What is the distance travelled by an electron revolving in the second orbit of \(\mathrm{Be}^{3+}\) ion, in 100 revolutions? (a) \(3.32 \times 10^{-8} \mathrm{~m}\) (b) \(5.29 \times 10^{-8} \mathrm{~m}\) (c) \(6.64 \times 10^{-8} \mathrm{~m}\) (d) \(1.16 \times 10^{-8} \mathrm{~m}\)

Bohr's model represents a major milestone in the understanding of atomic structure. Introduced by Niels Bohr in 1913, it was the first to incorporate quantum theory and proposed that electrons orbit the nucleus at specific distances. This marked a fundamental shift from the earlier plum pudding model and paved the way for modern atomic physics.

The model introduces the concept of quantized orbits, where electrons can only inhabit certain predefined energy levels. An electron in a hydrogen-like atom moves in a circular orbit around the nucleus, and this is determined by the principal quantum number, denoted as 'n'. The energy associated with an orbit is negative, indicating that energy must be supplied to move an electron to a higher orbit (or energy level).

The Bohr model also quantifies the radius of each orbit for a hydrogen-like atom using a simplified formula. In physical chemistry problems, knowing this formula is crucial for calculations involving electron positions and movements within an atom.

Quantum mechanics is the branch of physics that deals with the mathematical description of the motion and interaction of subatomic particles. It encompasses the behavior of these particles at the nanoscopic scale, which differ significantly from those we experience in macroscopic systems.

Where classical physics fails to explain the peculiar behavior of electrons in an atom, quantum mechanics thrives. In quantum mechanics, we encounter the probabilistic nature of particles. Particles like electrons are described by wave functions, which give the probability of finding a particle in a particular region of space. Quantum mechanics is fundamental in understanding and predicting atomic and molecular phenomena that are explained in your textbook exercises.

The study of atomic structure lies at the heart of chemistry and physics, dealing with the composition and arrangement of electrons, protons, and neutrons within an atom. The core of an atom is its nucleus, which is positively charged due to the presence of protons and carries the most mass because of the additional neutrons.

Electrons, being negatively charged, surround the nucleus in different energy levels or shells, whose theoretical arrangement is predicted by quantum mechanics and Bohr's model. The chemical properties of an element are largely due to the arrangement and interactions of these electrons. For instance, in the exercise where the distance travelled by an electron in a Beryllium ion is calculated, the atomic structure plays a critical role in determining the electron's path around the nucleus.

When solving physical chemistry problems, it's important to understand how mathematical concepts and physical principles come together. These problems, like calculating the electron orbit distance in the example given, rely on a deep understanding of the theoretical models that describe the behavior of atoms and molecules.

Calculating the orbit distance travelled by an electron normally involves using the principles of Bohr's model to derive the radius and then, in turn, calculate the orbit's circumference. Cognizance of constants, such as Planck's constant and the Bohr radius, is paramount when solving these problems. The acquired radius is then utilized to find the circumference, which, when multiplied by the number of revolutions, will give the total distance travelled by the electron. This ability to connect theoretical concepts with problem-solving techniques is key in mastering physical chemistry.