Suppose you perpare a jar of salt water and another of sugar water and invert one on top of the other with a divider between them, and then carefully remove that divider so the liquids are in contact.
Will the concentrations of salt and sugar reach equilibrium, despite the fact that the salt or sugar in the bottom jar has to overcome gravity to rise into the upper jar? How would you calculate the relevant forces here?
Osmotic pressure is not really relevant here. In osmosis, only the solvent can move. In your scenario, both the solvent and the solutes can move. You are asking about ordinary mixing.
It will always be possible for equilibrium to be reached eventually, but in general we don't know how to calculate from first principles how long it will take. The solute particles in the bottom jar are not confined to the bottom jar; they have kinetic energy and will occasionally cross over into the top jar. The mixing will be more rapid at higher temperatures.
At equilibrium, the solute will have a very slightly higher concentration closer to the ground. The heavier the solute is, the more pronounced the gradient. If we treat the solution as an ideal solution, we can calculate the difference in concentrations explicitly using the Boltzmann distribution. For example, let's assume that each jar has a height of 10 cm. The mass of a sucrose molecule is $5.68 \times 10^{-25}$ kg. Let's assume an ambient temperature of 293 K. The ratio of probabilities for it to be found in the bottom jar rather than the top jar is $\exp(\frac{mgh}{kT})$ which comes out to about 1.000138. That means the sugar will be more concentrated in the bottom jar by a ratio of 1.000138; a very slight difference. You might be able to measure it using a very sensitive spectrophotometer.
The Law of Gravity states that all matter is attracted to other matter, from the subatomic to the cosmic levels. Furthermore, an object with a large mass (such as the planet earth) will pull objects with less mass towards it. But, with diffusion and also with osmosis the opposite is happening. Instead of gravitating toward one another, the molecules move away from one another. Note, both diffusion and osmosis do not require energy.
The test: add a cube of sugar to a glass of water and observe what happens. Newton's Law of gravity suggests the lump of sugar cube will be pulled to the bottom of the glass. But that's not what happens. Furthermore, the sugar cube always dissolves in the water and more importantly the molecules that made up the sugar cube evenly disperse through the glass of water (i.e. diffusion).
Why ? What caused the molecules that made up the sugar cube to lose their attraction to another. Even if the sugar cube breaks up, the force of earth's gravity pulls all the grains of sugar to the bottom of the glass. If the answer is that gravity is weak compared to the force of osmosis, then we're in big trouble. Why? We need gravity to be strong in order to anchor our feet to the earth that is shaped like a ball, titled at 23.439281 degree angle, spinning at 1,000 mph, and hurtling through space. But if gravity is strong, then we should expect to find all of the grains of sugar at the bottom of the glass and not evenly dispersed throughout the water. Conclusion: something is not right about Newton's Law of Gravity because it can not accurately predict what will happen when a sugar cube is added to a glass of water.
