
Laura I. answered 10/25/15
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For the plasma membrane, the key thing is the understand that the membrane allows molecules to cross in order to have equilibrium, having the same concentration of molecules between the inside and outside of the membrane so it's balanced (equilibrium is a form of the word equal)
For osmosis, water molecules are seeking to achieve equilibrium so the concentrations are the same. It is basically the diffusion of water (meaning that it moves from a high concentration to a low concentration). The cell will easily allow water through.
Active transport involves proteins that are embedded in the membrane so molecules can travel through. It however moves against the gradient therefore requiring energy (it moves from low to high concentration instead). It is active because of the extra energy needed to achieve this.
Like Active Transport, Facilitated Diffusion is when molecules move through the membrane through integral proteins. However, this cellular transport goes with the gradient. It is facilitated diffusion because it is diffusion, but facilitated or made easier by use of proteins since the molecules might be too large to enter the pores of the plasma membrane.
Endocytosis and exocytosis occurs when molecules travel through the cytoplasm enclosed by part of the membrane as a bubble. Endocytosis involves a molecule that drifts into the cytoplasm after the membrane "pinches" around it to make a vacuole; exocytosis does the opposite by having the molecule move from within the cell to the plasma membrane. Think of Endo- as "enter" and Exo- as "exit"