Question

How does the Bohr theory account for the observed phenomenon of the emission of discrete wavelengths of light by excited atoms?

Short answer

The Bohr theory accounts for the observed phenomenon of the emission of discrete wavelengths of light by excited atoms by postulating that electrons in atoms are restricted to specific energy levels. When an electron moves from a higher energy level to a lower one, it releases the excess energy in the form of a photon of light, with an energy corresponding to the energy difference between the two levels. Using the relationship \(E = \dfrac{hc}{\lambda}\), where \(E\) is the energy of the emitted photon and \(\lambda\) is the wavelength, we can demonstrate that these emitted wavelengths are discrete and unique for each element, based on their specific energy transitions.

Step-by-step solution

Understanding the Bohr Model

The Bohr model is a simplified description of an atom, primarily focusing on hydrogen. In this model, electrons orbit the positively charged nucleus in specific circular energy levels. Each energy level corresponds to a specific energy value. Electrons in higher energy levels have higher energies while electrons in lower energy levels have lower energies. These energy levels are quantized, meaning that only specific energy levels are allowed.

Electrons Transitioning between Energy Levels

In the Bohr model, when energy is provided to an atom, electrons can absorb that energy and transition to higher, less stable energy levels. This process is called excitation. Conversely, when an electron moves from a higher to a lower energy level, it releases the excess energy in the form of a photon of light. The energy difference between the initial and final states of the electron determines the energy of the emitted photon.

Relationship between Photon Energy and Wavelength of Light

The energy of the emitted photon is related to the wavelength of light as follows: \(E = h\nu = \dfrac{hc}{\lambda}\) Here, \(E\) is the energy, \(h\) is the Planck's constant (\(6.626 \times 10^{-34}\ \text{J}\cdot\text{s}\)), \(\nu\) is the frequency of light, \(c\) is the speed of light (\(3\times 10^8\ \text{m/s}\)), and \(\lambda\) is the wavelength. When electrons transition between specific energy levels, the energy of the emitted photons corresponds to specific wavelengths. This is why the emitted light has discrete wavelengths.

Calculating Wavelengths for Emitted Light

The energy difference between initial and final energy levels in a hydrogen atom can be determined using the following formula: \(\Delta E = -13.6\ \text{eV} \left(\dfrac{1}{n_f^2} - \dfrac{1}{n_i^2}\right)\) Here, \(n_i\) and \(n_f\) are the initial and final principal quantum numbers of the energy levels, and \(\Delta E\) is the energy difference between them. Using the relationship between the energy and the wavelength established in Step 3, we can calculate the wavelengths of light emitted during these specific transitions.

Hydrogen Emission Spectrum

As electrons transition between energy levels in a hydrogen atom, they emit specific wavelengths of light that form the hydrogen emission spectrum. The unique wavelengths correspond to specific energy transitions and are discrete in nature. This is how the Bohr theory accounts for the observed phenomenon of the emission of discrete wavelengths of light by excited atoms.

Key concepts

Photon Emission

Photon emission is a fascinating process that occurs when electrons within an atom drop from a higher energy level to a lower one. During this transition, the electron releases energy in the form of a photon. The energy of the emitted photon is directly proportional to the difference in energy between the two levels. This means that the larger the energy gap between levels, the more energetic (and shorter in wavelength) the emitted photon will be.

It's important to understand that photons are discrete packets of energy, hence the term 'quantum'. This is why the light emitted from atoms appears in specific wavelengths, rather than a continuous spectrum. These discrete wavelengths make atomic emission spectra extremely valuable tools in analytical chemistry and physics for identifying substances.

Energy Levels

In the Bohr model of the atom, electrons exist in distinct energy levels, also known as shells. These levels can be envisioned as concentric orbits around the atom's nucleus, somewhat like planets orbiting the sun. What's unique about these levels is their quantized nature. Only certain orbits are permissible and electrons cannot exist in between these levels.

• Lower energy levels are closer to the nucleus and electrons residing here are more stable.
• Higher energy levels are further away and electrons here are more easily excited to jump to another level.

An atom becomes excited when an electron absorbs energy and moves to a higher energy level. However, this state is unstable, and the electron eventually falls back to a lower energy level, releasing energy in the form of light. The Bohr model accurately explains these transitions and the resulting emitted radiation, helping us understand many atomic behaviors and phenomena.

Hydrogen Spectrum

The Hydrogen spectrum is an excellent illustration of photon emission where light is emitted at discrete wavelengths. When electrons in a hydrogen atom transition between energy levels, they emit light at specific wavelengths that correspond to those energy differences in the Bohr model. These spectral lines can be categorized into series like the Lyman, Balmer, and Paschen series, each corresponding to electron transitions ending at different energy levels.

• Lyman series: Transitions to the ground state, emitting ultraviolet light.
• Balmer series: Transitions to the first excited state, producing visible light.
• Paschen series: Transitions to the second excited state, resulting in infrared light.

This hydrogen emission spectrum is essential for understanding atomic structure. It supports Bohr's theory by demonstrating that electrons occupy quantized energy levels. Additionally, it provides a reliable method for identifying elements due to the uniqueness of each element's spectral lines.