Final answer:
The proteins of the electron transport chain in a eukaryotic cell are located in the mitochondrial inner membrane, which has folds called cristae to increase surface area for ATP synthesis.
Step-by-step explanation:
In a eukaryotic cell, most of the proteins for the electron transport chain are located in the mitochondrial inner membrane. This includes a series of electron carriers and ion pumps that are crucial for ATP synthesis. The inner membrane contains folds known as cristae which increase the surface area for these biochemical reactions to take place. It is this strategic location that allows the efficient production of ATP, the cell's energy currency. Meanwhile, prokaryotic cells have their electron transport systems embedded in the cytoplasmic membrane, as they lack mitochondria.