Question

If \(\mathrm{r}_{0}\) be the radius of first Bohr's orbit of \(\mathrm{H}\) atom, the de-Broglie's wavelength of an electron revolving in the third Bohr's orbit will be: (1) \(2 \pi r_{0}\) (2) \(4 \pi \mathrm{r}_{0}\) (3) \(6 \pi \mathrm{r}_{0}\) (4) \(\pi \mathrm{r}_{0}\)

Short answer

The de-Broglie wavelength is \(6 \pi r_{0}\). Hence, the correct option is (3) \(6 \pi r_{0}\).

Step-by-step solution

Identify the relevant formula

The de-Broglie wavelength (\(\lambda\)) for an electron in the nth Bohr orbit can be given by: \(\lambda = \frac{2 \pi r_{n}}{n}\), where \(r_{n}\) is the radius of the nth orbit.

Express the radius of the third orbit

For the third Bohr orbit (n=3), the radius can be written as: \(r_{3} = 9 r_{0}\). This is because \(r_{n} = n^2 r_{0}\).

Substitute \(n = 3\) and \(r_{3}\) into the formula

Plug in the values: \(\lambda = \frac{2 \pi r_{3}}{3} = \frac{2 \pi \times 9 r_{0}}{3} = 6 \pi r_{0}\).

Key concepts

de-Broglie wavelength

The de-Broglie wavelength is a fundamental concept that merges classical and quantum physics.
It stems from the idea that particles, such as electrons, can exhibit wave-like behavior. De-Broglie proposed that the wavelength (\(\lambda\)) of a particle is inversely proportional to its momentum (\(p\)). The formula is given by: \[ \lambda = \frac{h}{p} \] where \(h\) is Planck's constant.

For electrons in Bohr's orbits, their momentum can be related to their orbit. In the nth Bohr orbit, the de-Broglie wavelength is given by: \[ \lambda = \frac{2 \pi r_n}{n} \] where \(r_n\) is the radius of the nth orbit.

This proposes that electrons in different orbits have distinct wavelengths, which is crucial for understanding their behavior and energy levels in an atom.
In the provided exercise, for the third Bohr orbit, the calculation confirmed the wavelength as \(6 \pi r_0\), showcasing these principles.

Bohr radius

The Bohr radius (\(\mathrm{r}_{0}\)) is a key concept in atomic physics.
It represents the most probable distance between the nucleus and the electron in a hydrogen atom's ground state (first orbit). The Bohr radius is given by: \[ r_0 = \frac{h^2}{4 \pi^2 m_e e^2} \] where \(h\) is Planck’s constant, \(m_e\) is the electron mass, and \(e\) is the elementary charge.

In the context of Bohr’s model, the radius for the nth orbit is given by: \[ r_n = n^2 r_0 \] This means that for the third orbit (n=3), its radius would be: \[ r_3 = 9 r_0 \] Understanding this helps in computing the de-Broglie wavelength for electrons in different orbits as seen in the exercise.

Quantum mechanics

Quantum mechanics is a foundational theory in physics.
It describes the behavior of particles at atomic and subatomic scales, where classical mechanics no longer applies.

Some core principles of quantum mechanics include:

• Wave-particle duality: Particles, such as electrons, have both particle-like and wave-like properties.
• Quantization: Certain properties, like energy, can only take discrete values.
• Uncertainty Principle: It is impossible to simultaneously know the exact position and momentum of a particle.

Bohr's model and de-Broglie's hypothesis form part of early quantum theory.
Bohr's model quantizes electron orbits and energies, while de-Broglie introduced the concept of matter waves.
The principles of quantum mechanics provide the framework for understanding atomic structures and interactions as illustrated in the exercise involving the de-Broglie wavelength.