The passage of electrons down the electron transport chain appears to provide the energy for the active transport of hydrogen ions across the membrane from the mitochondrial matrix. What is the purpose of transporting the transporting the hydrogen ions across the membrane?
The high concentration of positively charged hydrogen ions in this outer chamber causes the H+ ions to flow back into the mitochondrial matrix through an ATP synthase protein complex embedded in the inner mitochondrial membrane. The flow of H+ ions (protons) through this complex constitutes a proton-motive force that is used to drive ATP synthesis.
Short answer: to drive ATP synthesis from ADP