The energy of an electron moving in \(n^{\text {th }}\) Bohr's orbit of an element is given by \(E_{n}=\frac{-13.6}{n^{2}} Z^{2}\) \(\mathrm{eV} /\) atom \((Z=\) atomic number \()\). The graph of \(E\) vs. \(Z^{2}\) (keeping "n" constant) will be :

The Bohr atomic model was a pioneering concept introduced by Niels Bohr in 1913 to explain how electrons can have stable orbits around the nucleus. According to this model, electrons circle the nucleus in specific paths, known as orbits or shells, without losing energy. These orbits correspond to certain discrete energy levels, which means that an electron can only occupy certain areas within the atom at particular energy states.

In the Bohr model, the energy of an electron is quantized, i.e., electrons can only gain or lose energy by jumping from one permitted orbit, or energy level, to another. This is done through the absorption or emission of light with energy corresponding to the difference between these levels. This concept was revolutionary because it combined classical physics with quantum mechanics, providing a stepping stone towards the more modern and comprehensive quantum mechanical model of the atom.

Electron energy levels, also known as electron shells, are an essential principle of the Bohr atomic model. In the context of physical chemistry for competitive exams like JEE, understanding these levels is crucial. Electrons are arranged in atoms within certain energy levels which are at different distances from the nucleus.

The energy of an electron within an atom is negative, meaning that the electron is in a bound state relative to the positively charged nucleus. The more negative the energy, the more stable the electron is within that energy level. When electrons move to a higher energy level, they become less negative, which requires an input of energy. Conversely, when they fall to a lower energy level, they release energy in the form of photons.

The exercise provided exemplifies this by defining the energy of an electron in a Bohr orbit with respect to the atomic number, emphasizing the significance of energy quantization in these levels.

The atomic number, denoted as Z, is a fundamental concept in chemistry and physics that signifies the number of protons found in the nucleus of an atom of an element. It uniquely identifies a chemical element and determines the element's position in the periodic table. Additionally, the atomic number provides insight into the electronic configuration of the atom since, in a neutral atom, the number of electrons equals the number of protons.

As our exercise suggests, the energy levels of electrons within an atom are directly affected by the atomic number. This is because the energy levels are dependent on the electrostatic force of attraction between the positively charged nucleus and the negatively charged electrons. With an increase in the atomic number, the strength of this attraction grows, and therefore, the energy of an electron becomes more negative, as indicated by the given formula.

Physical chemistry is one of the branches of chemistry that deals with the study of how matter behaves on a molecular and atomic level and how chemical reactions occur. It is a subject that plays a pivotal role in the Joint Entrance Examination (JEE), an engineering entrance assessment conducted in India. Concepts like the Bohr atomic model and electron energy levels are foundational in the physical chemistry part of the JEE syllabus.

For aspirants of competitive exams like the JEE, understanding and applying principles like the energy of electrons in Bohr's orbits is integral in solving problems and making predictions about atomic interactions. Equations like the one provided in the exercise are frequently used in the JEE to test students' grasp of these concepts. A clear understanding of atomic structure, including energy levels and the significance of the atomic number, is essential for success in physical chemistry and the JEE.