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To understand reverse osmosis, one must understand first understand what is osmosis. Osmosis is the movement of water from high to low concentration across a membrane that is freely permeable to water but not to dissolved substances. See Figure A.

Figure A.
As seen in Fig. A, the salt in the water is unable to pass through the membrane and it causes an unbalanced amount of water molecules on each side of the membrane. The natural tendency is for water to move from left (high concentration) to right (low concentration) of water. This creates what is called osmotic pressure which we will later discuss. As soon as the amount of water molecule level is even on both sides than the movement will stop and reach equilibrium.
Reverse osmosis is the exact opposite as the name implies. See Figure B.

Figure B
Due to water`s natrual tendency to move from high to low concentration. Pressure is required to force ONLY the water through the semi-permeable membrane. How much pressure is required? The pressure exerted must be higher than the osmotic pressure. Osmotic pressure is the potential energy of the water - the required amount of energy to force the water through semi-permeable membrane.
The same principle applies to the actual reverse osmosis system. The membrane is either made of cellulose triacetate (CTA) or thin-film composite (TFC). The pressure exerted comes from the feed water pressure.
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