That the solvent is aproic is far more important for S
N2 reactions than polarity. Polarity in S
N1 reactions shows a relatively large effect on rate because there are charged intermediates to solvate. If you consider the transition state [TS] for the rate determining step of S
N1:
R-X
[R
δ+---X
δ-]
R
+ + X
-The rate of the reaction is dependent on the difference in energy between the starting material and the TS (activation energy). Since there is a much greater degree of charge separation in the TS compared to the starting material, a more polar solvent stabilises the TS more than it stabilises the starting material and the activation energy drops.
For S
N2 we can play the same game:
Nu
- + R-X
[Nu
δ----R---X
δ-]
Nu-R + X
-Now if we compare the starting materials, with -ve charge localised on the nucleophile, to the TS, which has the charge spread over the nucleophile and the leaving group, we see a slight decrease in charge separation in the TS compared to the starting materials. As a result, S
N2 reactions with charged nucleophiles slightly favour solvents of lower polarity (which is not what most people will tell you). There is a catch though, there is a reason people do not run these reactions with hexane as the solvent. The solvent must be polar enough to dissolve the nucleophile, so the least polar solvent you can normally get away with in this respect is THF or acetone, but commonly DMF is needed to dissolve the nucleophile salt.
For S
N2 with neutral nucleophiles (e.g. R
3P, R
2S etc.), you can do the same analysis to show that there is an increase in charge separation in the TS compared to the starting materials, and that a more polar solvent will increase reaction rate.
For a nice explanation of these concepts, I recommend: Peter Sykes - A Guidebook to Mechanism in Organic Chemistry