How much energy is required to ionise a H-atom if the electron occupies \(n=5\) orbit? Compare your answer with the ionization energy of \(\mathrm{H}\) -atom (energy required to remove the electron from \(n=1\) orbit).

The principal quantum number, symbolized as 'n', is fundamental in understanding the behavior of electrons within an atom. It originates from the early days of quantum mechanics, where it was found that electrons occupy certain discrete energy levels.

Each electron in an atom is described by a set of quantum numbers, and the principal quantum number is one of them. Specifically, it determines the energy level or shell in which an electron resides. As 'n' increases, the electron is further from the nucleus and possesses higher energy. The values of 'n' are positive integers — 1, 2, 3, etc. Understanding the role of 'n' is crucial when calculating the ionization energy of an atom, as it directly impacts the energy calculation.

Hydrogen, being the simplest atom with only one electron, serves as a model for explaining the energy levels of electrons governed by quantum mechanics. Its energy levels can be represented by the formula \( E_n = -\frac{13.6 \text{eV}}{n^2} \) where \( E_n \) denotes the energy of the n-th level measured in electron volts (eV) and \( n \) is the principal quantum number.

This formula shows that the energy of an electron is quantized and negative, indicating that it is bound to the nucleus. The magnitude decreases (becomes less negative) as 'n' increases, implying that electrons in higher energy levels are less tightly bound and easier to ionize. This model is a cornerstone in understanding the behavior of electrons in atoms and lays the foundation for more complex quantum systems.

Quantum mechanics is not just a field in physics but a significant framework in chemistry, especially when studying the structure and behavior of atoms and molecules. It provides the mathematical underpinnings for the electron configuration of atoms, chemical bonding, and reaction dynamics.

At its core, quantum mechanics challenges classical intuitions by showing that at the atomic level, energy is not continuous but quantized. Electrons do not orbit the nucleus in defined paths but exist within probabilistic regions called orbitals. Understanding quantum mechanics is essential for predicting how atoms and molecules will interact, making it one of the pillars of modern chemistry.

Ionization energy refers to the amount of energy required to remove an electron from an atom or ion. When discussing the hydrogen atom, the ionization energy can be found with the energy levels formula \( E_n = -\frac{13.6 \text{eV}}{n^2} \).

To calculate the ionization energy, you need to consider the difference in energy between the initial state of the electron and the final state, where the electron is free and no longer bound to the nucleus (essentially zero energy).
For example, for an electron in orbit 'n=5':
\begin{align*} Ionization \ Energy &= 0 \text{ eV} - (Energy \ at \ n=5) \ &= 0 \text{ eV} - (-0.544 \text{ eV}) \ &= 0.544 \text{ eV} \end{align*}
Thus, the electron in the 'n=5' state requires less energy to be ionized compared to an electron in the ground state 'n=1', which requires 13.6 eV.