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Brønsted basic strength depends on the availability of the nitrogen lone pair for accepting: $$\mathrm{H^+}$$
Basicity is mainly influenced by: $$\mathrm{+I\ effect}$$ and $$\mathrm{Steric\ Hindrance}$$
In: $$\mathrm{N(CH_2CH_3)_3}$$ (triethylamine), three ethyl groups increase electron density on nitrogen through: $$\mathrm{+I\ effect}$$
However, the bulky ethyl groups create significant steric hindrance around nitrogen, reducing protonation and solvation.
Hence, its basic strength decreases.
In:$$\mathrm{HN(CH_2CH_3)_2}$$ (diethylamine), two ethyl groups provide strong: $$\mathrm{+I\ effect}$$ with much lower steric hindrance.
Thus, secondary amines are generally stronger bases than ordinary tertiary amines in aqueous medium.
In: $$\mathrm{1\text{-}Methyl\text{-}3\text{-}pyrroline}$$ the adjacent: $$\mathrm{sp^2}$$ hybridised carbon atoms exert a slight: $$\mathrm{-I\ effect}$$
which decreases electron density on nitrogen. Hence, it is less basic than saturated aliphatic amines.
In: $$\mathrm{Quinuclidine}$$ the nitrogen lone pair is highly available. The bicyclic cage structure provides strong $$\mathrm{+I\ effect}$$ while preventing steric crowding around the lone pair.
Thus, protonation occurs very easily and the conjugate acid formed is highly stable.
Therefore, the strongest Brønsted base is:
$$\mathrm{Quinuclidine}$$
Correct option:
$$\mathrm{D}$$
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