In prokaryotes, there are no membrane-bound organelles and no nuclear membranes. in addition, there are no mitochondria. where are the electron transport chain and the atp synthase working? what physical place are they in a prokaryotic cells?

Prokaryotes lack membrane-bound organelle like mitochondria and chloroplast which is used by eukaryotes to synthesize ATP but ATP synthesis also takes place in prokaryotes. The electron transport chain and ATP synthetase of prokaryotic cells are embedded in the cell's plasma membrane of the prokaryotes.

The ATP synthesis in prokaryotes is similar to eukaryotes. NADH releases electrons to the electron carriers that are embedded in the plasma membrane which causes the translocation of H+ ions to the cell membrane outer face against the concentration gradient.

Then these electrons pass through the ATP synthetase which generates ATP for prokaryotic cell. So cell membrane is the physical place where ETC and ATP synthetase are present.