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Why is $\ce{N2}$ (dinitrogen) not a Lewis base even with two lone pairs?

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Actually, it is, but a very weak one.

The lone electron pairs in dinitrogen are in $\mathrm{sp}$ hybridization state, and nitrogen is quite electronegative, so they are not easily donated. Still, if a Lewis acid is strong enough, dinitrogen may form a complex.

It is known that dinitrogen adsorbs on acidic $\ce{OH}$-groups in zeolites, forming complex with IR-active frequency. The nature of the complex may be debated, as no full proton transfer occurs, but the electron pairs are definitely involved.

Diazonium salts may be considered as a lewis-acid complex of a carbocation and dinitrogen molecule. While no direct synthetic route exists from dinitrogen to diazonium salt (because typically a stronger Lewis base is available), they typically decompose with formation of dinitrogen molecule.

Significant amount of dinitrogen complexes with transition elements are known. Typically they are obtained indirectly, by forming dinitrogen ligand in-place, though.

Finally, adsorption of dinitrogen on the surface of ammonia synthesis catalysts starts as a Lewis-interaction. Similarly, nitrogen adsorption on cations in zeolites is known.

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