ATP synthase can indeed produce more than 100 ATP molecules per second, and in the process, it only requires a few -- three or four -- protons, per ATP. These protons pass down a gradient through the membrane. Hence, the protein is membrane-bound. The protons cause the rotor of 10-14 subunits to spin. The protein's head itself has six subunits, three of which have ADP binding and phosphate binding sites.

The electron transport chain moves high energy electrons through its complexes in order to create a proton gradient across the mitochondrial inner membrane.  The ATP synthase then uses this gradient to pass hydrogen atoms through it.  Because this is a favorable movement, it can be coupled to unfavorable processes such as conversion of ADP to ATP.  

ATP synthase catalyzes the reaction that shows ADP and the phosphate group forming ATP.  The hydrogen in the reactant side is the one involved in the proton gradient, and water is a byproduct of the reaction.

ATP synthase is located in the inner mitochondrial membrane. It has an F0 portion within the membrane and an F1 portion in the matrix. The F1 portion has a hexameric ring structure and is responsible for the creation of ATP from mechanical energy. The alpha, beta, and gamma subunits are all parts of the F1 portion of ATP synthase, however it is only the alpha and beta subunits that form the ring. Further, the beta subunit is the part of the ring that is considered to be catalytic.

First, let's consider the half reactions involved to determine . 

This overall reaction involves the donation of 2 electrons, so

 is defined as . The reaction we drew earlier is shown below:

We can see that  was oxidized and  was reduced. Find .

 is Faraday's constant, and is defined as: 

Solve for 

A positive Gibbs free energy implies that the process in question should be unfavorable under normal conditions.  The only process listed that is unfavorable and requires an input of energy is the pumping of hydrogen ions into the intermembrane space.  This occurs during the electron transport chain.

The phases of cellular respiration are glycolysis, pyruvate dehydrogenase complex, Krebs cycle, electron transport chain. Glycolysis produces a net total of 2 ATP, the Krebs cycle produces 1 GTP that is converted to ATP in another process, and the electron transport chain is where almost all of the ATP made in cellular respiration is formed. However, during the pyruvate dehydrogenase complex phase of cellular respiration, pyruvate is converted to acetyl-CoA as a preparation for the Krebs cycle, but no ATP is created.

The correct answer is cyanide. This compound acts to inhibit cytochrome C oxidase, otherwise known as Complex IV of the electron transport chain. By inhibiting this complex, cyanide effectively halts the flow of electrons through the chain. Consequently, protons are not able to be pumped from the matrix to the intermembrane space and thus, a proton gradient cannot be established. Without a proton gradient, protons will not flow through ATP synthase, hence no ATP will be produced.

Oligomycin, on the other hand, acts to inhibit ATP synthase, which means that ATP will not be able to be produced. However, electrons are still able to flow through the chain, which means that protons are still able to be pumped across the inner membrane.

Dinitrophenol is a relatively nonpolar compound that is able to situate itself into the inner mitochondrial membrane. In doing so, it is able to dissipate the proton gradient by allowing protons to essentially be transported from the intermembrane space to the matrix without traversing through ATP synthase. Even though protons can still flow through ATP synthase to generate ATP in this scenario, the proton gradient won't be nearly as potent because they now have an alternative route to the matrix.

Methotrexate acts to competitively inhibit the enzyme known as dihydrofolate reductase. This enzyme has nothing to do with the electron transport chain, and thus will have no effect on ATP synthesis. Commonly, this drug is used as an anti-cancer agent because its substrate, dihydrofolate, is a compound that is used in the syntheis of thymine nucleotides for DNA synthesis. Inhibiting dihydrofolate reductase effectively reduces the production of thymine, which can negatively impact DNA replication in rapidly dividing cancer cells.

Likewise, allopurinol has nothing to do with the electron transport chain. This drug acts as an inhibitor of the enzyme xanthine oxidase, which is responsible for synthesizing uric acid. High levels of uric acid can lead to the development of gout, and thus, this drug is typically used to help treat people suffering from gout.

Potassium cyanide inhibits cellular respiration by acting on mitochondrial cytochrome c reductase (leading to hypoxia and death). Rotenone also affects oxidative phosphorylation, by inhibiting electron transfer from cytochrome 1 to ubiquinone, making it a potent insecticide. Oligomycin inhibits ATP synthase, also slowing flow of the electron transport chain. Fructose 2,6-biphosphate affects the activity of enzymes regulating glycolysis and gluconeogenesis. Dinitrophenol dissipates the proton gradient across mitochondrial membranes, and shuttles protons across them, inhibiting ATP production. 

Aerobic respiration is a process that utilizes the electron transport chain in order to oxidize glucose into energy. If a chemical were added that inhibited the electron transport chain, the cell would no longer be able to fully oxidize glucose. Therefore, oxygen consumption will decrease. Furthermore, since the cell is now in a situation in which it is not able to make as much energy per glucose molecule as before, it will need to increase its consumption of glucose in order to generate enough energy through anaerobic respiration alone.