Step 1: Recall the trend of first ionization enthalpy in the second period.
In the second period, the elements are:
\[
Li,\; Be,\; B,\; C,\; N,\; O,\; F,\; Ne
\]
Generally, first ionization enthalpy increases from left to right across a period because atomic size decreases and effective nuclear charge increases.
Step 2: Identify the element with the second lowest ionization enthalpy.
Among the second period elements, Lithium (\(Li\)) has the lowest first ionization enthalpy.
The next higher value is for Boron (\(B\)).
Hence, the element \(X\) having the second lowest first ionization enthalpy is:
\[
X = B
\]
Step 3: Identify the element with the second highest ionization enthalpy.
Neon (\(Ne\)) has the highest first ionization enthalpy in the second period because it has a completely filled stable electronic configuration.
The second highest value is for Fluorine (\(F\)).
Thus,
\[
Y = F
\]
Step 4: Match with the given options.
The correct pair is:
\[
(B,\;F)
\]
which corresponds to option (1).
Step 5: Final conclusion.
Hence, the correct answer is:
\[
\boxed{(1)\; B,\;F}
\]