The energy of the second Bohr orbit in the hydrogen atom is \(-3.41 \mathrm{eV}\). The energy of the second Bohr orbit of \(\mathrm{He}^{+}\) ion would be : (a) \(-0.85 \mathrm{eV}\) (b) \(-13.64 \mathrm{eV}\) (c) \(-1.70 \mathrm{eV}\) (d) \(-6.82 \mathrm{eV}\)

In an atomic model, especially in the context of the Bohr model, atoms are theorized to have distinct layers of energy where electrons reside. These layers are often referred to as shells or orbits, and each one is associated with a specific energy level. As an analogy, imagine the atom as a solar system where the nucleus is the sun and the electrons are planets, each orbiting in their unique pathways.

The energy levels in an atom are quantized, meaning electrons can only exist in certain regions with specific energies, and not in between. This can be likened to a staircase where you can stand on a step but not in the air between the steps. In the case of the hydrogen atom, the energy levels are calculated using a formula derived from the Bohr model, which incorporates constants such as Planck's constant, the mass and charge of an electron, and the permittivity of free space. This quantization is critical to the unique chemical properties of each element and underpins much of modern chemistry and physics.

The principal quantum number, typically symbolized as 'n', plays a vital role in the Bohr model of the atom. It is an integer that determines the energy level or shell of an electron within an atom and is directly associated with the size and energy of the respective orbit.

Imagine a ladder, with each rung representing a different principal quantum number. Electrons can 'jump' from one rung to another, but they cannot stand in between the rungs. The higher the principal quantum number, the farther the electron is from the nucleus and the higher its energy. In the formula for calculating the energy levels, using a larger value for 'n' results in a less negative (meaning higher) energy value. The significance of the principal quantum number extends beyond the Bohr model as well and is one of the four quantum numbers used in modern quantum mechanics to describe the unique state of an electron.

Permittivity of free space, denoted as \( \epsilon_0 \), is a fundamental constant in physics that describes how electric fields interact with the vacuum of space. It can be thought of as the electrical resistance 'encountered' by an electric field in a perfect vacuum, playing a critical role in quantifying electrostatic interactions.

In the Bohr model, \( \epsilon_0 \) is included in the formula to calculate the energy levels of electrons within an atom. It ensures that the electrostatic force, and hence the energy calculations, are accurate. Understanding the permittivity of free space is not only crucial for atomic models but also for engineering applications, such as designing capacitors or understanding electromagnetic wave propagation in free space.