organic chemistry - Confusion about the SN1 and SN2 Mechanism

My book says, and I understand, that the $\mathrm{S_N1}$ mechanism is much faster for tertiary alkyl halides than primary. And the opposite for $\mathrm{S_N2}.$ This all makes sense. What I don't get is the conclusion it comes to next: hence, we predict that the primary alkyl halides will react primarily via $\mathrm{S_N2},$ and the tertiary via $\mathrm{S_N1}.$

The reason I get confused here is that nothing was said about the comparative rates for a primary alkyl halide via the $\mathrm{S_N1}$ or $\mathrm{S_N2}$ (they just compared primary and tertiary for $\mathrm{S_N1}$ and $\mathrm{S_N2}$). In my mind, which mechanism dominates for primary or tertiary should be dependent on the relative rates for that compound; not rates compared between primary and tertiary.

Is my argument sound? And if so, is there some literature you know of that explains this in such a manner?

Answer

Your reasoning is correct. It's the relative rates for a given alkyl halide that matter, and it's difficult to draw conclusions about those, because the relative rates are not fixed, as the $\mathrm{S_N2}$ rate is dependent also on the nucleophile concentration.

We can write the rate of the $\mathrm{S_N1}$ as $k_1[\text{alkyl halide}]$ and the rate of the $\mathrm{S_N2}$ as $k_2[\text{alkyl halide}][\text{nucleophile}]$, so the relative rate at a fixed concentration of the alkyl halide is given by

$$\frac{\mathrm{S_N2}}{\mathrm{S_N1}} = \frac{k_2[\text{nucleophile}]}{k_1}.$$

The identity of the nucleophile will be a factor as well, since that will affect $k_2$.

So all in all, you can't make a general conclusion that one reaction type will always go faster than the other, although it is likely that there are alkyl halides for which one reaction type is faster in a large majority of cases.