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$\ce{ N2F2}$ exists in two forms cis and trans. Today my chemistry teacher tells me that among the two only cis form takes part in reactions and that it is more reactive. But I have read elsewhere that trans is less stable. As far as I understand less stable compounds are usually more reactive. So why this discrepancy?

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According to Dinitrogen Difluoride Chemistry Inorg. Chem., 2010, 49 (15), pp 6823–6833:

This increased reactivity of the cis-isomer can be readily explained. In both $\ce{N2F2}$ isomers, each nitrogen atom possesses a sterically active free valence electron pair. The transformation of $\ce{N2F2}$ to $\ce{N2F+}$ requires the conversion of the $\ce{N=N}$ double bond into a $\ce{N#N}$ triple bond by an overlap of electrons from these two free valence electron pairs. Obviously, this overlap is greatly facilitated in cis-$\ce{N2F2}$, where these free pairs are on the same side of the molecule.

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