organic chemistry - Does nitrogen inversion affect the basicity of amines?

If I were to compare the basic strength of 1-azabicyclo[2.2.1]heptane and triethylamine:

Can I say that 1-azabicyclo[2.2.1]heptane is more basic than triethylamine because the lone pair of electrons is less available in the latter due to rapid nitrogen inversion? Nitrogen inversion is not possible in the bicyclic amine.

There is not much difference in the inductive effect. In both cases, the nitrogen atoms are $\mathrm{sp^3}$ hybridized.

Answer

Background

Amine basicity correlates with, among other things, the hybridization of the nitrogen orbital that is holding the lone pair of electrons. The less s-character in this orbital, the more basic the amine. So, as the following figure indicates, a trialkylamine (the lone pair is in an $\ce{sp^3}$ orbital) is more basic (less acidic or higher $pK_{a}$) than the lone pair in pyridine (the lone pair is in an $\ce{sp^2}$ orbital), which is less acidic then the lone pair in a nitrile (the lone pair is in an $\ce{sp}$ orbital).

(image source)

The rate of nitrogen inversion also correlates with, among other things, hybridzation. But in cases where these "other things", such as ring constraints, limit nitrogen inversion, then the rate of nitrogen inversion may not correlate with hybridization. So the rate of nitrogen inversion does not always correlate with hybridization and is therefore not a good indicator of lone pair "availability".

Answer

Looking at your two compounds, a reasonable first guess might be that the nitrogens in these compounds are both roughly $\ce{sp^3}$ hybridized. Therefore, even though one compound can undergo nitrogen inversion and the other can't, their hybridzations are similar and we would expect similar basicities.

We could refine this prediction by noting that in the bicyclic compound, due to geometric constraints, the carbon-nitrogen bonds around nitrogen are slightly pulled back a bit from the tetrahedral angle. This would increase the p-character in these bonds (as a bond angle moves from 109° to 90° the p-character increases with 90° being pure p). As a result, the p-character in the lone pair orbital decreases and the s-character increases. As mentioned above, more s-character in the lone pair orbital leads to lower basicity.

Therefore we might expect both compounds to have similar basicity with the bicyclic compound perhaps being a bit less basic.

This analysis is corroborated by the $\mathrm pK_\mathrm a$ values of the conjugate acids (a more positive number means that the amine is more basic). According to J. Org. Chem. 1987, 52 (10), 2091–2094, quinuclidine has a $\mathrm pK_\mathrm a$ of 10.90, and 1-azabicyclo[2.2.1]heptane a $\mathrm pK_\mathrm a$ of 10.53. The decrease in basicity can be attributed to the increased strain and greater s-character of the lone pair. Triethylamine itself has a $\mathrm pK_\mathrm a$ of 10.75 (Evans table), making it marginally more basic than 1-azabicyclo[2.2.1]heptane.