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Exactly one of these statements is false. Which one?

(a) When ammonia dissolves in water, the latter behaves as a Lewis base.
(b) The conjugate acid of $\ce{H-}$ is $\ce{H2}.$
(c) Boric acid is a monobasic acid.
(d) Ammonia behaves as an acid when it reacts with sodium metal to give sodamide and hydrogen.

The key says (d) is the false statements. How? I think (a) is the one.

(d) $\ce{2NH3 + 2Na -> 2NaNH2 + H2}$

The ammonia here loses a proton, which is characteristic for acids. In addition, it attaches to the electropositive sodium.

(a) On the other hand, when ammonia dissolves in water:

$\ce{NH3 + H2O <=> NH4+ + OH-}$

Ammonia gains a proton, which is characteristic for bases.

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Your reasoning is correct. Perhaps the answer key is mistaken.

For part (a), in the reaction

$$\ce{NH3 + H2O <=> NH4+ + OH-}$$

water is donating a proton $(\ce{H+}$ ion) and hence is behaving as a Brønsted acid. Since all Brønsted acids are Lewis acids, water is behaving as a Lewis acid.

For part (d) as well, the half reactions are:

$$ \begin{align} \ce{2 NH3 &-> 2 NH2- + 2 H+} \\ \ce{2 Na &-> 2 Na+ + 2 e-} \end{align} $$

It's clear that $\ce{NH3}$ is acting as an acid and $\ce{Na}$ is acting as a Lewis base by donating electrons. The $\ce{NH2-}$ ion pairs up with the $\ce{Na+}$ ion to form $\ce{NaNH2}$ and hydrogen accepts the electrons and is liberated as $\ce{H2}.$

EDIT: As pointed out by Mithoron in the comments, the reduction of ammonia is not so simple. Sodium dissolves in ammonia to give a deep-blue solution containing ammoniated electrons (reference here):

$$\ce{Na + \text{(x+y)}NH3 <=> Na(NH3)_x^+ + e(NH3)_y^-}$$

The $\ce{e(NH3)_y^-}$ eventually loses a hydrogen atom to give $\ce{NH2-}$, which then combines with the $\ce{Na+}$ giving the product.

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    $\begingroup$ It's not so simple. Mechanism for ammonia reduction is more like: $\ce{Na + NH3 -> Na+ + NH3-}$ then $\ce{NH3- -> NH2- + H}$. $\endgroup$
    – Mithoron
    Jan 3 at 14:12
  • $\begingroup$ The question then is even wronger! There are two false answers: a and d. $\endgroup$ Jan 3 at 14:40
  • $\begingroup$ @Mithoron Good point, I forgot about ammoniated electrons. Updated the answer now $\endgroup$ Jan 3 at 17:51
  • $\begingroup$ Yes, unfortunately that means that the statement is indeed wrong. Moreso the statement about water can be considered true as H2O acts as also as acceptor of hydrogen bond, partially donating its lone pair - such coordination is pretty weak, but for example cationic complexes of noble gases also have rather weak bonds, while being clearly Lewis adducts. $\endgroup$
    – Mithoron
    Jan 4 at 0:45

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