Final answer:
The false statements regarding the transport of uncharged molecules across a membrane are that Keq is always equal to one and ΔG is independent of a gradient. Keq varies and ΔG does depend on a gradient, and facilitated transport often requires a transport protein.
Step-by-step explanation:
When transporting uncharged molecules across the membrane, the false statement is that Keq is always equal to one and that ΔG is not dependent upon a gradient. The equilibrium constant (Keq) for a reaction can vary and is not always equal to one. The change in free energy (ΔG) for the transport of molecules is indeed dependent on a concentration gradient. Only under standard conditions (ΔG°), which means when a concentration gradient does not exist, the ΔG° can equal zero, but this is not the case for the actual movement of molecules which often occurs in non-standard conditions in living organisms.
Movement of molecules often requires a transport protein, especially when they cannot easily diffuse through the lipid bilayer, such as polar molecules like glucose and amino acids. These molecules use facilitated transport, moving down their concentration gradient through specific protein channels without expending energy.