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Polar solvents love polar solutes to be dissolved in it and non polar with non polar. Often said as like dissolves like.
Okay, polar loving polar can be understood with help of the facts: same polar nature, same kind of interactions etc.
But how will you explain the application of this rule with organic or non-polar solvents?
Solomons’ and Fryle’s Organic Chemistry says that there are some unfeasible entropy changes when polar solutes are added in non polar solvents and vice versa.
Why so?
The reason behind this is the hydrophobic effect. Everyone has seen it if they pour a spoonful of vegetable oil into a pot of water, e.g. to cook pasta. As long as nothing is disturbing the vegetable oil, it will collect itself together in one big bubble rather than form many small bubbles.
Polar solvents will always be arranged in a way that positively polarised areas are close to negatively polarised areas in the neighbouring solvent molecule. Hydrogen bonds — especially in water — are nothing but the extension of this concept to even more polarisation. Polar molecules, as you stated, will fit together with this scheme well. Unpolar compounds mixed in do not. As soon as you add an unpolar compound to a polar solvent, you are creating a type of artificial boundary and only the areas of the solvent molecules that are neither particularly positive or negative will be happy to be in the near vicinity. That means that you have a much higher ordering of the solvent molecules where they hit an unpolar compound, because one direction is basically doomed to be neither positive nor negative.
This ‘solvent wall’ will form no matter whether the unpolar island is large or small. However, it will also have almost the same thickness, no matter how big the contained part is. Therefore, if multiple unpolar molecules clump together, the overall number of polar molecules constrained in that wall is lower — an entropic gain. Thus, unpolar compounds tend to either not dissolve or precipitate out of polar solvent solutions.
This also works in the opposite direction. A polar molecule will prefer, for energetic reasons, to have a polar neighbour. However, an unpolar solvent cannot provide that polar environment. A neighbouring undissolved polar molecule, however, can. Here again, the polar compounds rather stay clustered together as it allows the central molecules to be more disordered while only a small layer on the border must take care to interact as well as possible with the unpolar solvent.
Note that this answer assumes some kind of black/white dichotomy. In reality, most compounds are somewhere on a scale from absolutely unpolar to very polar and are able to adapt to a wide range of organic solvents. Conversely, some organic solvents have the reputation of dissolving almost anything organic; most notably dichloromethane. However, the solubilities may greatly vary.
I would be quite dubious in ascribing much validity to answers which use the word "unpolar", since that word indicates either a lack of familiarity with chemical terminology or a lack of respect for it.
The way I consider solvation (for two pure chemical compounds) is that in their pure state the interactions are between molecules of themselves. For two compounds, A & B, call them A2A and B2B interactions. These interactions can be either weak or strong (and this can be determined using Thermodynamic experimentation). When considering solubility, the added term is the A2B interactions. When A2A and B2B interactions are weak, then even weak A2B interactions should result in substantial interactions (and hence substantial mixing). When A2A and B2B interactions are both strong, A2B interactions must also be strong to get a homogeneous equilibrium phase. So we should look at these interactions relative to the strongest. If A2A interactions are strong (say NaCl) and B2B interactions are weak (say n-hexane) then we'd predict that the A2B interactions are also going to be (very) weak, and that the solubility of NaCl in hexane is going to be small (negligible, actually). So in this case the strong interaction in the (non)solute (NaCl) dominates. But instead of NaCl, consider C20H42 mixed into n-hexane. We'd predict weak A2A and B2B interactions and so, since we know all matter interacts electromagnetically, we'd assume A2B interactions are going to be relatively significant, and so the icosane will be soluble in the hexane. But what about icosane in water? Here we'd have weak A2A interactions and strong B2B interactions. What about A2B? Well, looking at the molecules, there is no obvious specific interactions which could add to the interaction energy, so I'd assume that the water isn't going to substantially interact with the icosane, and the solubility will be small. You'll note that it is the stronger interaction, whether it is in the solvent or the (potential) solute which determines the 'baseline'. The hetero-interaction (A2B) must be similar in magnitude. The easiest way for that to occur is for both A and B to have similar polarity. Hence, like dissolves like. (with lots of exceptions for specific interactions which "like" doesn't cover (h-bonding, polarizability, steric effects, etc.) [Added clarification: A2A is an abbreviation of "a molecule of A interacting with another molecule of A" or "A-to-A interactions"]
