What is the wavelength of light (in nm) emitted when an electron moves from the conduction band to the valence band in a sample of diamond, which has a band gap of 5.5 eV?

Name: Chemistry: A Molecular Approach
Author: Tro
ISBN: 9780134112831

검증된 단계별 안내

Convert the band gap energy from electron volts (eV) to joules (J) using the conversion factor: 1 eV = 1.602 x 10^{-19} J.

Use the energy-wavelength relationship given by the equation: E = \(\frac{hc}{\lambda}\), where E is the energy in joules, h is Planck's constant (6.626 x 10^{-34} J·s), c is the speed of light (3.00 x 10^8 m/s), and \(\lambda\) is the wavelength in meters.

Rearrange the equation to solve for wavelength: \(\lambda\) = \(\frac{hc}{E}\).

Substitute the values for h, c, and the converted energy E into the equation to calculate the wavelength in meters.

Convert the wavelength from meters to nanometers by multiplying by 10^9, since 1 meter = 10^9 nanometers.

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Band Gap Energy

The band gap energy is the energy difference between the conduction band and the valence band in a solid material. It determines the electrical conductivity and optical properties of the material. In semiconductors and insulators, a larger band gap typically means that the material is less conductive and can absorb higher energy photons, which is crucial for understanding electron transitions.

Photon Energy and Wavelength Relationship

The energy of a photon is inversely related to its wavelength, described by the equation E = hc/λ, where E is energy, h is Planck's constant, c is the speed of light, and λ is the wavelength. This relationship allows us to calculate the wavelength of light emitted when an electron transitions between energy bands, as the energy lost during the transition is emitted as a photon.

Electron Transition in Semiconductors

In semiconductors, when an electron moves from the conduction band to the valence band, it releases energy in the form of light (photons). This process is fundamental in understanding how materials like diamond emit light when excited, and the energy of the emitted light corresponds to the band gap energy of the material, which in this case is 5.5 eV for diamond.