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The answer depends on several factors associated with the system in question. One would need to know information about the membrane across which osmosis is occurring and the concentration gradient across the membrane. I will answer the question to give a general understanding.
First, we assume that the membrane is permeable to water as osmosis is the movement of water across a membrane. If this is true, we then look at the concentration gradient-where are solutes (dissolved substances) in higher concentration/more abundant?
Assume that we have a decalcified egg (no hard shell on the outside so it looks like a cell with a semi-permeable membrane) and we put it into distilled water (there are NO solutes in distilled water). Because the inside of the egg does include solutes (proteins, ions, etc.) and the environment outside of the egg is distilled water and contains no solutes, we can say that the egg or cell is hypertonic to its environment. This simply means that there is a higher concentration of dissolved particles inside the egg than outside. This also helps us to determine which way the water will move by osmosis.
Water will always move from a hypotonic (lower level of solutes) to a hypertonic solution (assuming the the membrane is permeable) until, equilibrium or a balanced concentration on either side of the membrane is achieved. In our example, the water will move from the environment (distilled water) into the egg as there is a higher concentration of solutes inside of the egg than in the distilled water. Initially, when the concentration gradient is steep (large difference in concentrations of solutes on either side of the membrane) there is high osmotic pressure, a pulling force on the water toward the hypertonic side, trying to equilibrate the concentrations. The water will rapidly move into the egg/cell. This could be your 5 minute point.
If the concentrations on either side of the membrane were becoming more similar, then the flow of water would slow down. Even when concentrations of solutes are equal on both sides of the membrane, however, osmosis still occurs with equal volumes of water continuously moving in both directions across the membrane. This could be the case at 1 hour assuming that the concentrations on either side of the membrane are becoming more or less equal by that point.
In the egg example, assuming that the membrane is not permeable to any of the solutes inside of the egg, the water would keep entering the egg (as the inside would have to approach 0% solute to equilibrate with its environment) and it would increase in size until the membrane could stretch no more. The egg in this case might explode or "lyse."
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