The movement of protons during ATP synthesis in mitochondria takes...

The oxysome or ATP synthase consists of two major components, $F_0$ and $F_1$.

The $F_0$ portion is an integral membrane protein complex embedded in the inner mitochondrial membrane. It forms a transmembrane channel that facilitates the diffusion of protons from the intermembrane space (outer chamber) back into the mitochondrial matrix (inner chamber).

The $F_1$ portion is a peripheral membrane protein complex situated on the matrix side of the inner mitochondrial membrane, where it catalyzes the synthesis of ATP from ADP and inorganic phosphate.

Answer: inner mitochondrial membrane.