Take a known homogeneous mixture like air (oxygen and nitrogen) for example. The two have different atomic masses, so the molecular weight is different. If I let the solution sit there for a long time, eventually they should separate due to their different mass.
It seems like this logic can be generalized to every other solution, so why are there categorizations like homogeneous mixtures and heterogeneous mixtures, when they just seem to be more extreme versions of another?
You’re actually asking a very interesting question and many people put little thought into it. It is true, the words heterogenous and homogenous when applied to mixtures may be considered misleading. For example, gases will actually separate to a certain extent (not perfectly, but noticeably) if they are not disturbed. All of us will have heard that ‘carbon dioxide is heavier than air and thus drops to the bottom of a container’. Any many of us will have seen an experiment with a number of candles at different heights in a container where the lower candles are extinguished first due to lack of oxygen.
In the atmosphere, however, gases mix much better due to winds. They whirl around the gases to re-mix them. Hence why even though nitrogen is lighter than oxygen both gases are found in relatively equal proportions at different heights.
Similarly, a solution of $\ce{NaCl}$ in water is often thought to be fully homogenous. However, there are actually separation techniques that make use of a $\ce{NaCl}$ concentration gradient in centrifugation: the more dense part of the solution with a relatively higher $\ce{NaCl}$ content drops to the bottom and different biomolecules are held up at exactly the concentration that matches their density.
Therefore, it seems reasonable to say ‘there really only is one type of mixture which will separate itself given enough influence of gravity’. I’m not sure whether this fully holds true, though.
First and foremost, it makes a difference whether a mixture consists of macroscopicly separated domains with a high relative concentration of one component or whether the components are randomly distributed at a molecular level. The latter is the typical description of a homogenous solution, the former is a description of two substances that don’t mix even though they are finely dispersed (e.g. fat and proteins in milk). A well-mixed sodium chloride solution in a closed vessel will not separate at any reasonable timescale while even homogenised milk will, given enough time, separate into its components because it is entropically favourable.
Secondly, there are actual physical differences between the two. For example, heterogenous solutions such as colloids can be detected by shining a focussed light beam through them. A homogenous solution will not scatter light but in a heterogenous solution you can follow the path of the light through it due to scattering. The low-level physical explanation is that of microscopically different refraction indices between the colloidal particles and the solution around it. A ‘phase’ (and a homogenous solution is defined as a single phase) is defined such that all intrinsic parameters (density, refraction index, …) are constant throughout the entire solution. Thus, while the solution looks homogenous, it is distinct from a homogenous solution at a molecular level.
Therefore also, the carbon dioxide that is collected at the bottom of the tub extinguishing candles as mentioned above has actually just not been mixed properly with the surrounding air. If there is a fan, whirling the air in the tub around, the candles are expected to be extinguished at approximately the same time. And if one hears about ‘more dense sea water’ that travels along the ocean ground, that is simply due to incomplete mixing of the different water levels.
Conclusion: While indeed it may seem that the distinction between homogenous and heterogenous is arbitrary and examples can be found that seem to support the view, there is a well-defined physical difference between the two and the examples used are actually examples of not properly homogenous solutions.
The separation of gases won't occur in a laboratory scale experiment, but does occur on a planetary scale. So at sea level the atmosphere will have slightly richer oxygen content than at the outer reaches of the atmosphere. The effect is most pronounced for helium. Helium released into the atmosphere migrates upward in the atmosphere until it reaches the fringe of outer space. Then an occasional bump away from earth will be with enough force to enable the helium molecule (an atom) to reach escape velocity and the helium "boils off" the planet. Thus the atmosphere retains no helium.
For a "homogeneous mixture" no matter what sample i take the sampless will all have the same composition. This would be like a solution of sodium chloride.
For a "heterogeneous mixture" I could, theoretically at least, take samples that would differ in composition. Milk could be a good example. The fat globules are very small, but theoretically at least, I could get an "aqueous" sample and a "fat" sample by judicious sampling.
