To determine the formula of the compound, we first need to understand the concept of empirical formula. The empirical formula represents the simplest whole-number ratio of atoms of each element in a compound. Given that the compound contains nitrogen (N) and oxygen (O) in a ratio of 28 g of nitrogen to 80 g of oxygen, we can use the following steps to find the empirical formula:

Step 1: Convert the masses of nitrogen and oxygen to moles using their atomic masses. The atomic mass of nitrogen (N) is approximately 14 g/mol, and the atomic mass of oxygen (O) is approximately 16 g/mol.

$$ \text{Moles of N} = \frac{28 \, \text{g}}{14 \, \text{g/mol}} = 2 \, \text{moles} $$

$$ \text{Moles of O} = \frac{80 \, \text{g}}{16 \, \text{g/mol}} = 5 \, \text{moles} $$

Step 2: Determine the simplest whole-number ratio of the moles of nitrogen to the moles of oxygen by dividing both by the smallest number of moles:

$$ \text{Ratio of N : O} = 2 : 5 $$

This ratio suggests that for every 2 moles of nitrogen, there are 5 moles of oxygen. Thus, the empirical formula of the compound can be represented as $N_2O_5$.

Therefore, the correct option is Option C: $N_2O_5$.