Question

Which wavelength of light (in \(\mathrm{nm}\) ) is emitted if an electron moves from the conduction band to the valence band in a sample of diamond (diamond has a band gap of \(5.5 \mathrm{eV}\) )?

Short answer

The wavelength of light emitted is approximately \(337.5 \mathrm{nm}\).

Step-by-step solution

Convert Energy from eV to Joules

To find the wavelength, we first need to convert the energy of the band gap given in electron volts (eV) to joules (J). We use the conversion factor where 1 eV equals to approximately 1.602 x 10^-19 joules. Energy in joules (E) is calculated as: \(E = (5.5 \text{ eV}) \times (1.602 \times 10^{-19} \text{ J/eV})\).

Use Planck's Equation

Planck's equation relates the energy (E) of a photon to its wavelength (\(\lambda\)). The equation is \(E = \frac{hc}{\lambda}\), where h is Planck's constant (\(6.626 \times 10^{-34} \text{ J s}\)) and c is the speed of light (\(3.00 \times 10^{8} \text{ m/s}\)). Using the energy computed from Step 1, we can find the wavelength as \(\lambda = \frac{hc}{E}\).

Calculate the Wavelength and Convert to Nanometers

Substitute the values into the wavelength equation and solve for \(\lambda\). After obtaining the wavelength in meters, convert it to nanometers by multiplying it with \(10^{9}\), since 1 meter equals 1 billion nanometers (\(1 \text{ m} = 10^9 \text{ nm}\)).

Key concepts

Electron Volts to Joules Conversion

To understand the energy transitions in a material such as diamond, it's crucial to convert the energy levels from electron volts (eV) to joules (J), the standard unit of energy in the field of physics. This conversion helps us to apply the fundamental physics equations that are usually expressed in the International System of Units (SI).

An electron volt is the amount of kinetic energy gained or lost by a single electron when it moves through an electric potential difference of one volt. To convert electron volts to joules, we use the conversion factor: 1 eV is equivalent to approximately 1.602 x 10^-19 joules. So, if an electron transitions and gives off 5.5 eV of energy, as in the case of diamond, we multiply 5.5 eV by the conversion factor to get the energy in joules (J).

This step is essential for connecting the microscopic quantum world of electron transitions to the macroscopic world where Planck's equation and other classical physics laws apply.

Planck's Equation

Planck's equation is fundamental for understanding the relationship between the energy of photons and their corresponding wavelengths. It is elegantly simple and is stated as
\(E = \frac{hc}{\lambda}\)
, where \(E\) represents the energy of the photon in joules (J), \(h\) is Planck's constant (approximately 6.626 x 10^-34 J s), \(c\) is the speed of light in a vacuum (approximately 3.00 x 10⁸ m/s), and \(\lambda\) is the wavelength of the photon in meters.

With the energy in joules from the previous conversion, we apply Planck's equation to find the photon's wavelength. The equation reveals that as the energy of the photon increases, the wavelength decreases, reflecting an inverse relationship. This concept is key to analyzing emitted light from electronic transitions, like those occuring within the band gap of materials such as diamond.

Photon Energy and Wavelength Relationship

The photon energy and wavelength relationship is integral to understanding electromagnetic radiation and its interactions with matter. As derived from Planck's equation, there is an inverse relationship between the energy of a photon and its wavelength: as the wavelength of light increases, the energy decreases, and vice versa.

This relationship is given by the expression
\(E = \frac{hc}{\lambda}\)
, where each symbol retains its previous meaning. When electrons transition between energy bands in materials, they emit photons with specific energies corresponding to the band gap; the energy of these photons can then be used to determine their wavelengths. For instance, a photon emitted as a result of an electron moving from the conduction band to the valence band in diamond will have a specific wavelength determined by the material's characteristic band gap energy.

To complete the picture, wavelengths of light are often expressed in nanometers (nm) for convenience, especially in the context of optical phenomena. By converting the wavelength from meters to nanometers, we make the resultant values much easier to interpret and compare with common reference scales in spectroscopy and photonics.