Consider the oxidation of NADH by molecular oxygen as carried out via the electron-transport pathway \\[ \mathrm{NADH}+\mathrm{H}^{+}+\frac{1}{2} \mathrm{O}_{2} \longrightarrow \mathrm{NAD}^{+}+\mathrm{H}_{2} \mathrm{O} \\] a. What is the standard free energy change \(\left(\Delta G^{\circ}\right)\) for this reaction if \(\mathscr{E}_{\mathrm{o}}^{\prime}\left(\mathrm{NAD}^{+} / \mathrm{NADH}\right)=-0.320 \mathrm{V}\) and \(\mathscr{E}_{\mathrm{o}}^{\prime}\left(\mathrm{O}_{2} / \mathrm{H}_{2} \mathrm{O}\right)=\) \\[ +0.816 \mathrm{V} \\] b. What is the equilibrium constant \(\left(K_{\mathrm{cq}}\right)\) for this reaction? c. Assume that (1) the actual free energy release accompanying NADH oxidation by the electron-transport pathway is equal to the amount released under standard conditions (as calculated in part \(a),(2)\) this energy can be converted into the synthesis of ATP with an efficiency \(=0.75\) (that is, \(75 \%\) of the energy released upon NADH oxidation is captured in ATP synthesis), and (3) the oxidation of 1 NADH leads to the phosphorylation of 3 equivalents of ATP. Under these conditions, what is the maximum ratio of [ATP]/ \([\mathrm{ADP}]\) attainable by oxidative phosphorylation when \(\left[\mathrm{P}_{\mathrm{i}}\right]=2 \mathrm{m} M ?\) (Assume \(\Delta G^{\circ \prime}\) for ATP synthesis \(=+30.5 \mathrm{kJ} / \mathrm{mol}\).)

Step by step solution

01

Calculating the Standard Cell Potential (E°)

Given that, \(E_{o}^{'}(\mathrm{NAD}^{+} / \mathrm{NADH}\) = -0.320 V and \(E_{o}^{'}(O_{2} / H_{2} O)\) = +0.816 V, then we can calculate the standard cell potential (E°) using the formula: \(E° = E_{o}^{'}(O_{2} / H_{2} O) - E_{o}^{'}(\mathrm{NAD}^{+} / \mathrm{NADH})\)

02

Calculating the Standard Free Energy Change ΔG°

The standard free energy change can be determined using the formula \(\Delta G° = -nFE°\), where \(n\) is the number of electrons transferred, \(F\) is Faraday's constant (96485 C/mol) and \(E°\) is the standard cell potential. In this case, two electrons are transferred.

03

Determining the Equilibrium Constant (Kcq)

The equilibrium constant can be calculated using the formula \(\Delta G° = -RT \ln K_{cq}\), where \(R\) is the gas constant (8.314 J/mol.K), \(T\) is the absolute temperature in Kelvin (298.15 K in our case) and \(\Delta G°\) is the standard free energy change.

04

Calculating the Maximum Ratio of ATP/ADP

We are given that the efficiency of the process is 75% (0.75) and that the oxidation of 1 NADH molecule leads to the creation of 3 ATP molecules. We can calculate the maximum ratio of ATP to ADP using the formula \(\Delta G = \Delta G°' + RT \ln [\text{Products}]/[\text{Reactants}]\) where \(\Delta G°' = +30.5 \text{kJ/mol}\) is the standard free energy for ATP synthesis, and assuming \([\text{P}_\text{i}]=2 \text{mM}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

NADH Oxidation

NADH oxidation is a fundamental part of the cellular respiration process, where NADH (Nicotinamide adenine dinucleotide + Hydrogen) is oxidized to NAD+ (Nicotinamide adenine dinucleotide). This reaction occurs in the electron transport chain, located within the mitochondria of the cell. During this process, electrons from NADH are passed through a series of proteins that make up the electron transport chain, eventually being transferred to molecular oxygen (O₂), which is reduced to water (H₂O).

This flow of electrons generates a proton gradient across the mitochondrial membrane, providing the energy for ATP synthesis. The overall chemical reaction can be represented as:
\[\mathrm{NADH} + \mathrm{H}^{+} + \frac{1}{2} \mathrm{O}_{2} \longrightarrow \mathrm{NAD}^{+} + \mathrm{H}_{2}\mathrm{O}\]
A firm understanding of this process is crucial as it's the primary method by which cells generate usable energy, and knowing the mechanics of NADH oxidation helps students grasp the intricacies of cellular respiration.

Standard Free Energy Change

The standard free energy change (\(\Delta G^\circ\)) of a reaction measures the energy released or absorbed under standard conditions (1 atm pressure, 25^oC temperature, and 1 M concentration of reactants). It's a critical concept because it indicates the spontaneity of a reaction; negative values signify that the reaction can occur spontaneously. The standard free energy change is interconnected with cell potential (\(E^\circ\)) in redox reactions, as shown in the formula:\[\Delta G^\circ = -nFE^\circ\]
Where:

• \(n\) is the number of moles of electrons transferred,
• \(F\) is the Faraday constant (96485 C/mol), and
• \(E^\circ\) is the standard cell potential.

By calculating the standard free energy change, we can predict the direction and extent of chemical reactions, which is vital when considering the efficiency of processes like ATP synthesis in cells.

Equilibrium Constant

The equilibrium constant (\(K_{eq}\)) of a chemical reaction is a quantitative measure of the ratio of concentrations of products to reactants at equilibrium. It helps predict the position of equilibrium and the extent to which reactants are converted to products. The relationship between the standard free energy change and the equilibrium constant is given by the thermodynamic equation:
\[\Delta G^\circ = -RT \ln K_{eq}\]
Where:

• \(R\) is the universal gas constant (8.314 J/mol∙K), and
• \(T\) is the temperature in Kelvin.

An essential point for students to note is that a large equilibrium constant implies that the forward reaction is favored, meaning products are predominantly formed under standard conditions. Conversely, a small equilibrium constant suggests that reactants are favored.

ATP Synthesis

ATP (adenosine triphosphate) synthesis is the process by which energy within a cell is stored in the high-energy phosphate bonds of ATP molecules. ATP is often referred to as the energy currency of life due to its role in storing and transferring energy within cells. The synthesis of ATP primarily occurs through oxidative phosphorylation, where the energy derived from the oxidation of nutrients, like NADH, is used to synthesize ATP from ADP (adenosine diphosphate) and inorganic phosphate (\(P_i\)).

The efficiency of this process is crucial for cellular energy economics, where typically three ATP molecules are created from the oxidation of one NADH molecule. The equation for the synthesis of ATP in terms of standard free energy is:\[\Delta G = \Delta G^{'\circ} + RT \ln \frac{[\text{ATP}]}{[\text{ADP}][\text{P}_i]}\]
This equation allows us to calculate the maximum ratio of ATP to ADP that can be produced by oxidative phosphorylation given a certain amount of free energy available. Understanding ATP synthesis is key for students as it ties the concepts of energy, enzymes, and metabolism together in a fundamental way that underpins all of biology.