Answer
The elements \( \mathrm{B}, \mathrm{Be}, \mathrm{C}, \mathrm{N} \), and \( \mathrm{O} \) can be ranked in order of increasing first ionization energy as follows:
\[ \mathrm{B} < \mathrm{Be} < \mathrm{C} < \mathrm{N} < \mathrm{O} \]
Work Step by Step
The reasoning behind this ranking is that as we move across a period in the periodic table from left to right, the number of protons in the nucleus increases, leading to a greater nuclear charge. This increased nuclear charge exerts a stronger pull on the outermost electrons, making it more difficult to remove them, and thus increasing the first ionization energy.
