Question

The time period of revolution of electron in H-atom is directly proportional to (a) \(r\) (b) \(r^{1 / 2}\) (c) \(r^{3 / 2}\) (d) \(r^{2}\)

Short answer

The time period of revolution of an electron in an H-atom is directly proportional to \(r^{2}\).

Step-by-step solution

Understand the Bohr Model of Hydrogen Atom

Recall Bohr's model of the hydrogen atom, where the electron orbits the nucleus in distinct paths called orbits or shells. The energy of the electron is quantized, and its orbital period depends on the size of the orbit.

Relate Orbital Period to Radius

According to Bohr's model, the velocity of the electron is given by the relationship: \(v = \frac{n \hbar}{m_er}\) where \(n\) is the principal quantum number, \(\hbar\) is the reduced Planck constant, \(m_e\) the mass of the electron, and \(r\) the radius of the orbit. The orbital period \(T\) is the time it takes for the electron to complete one revolution, which can be found using the circumference of the orbit \(2\pi r\) and the velocity of the electron \(v\).

Derive the Proportionality

Using the circumference \(C = 2\pi r\) and velocity \(v\): \(T = \frac{C}{v} = \frac{2\pi r}{\frac{n \hbar}{m_er}} = \frac{2 \pi m_e r^2}{n \hbar}\). Hence, the time period of revolution \(T\) is directly proportional to \(r^2\).

Key concepts

Quantum Mechanics

Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation for understanding atomic and subatomic processes and introduces some of the most challenging concepts in physics.

In the context of the Bohr model of a hydrogen atom, quantum mechanics plays a crucial role. It suggests that instead of traveling in random orbits, electrons inhabit quantized energy levels. These levels are defined by orbitals, where the position and momentum of the electron cannot be simultaneously known — a principle known as Heisenberg's Uncertainty Principle.

These discrete energy levels explain why atoms absorb or emit radiation at certain specific wavelengths. In Bohr's model, when an electron transitions between energy levels, it does so by absorbing or emitting a photon of a particular energy. This shifted the view of the atom from a mini solar system to a quantized system specific to quantum physics.

Atomic Physics

Atomic physics is the field of physics that studies atoms as an isolated system of electrons and an atomic nucleus. It primarily focuses on the arrangement and behavior of electrons in the shells around the nucleus.

In the Bohr model, the hydrogen atom is depicted with a positively charged nucleus and a negatively charged electron moving in a circular orbit around it. This model was revolutionary because it combined classical concepts with quantum mechanics by introducing quantization into the stable orbits of electrons.

However, Bohr's model is an over-simplification of the hydrogen atom. Atomic physics now uses the more complex Quantum Mechanical Model, which offers a better explanation of how electrons behave around the nucleus. Quantum mechanics allows for the prediction of probabilities of finding electrons in certain locations around the nucleus, rather than fixed orbits.

Electron Orbital Period

The electron orbital period is the time it takes for an electron to complete one full orbit around the nucleus of an atom. According to the Bohr model, the electron travels in circular paths—or orbits—with distinct energy levels.

Using Bohr's equation for the velocity of an electron and the equation for the circumference of its orbit, the orbital period can be derived. It is found that this period is directly proportional to the square of the radius of the electron's orbit, indicated as \( T \propto r^2 \) in the solution.

This relationship reveals a lot about the nature of the electron's movement. The larger the radius of the orbit, the longer it takes for an electron to complete a revolution. It’s an excellent example of how quantum mechanics, at a fundamental level, governs the behavior of subatomic particles like electrons within atoms.