Question

What is the meaning of stationary orbit of atom? Deriving formula for the energy of the electron in the stationary orbit of the hydrogen atom, mention its relation with the radius of the orbit of electron.

Hint:

The energy of an electron in a stationary orbit depends on the radius of the orbit and the quantum number \(n\). For the hydrogen atom, the energy levels are quantized and given by \(E_n = - \frac{13.6}{n^2}\) eV, where \(n\) is the principal quantum number.

Answer 1

The stationary orbit of an atom refers to the specific orbit of an electron in which the electron does not radiate energy. These orbits are defined by quantized energy levels and are stable, meaning that the electron remains in a given orbit without emitting or absorbing energy. These stationary orbits are described by Bohr’s model of the atom.
According to Bohr’s model, the electron in the hydrogen atom moves in circular orbits around the nucleus under the influence of the Coulomb force between the negatively charged electron and the positively charged proton. The energy of the electron in a stationary orbit can be derived using the following principles: 1. Coulomb Force and Centripetal Force: The centripetal force required to keep the electron in a circular orbit is provided by the electrostatic attraction between the electron and the proton. Using Coulomb’s law, the force is: \[ F = \frac{1}{4 \pi \epsilon_0} \frac{e^2}{r^2} \] Where:
- \(e\) is the charge of the electron,
- \(r\) is the radius of the orbit,
- \(\epsilon_0\) is the permittivity of free space.
2. Quantization of Angular Momentum: Bohr postulated that the angular momentum of the electron in a stationary orbit is quantized and given by: \[ m_e v r = n h / 2 \pi \] Where:
- \(m_e\) is the mass of the electron,
- \(v\) is the speed of the electron,
- \(r\) is the radius of the orbit,
- \(n\) is a positive integer (the principal quantum number),
- \(h\) is Planck’s constant.
3. Expression for Energy: The total energy of the electron in the orbit is the sum of its kinetic energy \(K\) and potential energy \(U\). The potential energy is given by: \[ U = - \frac{1}{4 \pi \epsilon_0} \frac{e^2}{r} \] The kinetic energy is half the magnitude of the potential energy (due to the virial theorem for Coulomb forces): \[ K = \frac{1}{2} \times \frac{1}{4 \pi \epsilon_0} \frac{e^2}{r} \] Thus, the total energy \(E\) is: \[ E = K + U = - \frac{1}{4 \pi \epsilon_0} \frac{e^2}{2r} \] 4. Relation with Radius: Using the quantization condition \(m_e v r = n h / 2 \pi\), we can derive the radius of the electron’s orbit as: \[ r_n = \frac{n^2 h^2 \epsilon_0}{\pi m_e e^2} \] Therefore, the energy of the electron in the \(n\)-th orbit is: \[ E_n = - \frac{13.6}{n^2}~\text{eV} \] This shows that the energy of the electron is inversely proportional to the square of the radius of the orbit.