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$\ce{NF4+}$ ion reacts with water as:

$$\ce{NF4+ + H2O -> NF3 + H2F+ + {1\over 2}O2}$$

My question is, does $\ce{NF4+}$ function as an acid or base in this reaction, and why?

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Neither. The reaction is NOT one of ion exchange; it is a redox reaction. Thus, $\ce{NF4+}$ works as electron acceptor here, like this:

$$\ce{NF4+ + 2e- -> NF3 + F-}$$

Since the reaction mechanism is unclear, it is possible that some of its stages may have acid-base character (for example, $\ce{F-}$ sometimes acts as a base), but $\ce{NF4+}$ certainly isn't acting primarily as base or acid here.

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