The ratio of velocity of the electron in the third and fifth orbit of \(\mathrm{Li}^{2+}\) would be : (a) \(3: 5\) (b) \(5: 3\) (c) \(25: 9\) (d) \(9 \because 25\)

A hydrogen-like atom is an atomic species with only one electron, much like a hydrogen atom, but it can be any ion with a similar structure — for instance, He+, Li2+, or even more highly charged ions. These types of atoms are called 'hydrogen-like' or 'hydrogenic' because they behave similarly in terms of their atomic structure and how electrons transition between energy levels. In a hydrogen-like atom, the single electron orbits around the nucleus, which contains one or multiple protons, and this system can be described using quantum mechanics. In educational contexts, such as physical chemistry for JEE (Joint Entrance Examination), understanding hydrogen-like atoms is fundamental as they form the basis for more complex structures and are also a crucial part of various spectroscopic and quantum mechanical calculations.

From the Bohr model of the hydrogen atom, which can be applied to any hydrogen-like atom, it’s seen that the allowed orbits for electrons are quantized, each associated with a specific energy level. Knowledge about these systems is particularly useful for solving problems related to atomic structure and spectral lines, key topics in physical chemistry syllabi.

The principal quantum number, symbolized as 'n', plays a pivotal role in the quantum mechanics of atoms. It determines the energy level or shell of an electron within an atom and, as such, is integral to the understanding of atomic structure. The principal quantum number begins from 1, which denotes the closest orbit to the nucleus, and increases with the electron’s energy and distance from the nucleus. In the context of the Bohr model, as well as the more advanced wave-mechanical model of the atom, the principal quantum number defines the gross energy level of an electron.

For the Joint Entrance Examination (JEE) which is an engineering entrance assessment in India, comprehending the role of the principal quantum number is crucial for subjects like physical chemistry. It's because this number not only determines the energy of the electron but also has implications on other quantities such as the radius of the electron's orbit and the electron's orbital velocity.

In a hydrogen-like atom, the electron orbital velocity refers to the speed at which an electron travels in its orbit around the nucleus. According to Bohr's theory, this velocity is inversely proportional to the principal quantum number, which implies an interesting point: as the electron moves to higher orbits with larger principal quantum numbers, its velocity decreases. This relationship allows the prediction of how fast an electron moves depending on which orbit it's in.

Understanding the relationship between principal quantum number and electron orbital velocity is a much-discussed topic in physical chemistry for competitive exams like the JEE. Problems often require students to calculate changes in velocity as the electron transitions between orbits, which is a direct application of Bohr's theory. As seen in the example provided, the ratio of electron velocities in different orbits is a typical kind of question that can be resolved using this concept.

Physical chemistry for JEE is a branch of chemistry that deals with the physical properties of molecules, the forces that act upon them, and the energy changes that occur during chemical reactions. It is an essential subject for students preparing for the JEE, as it forms one of the papers they have to excel at to secure a position in engineering colleges. Topics like atomic structure, quantum mechanics, thermodynamics, and kinetics require a strong understanding of fundamental principles and an ability to apply them to solve complex problems.

In terms of Bohr's theory and its applications, such as the velocity ratio of electrons in various orbits, mastering this topic can result in significant advantages during the exam. Providing students with exercises that invite them to apply the theory to atoms other than hydrogen, like the Li2+ ion in the example problem, ensures that they are better prepared for the types of questions they might encounter on the actual exam.