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Why is NH3 a lewis base and BCl3 a lewis acid?
By definition a Lewis acid is an electron-pair acceptor, and a Lewis-base is an electron pair donor. The classifications of ammonia and boron trichloride are clear.
Ammonia has a formal lone pair of electrons on the nitrogen atom. This can donate to Lewis acids (including boron trichloride) in a Lewis base/Lewis acid interaction.
On the other hand, the boron atom in boron trichloride has a formal, vacant ##p## orbital that can accept electron density and quaternize the boron.