A gaseous compound of nitrogen and hydrogen contains 12.5% (by mass) of hydrogen. The density of the compound relative to hydrogen is 16. The molecular formula of the compound is:

First, we are given that the compound contains 12.5% hydrogen by mass. This means that in 100 grams of the compound, there are 12.5 grams of hydrogen and the remaining mass is nitrogen. So, the mass of nitrogen is 100 - 12.5 = 87.5 grams.

Next, we find the number of moles of each element. The atomic mass of hydrogen (H) is 1 gram per mole, and the atomic mass of nitrogen (N) is 14 grams per mole.

Moles of hydrogen = mass of hydrogen / atomic mass of hydrogen = 12.5 / 1 = 12.5 moles.

Moles of nitrogen = mass of nitrogen / atomic mass of nitrogen = 87.5 / 14. Let's calculate that: 87.5 divided by 14 equals 6.25 moles. So, moles of nitrogen = 6.25 moles.

Now, we find the simplest ratio of nitrogen to hydrogen by dividing both mole values by the smallest number, which is 6.25.

Nitrogen ratio: 6.25 / 6.25 = 1.

Hydrogen ratio: 12.5 / 6.25 = 2.

So, the ratio of nitrogen to hydrogen is 1:2. Therefore, the empirical formula is NH$$_2$$.

The empirical formula mass of NH$$_2$$ is calculated as: atomic mass of N + 2 × atomic mass of H = 14 + 2 × 1 = 16 grams per mole.

We are also given that the density of the compound relative to hydrogen is 16. For gases, the relative density with respect to hydrogen is equal to the ratio of the molecular mass of the compound to the molecular mass of hydrogen gas (H$$_2$$). The molecular mass of H$$_2$$ is 2 grams per mole.

So, molecular mass of compound = relative density × molecular mass of H$$_2$$ = 16 × 2 = 32 grams per mole.

Now, we find the ratio of the molecular mass to the empirical formula mass: n = molecular mass / empirical formula mass = 32 / 16 = 2.

This means the molecular formula is twice the empirical formula: (NH$$_2$$)$$_2$$ = N$$_2$$H$$_4$$.

Looking at the options:

A. NH$$_2$$ (empirical formula, but not molecular)

B. N$$_3$$H

C. NH$$_3$$

D. N$$_2$$H$$_4$$

The molecular formula N$$_2$$H$$_4$$ corresponds to option D.

Hence, the correct answer is Option D.