Examples of a specific item that moves across a cell membrane using..

a) Diffusion
b) Facilitated Diffusion

Thanks in advance!

Water moves across by simple diffusion, from high concentration to low concentration. It is called osmosis, but it is still diffusion. That's also how oxygen gas and carbon dioxide gas move from red blood cell to body cells and from body cells to red blood cells, and out into alveoli in the lungs.
Glucose is an example of a substance that requires facilitated diffusion. Facilitated diffusion means that the substance cannot just cross the cell membrane based on concentration. It either needs a special channel or, in some cases, a pumping device to force it against the normal diffusion gradient; in other words, to pump it in or out of a cell regardless of the concentration of the substance on either side of the membrane.
Glucose requires the protein hormone insulin, which forms a molecular bridge and allows glucose to diffuse into cells from the blood plasma, especially into muscle cells because they require so much energy. Without insulin, glucose remains in the blood plasma. This is Type I (juvenile) Diabetes.
There are many other examples, including ions pumped in or out of the cell membrane to create an action potential in nerve cells, most often Na+ and K+ ions, but there are lots of other examples of facilitated diffusion. Put it in your browser and you'll see, if you need more examples.