For the elements Be, B, N and O, the order of increasing ionization energies are:

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To tackle the ionization energies for Be, B, N, and O, I start from the general periodic trend: ionization energy tends to rise across a period as effective nuclear charge increases, but there are small irregularities due to electronic configurations.
Option 1: 'N < O < Be < B' would imply nitrogen has the smallest first ionization energy among the four, which contradicts the observed values where Be is quite low because it has a stable filled 2s subshell. This option also places Be after O, which is inconsistent with Be being a lighter element with relatively low energy to remove an electron.
Option 2: 'Be < B <N < O' suggests nitrogen has a higher ionization energy than oxygen. In reality, oxygen's first ionization energy is about 1314 kJ/mol, while nitrogen's is about 1402 kJ/mol, so nitrogen actually requires more energy than oxygen, meaning this ordering is not correct.
Option 3: 'B < Be < O < N' claims that boron has the smallest energy, followed by beryllium, which is at odds with the known fact that Be, with a filled 2s subshell, typically has a lower first ionization energy than B, not the other way around. Thus this sequence misrepresents the relative stabilities of the initial electrons.
Option 4: 'O < N < B < Be' places oxygen with the smallest first ionization energy and beryllium the largest, which is directly contradicted by actual data: Be’s first ionization energy is lower than both B and N, and oxygen’s is not smaller than nitrogen’s in the way this option implies. The pattern here also reverses the expected ascending order across these elements.
Option 5: 'Be < B < O < N' aligns with the experimental values: Be has the lowest first ionization energy among the four, followed by B, then O, and finally N having the highest of the four. This matches the known electronic configurations and the slight irregularities due to half-filled and filled subshell considerations across the period.
In summary, evaluating each option against the known ionization energy data and the underlying electronic structure explains why the only sequence that fits is Be < B < O < N, while the others violate either the known numeric values or the reasoning about subshell stability.

For the elements Be, B, N and O, the order of increasing ionization energies are:

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