy

Gibbs free energy, represented by the symbol \(G\), is a thermodynamic quantity that provides valuable insight into the direction and spontaneity of a chemical reaction. In essence, it represents the maximum amount of work a thermodynamic system can perform at constant temperature and pressure. When the change in Gibbs free energy for a reaction, denoted as \(\Delta G\), is negative, the reaction proceeds spontaneously, indicating that it releases energy. Conversely, a positive \(\Delta G\) signifies that the reaction is non-spontaneous and requires energy input.

In the context of biochemistry, Gibbs free energy is crucial for understanding metabolic reactions, such as the breakdown of glucose or the synthesis of ATP. For example, the breakdown of glucose to carbon dioxide and water has a highly negative \(\Delta G\), reflecting its spontaneity and release of energy. This energy can then be harnessed to drive other non-spontaneous cellular processes, such as the synthesis of ATP from ADP, through coupled reactions.

The Role of Gibbs Free Energy in Coupled Reactions

Coupled reactions are a core concept in biochemical energetics. They allow a non-spontaneous reaction with a positive \(\Delta G\) to proceed by pairing it with a spontaneous one that has a sufficiently large negative \(\Delta G\) to overcome the energy barrier. The combined energy profile must result in an overall negative \(\Delta G\) for the reactions to proceed spontaneously together. By leveraging the concept of Gibbs free energy, one can calculate not just the feasibility but also the stoichiometry of these coupled processes.

Chemical Stoichiometry

Chemical stoichiometry is the quantitative relationship between reactants and products in a chemical reaction. It involves using balanced chemical equations to determine the proportions of substances that react and the amounts of products formed. In biochemical systems, understanding stoichiometry is indispensable for figuring out how many molecules of one substance are required to react with another, or in the case of coupled reactions, to ensure overall energy requirements are met.

For students approaching problems in chemical stoichiometry, there is a necessity to become adept not only in balancing chemical equations but also in grasping the implications of these balances in terms of material conversion and energy changes. The exercise provided illustrates how stoichiometry helps in determining the correct amounts of reactants needed for ATP synthesis through glucose oxidation. By juxtaposing the stoichiometry of the glucose breakdown with the synthesis of ATP, one can infer the efficiency and yield of the coupled reaction.

Calculating Stoichiometric Coefficients in Biochemical Reactions

Calculating the stoichiometric coefficients involves algebraic manipulation of the reaction equations. The goal is to find the values that satisfy the relationship between the changes in Gibbs free energy for the overall coupled process. Precision in this calculation is vital, as even a slight misstep can lead to incorrect interpretation of a reaction's energetics. Thus, students must practice these calculations diligently to become comfortable with stoichiometry in various biochemical contexts.

ATP Synthesis

ATP synthesis is a fundamental biological process where the molecule adenosine triphosphate (ATP) is produced. ATP serves as the primary energy currency of the cell, providing the power needed for various biological activities, from muscle contraction to DNA replication. In the provided exercise, ATP is synthesized from adenosine diphosphate (ADP) and inorganic phosphate (\(HPO_4^{2-}\)).

The synthesis of ATP is often coupled with exergonic (energy-releasing) processes, such as the oxidation of glucose. This strategy is a hallmark of cellular energy conservation, ensuring that the energy released from spontaneous reactions isn't wasted but instead utilized to drive the formation of ATP. The concept of ATP synthesis is central to understanding bioenergetics, cellular respiration, and metabolism.

Mechanisms of ATP Synthesis

ATP can be synthesized through several biochemical pathways, including oxidative phosphorylation and substrate-level phosphorylation. In oxidative phosphorylation, the energy released from the electron transport chain is used to create a proton gradient across the mitochondrial membrane, which in turn drives the ATP synthase enzyme to produce ATP. Substrate-level phosphorylation, on the other hand, directly transfers a high-energy phosphate group to ADP to form ATP, without the need for a gradient or the electron transport chain. The exercise depicts substrate-level phosphorylation coupled with glucose metabolism to energetically favor ATP production, demonstrating the interconnectedness of metabolic pathways in living cells.