Question

excited hydrogen atom emits a Photon of wave length \(\lambda\) in returning to the ground state The quantum number \(\mathrm{n}\) of excited state is given by (A) \(\sqrt{[}(\lambda . \mathrm{R}-1) /(\lambda \mathrm{R})]\) (B) \(\sqrt{[}(\lambda \mathrm{R}) /(\lambda \mathrm{R}-1)]\) (C) \(\sqrt{[\lambda R}(\lambda \mathrm{R}-1)]\) (D) \(\lambda \mathrm{R}(\mathrm{R}-1)\)

Short answer

The correct expression for the quantum number n of an excited hydrogen atom that returns to its ground state by emitting a photon with a wavelength λ is (B) \(n = \sqrt{\frac{(\lambda R)}{(\lambda R - 1)}}\).

Step-by-step solution

Recall the Rydberg formula for hydrogen

The Rydberg formula for hydrogen can be expressed as: \[ \frac{1}{\lambda} = R \left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right) \] where: - \(λ\) is the wavelength of the photon emitted, - \(R\) is the Rydberg constant for hydrogen, approximately equal to \(1.097 \times 10^7 \, \text{m}^{-1}\), - \(n_f\) is the final principal quantum number of the electron, - \(n_i\) is the initial principal quantum number of the electron. In this case, the hydrogen atom returns to its ground state (i.e., \(n_f = 1\)), and we need to find the expression for \(n_i\), referred to as \(n\).

Substitute the values in the Rydberg formula

Substitute \(n_f = 1\) into the Rydberg formula: \[ \frac{1}{\lambda} = R \left(\frac{1}{1^2} - \frac{1}{n^2}\right) \]

Simplify and solve for n

Now simplify the formula and solve for \(n\): \[ \frac{1}{\lambda} = R \left(1 - \frac{1}{n^2}\right) \] Divide both sides of the equation by R: \[ \frac{1}{\lambda R} = 1 - \frac{1}{n^2} \] Rearrange to isolate the term \(\frac{1}{n^2}\): \[ \frac{1}{n^2} = 1 - \frac{1}{\lambda R} \] Now, take the reciprocal of both sides: \[ n^2 = \frac{1}{1 - \frac{1}{\lambda R}} \] Finally, to obtain the expression for \(n\), take the square root of both sides: \[ n = \sqrt{\frac{1}{1 - \frac{1}{\lambda R}}} \] Comparing the given options, we find the correct expression is:

Answer

(B) \(n = \sqrt{\frac{(\lambda R)}{(\lambda R - 1)}}\)

Key concepts

Hydrogen Atom

The hydrogen atom is the simplest atom in the universe. It consists of just one proton and one electron. This simple structure makes it an essential model for understanding more complex atoms. In the hydrogen atom:

• The electron orbits the proton, which acts as the nucleus.
• The forces between the electron and proton are governed by electromagnetic forces.
• Despite its simplicity, the hydrogen atom displays various energy levels where the electron can occupy different orbits.

These energy levels are quantized, meaning the electron can only exist at certain fixed energies. When the electron absorbs or emits energy, it jumps between these levels, a process crucial for understanding phenomena such as photon emission.

Quantum Number

Quantum numbers are essential in describing the properties of atomic orbitals and the electrons in atoms. They specify various properties such as energy levels, orbital shapes, and orientations. For an electron in the hydrogen atom:

• The principal quantum number (n) indicates the energy level of the electron and its orbit size.
• Higher values of n correspond to higher energy levels and larger orbits.
• In the Rydberg formula provided in the exercise, the initial quantum number \(n_i\) and the final quantum number \(n_f\) determine the energy emitted as photons.

This concept helps in understanding how electrons transition between different energy levels, leading to photon emission.

Photon Emission

Photon emission occurs when an electron in an atom transitions from a higher energy level to a lower one. In the case of a hydrogen atom:

• As the electron falls to a lower energy level, it releases energy in the form of a photon.
• The energy of the emitted photon corresponds exactly to the energy difference between the two levels.
• The Rydberg formula allows us to calculate the wavelength of this emitted photon, connecting it to quantum numbers involved in the transition.

This concept is key to understanding why light has different wavelengths (colors) and why each element has a unique emission spectrum.

Wavelength

Wavelength refers to the distance between successive crests of a wave. In terms of photon emission from a hydrogen atom:

• The wavelength \(\lambda\) of the emitted photon is inversely related to the photon's energy.
• Shorter wavelengths correspond to higher energy photons, while longer wavelengths indicate lower energy photons.
• The Rydberg formula for hydrogen expresses the relationship between the energy levels (quantum numbers) of the electron and the wavelength of the emitted photon.

Understanding wavelength helps explain different phenomena in optics and even in applications like lasers and spectroscopy, where light properties are critical.