Final answer:
Sodium and potassium ions cross the membrane in opposite directions due to established concentration gradients, selective voltage-gated ion channels that open at different rates in response to membrane potential changes, and the inherent properties of depolarization and repolarization during an action potential.
Step-by-step explanation:
The ultimate cause of sodium and potassium ions crossing the membrane in opposite directions when their respective voltage-gated ion channels open is the establishment of concentration gradients and the selective permeability of the ion channels which are triggered by changes in membrane potential. Sodium ions flow into the cell, causing depolarization, due to a higher concentration outside the cell and a positive charge relative to the inside. Conversely, potassium ions flow out of the cell, repolarizing the membrane and restoring the negative resting potential, driven by both the concentration gradient and the membrane potential. These processes are facilitated by the gated ion channels which open and close in response to voltage changes across the membrane, thus ensuring proper sequential function during an action potential.
Different rates at which these channels open and their specific gating mechanisms also contribute to the directionality of ion movement. The voltage-gated Na+ channels open first and inactivate shortly after depolarization peaks, while the voltage-gated K+ channels open with a slight delay as the cell depolarizes and remain open longer, allowing potassium to exit the cell and aiding in repolarization.