Which are the two most important factors determining the movement of ions across the cell membrane?

The two most important factors determining the movement of ions across the cell membrane are: the permeability of the molecule in the membrane, and the existence of an energy source.

Explanation:

Ions are charged molecules that, no matter their size, cannot pass through the membrane by simple diffusion because of the nature of the lipid bilayer of the membrane.

Ions can move across the cell membrane in two different ways: passive transport and active transport. The disparity between these two relies on the usage of ATP (energy) - passive transport doesn't need ATP while active transport does. This is where the availability of a source of energy is important, because if there isn't, then active transport is impossible.

On the other hand, ions can move through the cell membrane without the use of energy by diffusing through the membrane with the help of specific membrane proteins that form channels for ions to pass through - and this is where permeability matters: if the cell membrane is permeable to a specific ion, it means that it has opened channels for that ion to use; and this ion will move from one fluid to the other (intracellular or extracellular) following its concentration gradient (for example, sodium is poorly concentrated in the inside of the cell, while is highly concentrated on the outside - this means that sodium will go through the membrane to get inside the cell and even the concentrations between the two fluids, but only if the membrane is permeable to sodium!).

multiply it by 5 in order for you to understand i cant give an exact answer.