This question asks us to identify the specific dihedral angle associated with the second least stable conformation of the $n\text{-butane}$ molecule. To answer this, we need to understand the different possible spatial arrangements (conformations) of $n\text{-butane}$ and their relative energy levels, which determine their stability.

Understanding n-Butane Conformer Stability

$n\text{-butane}$ is a molecule with the chemical formula $ \text{CH}_3\text{CH}_2\text{CH}_2\text{CH}_3 $. Like many alkanes, rotation can occur around the single carbon-carbon bonds. Rotation around the central C2-C3 bond leads to different arrangements of the atoms in space, known as conformers. These conformers are not different molecules; they are just different rotational forms of the same molecule. They have different energies due to factors like steric hindrance (repulsion between bulky groups) and torsional strain (repulsion between electron clouds of bonds). More stable conformers have lower energy, while less stable conformers have higher energy.

Key n-Butane Conformations and Dihedral Angles

We typically analyze the conformations of $n\text{-butane}$ by looking at the dihedral angle between the two methyl ($ \text{CH}_3 $) groups as we rotate around the central C2-C3 bond. The main conformers are:

• Anti Conformer: In this arrangement, the two methyl groups are positioned opposite each other, with a dihedral angle of $180^\circ$. This is the most stable conformation because the bulky methyl groups are as far apart as possible, minimizing steric repulsion.
• Gauche Conformer: Here, the methyl groups are separated by a $60^\circ$ dihedral angle. This is a staggered conformation, which is generally more stable than eclipsed ones, but it's less stable than the anti form because the methyl groups are closer together, leading to some steric strain.
• Eclipsed Conformers: In these arrangements, atoms on one carbon are directly aligned with atoms on the adjacent carbon. There are two types:
  • Fully Eclipsed (Syn-periplanar): This occurs when the two methyl groups eclipse each other, resulting in a $0^\circ$ dihedral angle. This is the least stable conformation due to maximum steric hindrance between the large methyl groups.
  • Partially Eclipsed (Anticlinal): This occurs when a methyl group eclipses a hydrogen atom on the other carbon. The dihedral angle is $120^\circ$ (or $240^\circ$, which is energetically equivalent). This is less stable than the staggered (anti and gauche) conformations because of the eclipsing strain, but more stable than the fully eclipsed conformation where the two methyl groups clash.

Ranking Conformer Stability

To find the second least stable conformer, let's rank them from most stable (lowest energy) to least stable (highest energy):

1. Anti ($180^\circ$): Most stable
2. Gauche ($60^\circ$): Less stable than anti
3. Partially Eclipsed ($120^\circ$): Less stable than gauche
4. Fully Eclipsed ($0^\circ$): Least stable

Looking at this order:

• The least stable conformer is the Fully Eclipsed ($0^\circ$).
• The second least stable conformer is the one just above it in instability, which is the Partially Eclipsed ($120^\circ$).

Dihedral Angle of the Second Least Stable Conformer

As identified above, the second least stable conformation of $n\text{-butane}$ is the Partially Eclipsed conformation. The characteristic dihedral angle for this conformation, where a methyl group eclipses a hydrogen atom, is $120^\circ$.

Therefore, the dihedral angle of the second least stable conformer of $n\text{-butane}$ is $120^\circ$.