|MadSci Network: Physics|
Thank you for a most interesting question. To summarise it, you asked whether the upper parts of a solution of a solute heavier than the solvent have a lower concentration of solute because of the gravitational field the solution is in, for example some heavy water diluted by ordinary water. If, in another example, a large molecule in water showed this effect to a significant extent, you then asked whether, if the solution was contained in a cube whose top and bottom faces were made from a semi-permeable membrane and the cube was suspended in water, the differences in concentration and therefore osmotic pressure will cause a “perpetual motion of the water”.
I hope you are not looking for a “perpetual motion machine”. You will be disappointed ! Let us look at the first question concerning the effect of gravitation on concentration. Before looking at liquids let’s look at gases, for example the atmosphere. We know that at high altitudes the atmospheric pressure is lower. The thermal energy of the gas molecules in the form of kinetic energy provides the energy required to lift molecules against the influence of gravity. In the case of the atmosphere the pressure drop as we go higher is only slight over several metres and drops to half its sea- level value at about 5500m.
The dependence of pressure on height increases as the molecular weight increases and the height differences needed for noticeable changes is reduced. The effect is the basis of distillation methods of separation. For lighter molecules than air the dependence is lower for example for water molecules. Despite this, distillation has been used to separate heavy water from normal water so even very small differences can be useful.
Going back to liquids and solutions we must recognise some differences and similarities. In the case solutions, the solute molecules have the same thermal energy as they would have in the gas phase. They do not move freely in solution being confined and jostled by solvent molecules. Also, their weight is counteracted by their buoyancy so the effect of gravity is reduced and even reversed if the molecule displaces an amount of solvent which is heavier than the molecule itself. In the case of heavy water, the difference in weight is provided by 1 extra neutron above the 8 neutrons and 10 protons in normal water and its behaviour as a solute in water is more like a gas with a molecular weight of 1. It would be therefore very difficult to detect any change of concentration from the top to the bottom in a solution of any reasonable size.
Because of the effect of buoyancy even heavy molecules tend to be evenly distributed. Nevertheless we should expect to be able to observe differences. They certainly are noticeable in the case of colloidal particles which can of course contain millions of atoms. J.Perrin used this effect to estimate Avogadro’s Number ( the number of atoms or molecules in one mole ). He counted the number of colloidal particles using a microscope which focussed on a very thin layer of liquid and then in another lower layer. The amazing thing was that the two layers were only 0.1mm apart ! Not much gravitational difference there. Despite this he was able to make a reasonable estimate. In the absence of any calculations on my part, I think we can assume that small effects of gravity on concentration may be apparent in some cases of molecular solutions so let’s look at the case of the cube with solution inside.
Firstly, you call it a cube so we’ll assume it stays a cube after the box containing the solution has been immersed in water. In other words the semi-permeable membrane is rigid and allows water to flow through it while preventing the large molecule from doing so. To predict what will happen we need to understand osmotic pressure. This website had a very good explanation I thought. http://members.tripod.com/~urila/index.htm
I mentioned earlier that solute molecules are seen to behave as though they were gas molecules with the same weight and thermal energy. Even though their motions are controlled more by the solvent, they still act like gas molecules in exerting pressure and it is this pressure which is osmotic pressure. On the other hand the water can flow freely through the membrane in both directions and there is no net pressure due to water on either side of the membrane. This is true of both membranes in our box and therefore there is no net flow.
This may seem at odds with the view that solvents flow through membranes in order to lower the solute concentrations but I believe that is the wrong way to describe what is happening. Solvent will flow through the membrane in some situations. For example is our box had flexible membranes, they would be pushed out by the solute’s osmotic pressure. The solvent would flow though the membrane as it expands. The consequence of this flow is dilution but the cause is the movement of the membrane under pressure from within.
In another example a U-tube with a membrane across the tube at the bottom provides a volume either side of the membrane. Solvent is added to the tube and reaches the same distance up the arms of the tube. Now solute is introduced on one side. The solvent flows into the side containing the solute raising the surface on that side and lowering it on the other. The hydrostatic pressure due to the difference in heights is a measure of the osmotic pressure. The reason for the movement of the solute though is the fact that the solute, because it cannot escape through the air/solution surface exerts upward pressure on the surface which moves against gravity pulling the column of liquid through the tube until equilibrium is reached.
Although therefore solvent flow is a consequence of osmotic pressure it is not always the case. Thanks again for the question. It was along one and I thought it deserved a long answer !
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