Cinnabar (HgS) was utilized as a pigment known as vermillion. It has a band gap of 2.20 eV near room temperature for the bulk solid. What wavelength of light (in nm) would a photon of this energy correspond to?

Verified step by step guidance

1. Step 1: Understand the relationship between energy and wavelength. The energy of a photon is related to its wavelength by the equation: \( E = \frac{hc}{\lambda} \), where \( E \) is the energy in electron volts (eV), \( h \) is Planck's constant (\( 6.626 \times 10^{-34} \) J·s), \( c \) is the speed of light (\( 3.00 \times 10^8 \) m/s), and \( \lambda \) is the wavelength in meters.
2. Step 2: Convert the energy from electron volts to joules. Since 1 eV = \( 1.602 \times 10^{-19} \) J, multiply the given energy (2.20 eV) by this conversion factor to get the energy in joules.
3. Step 3: Rearrange the equation \( E = \frac{hc}{\lambda} \) to solve for wavelength \( \lambda \). This gives \( \lambda = \frac{hc}{E} \).
4. Step 4: Substitute the values for \( h \), \( c \), and the energy in joules into the equation to calculate \( \lambda \).
5. Step 5: Convert the wavelength from meters to nanometers by multiplying the result by \( 10^9 \), since 1 meter = \( 10^9 \) nanometers.

Key Concepts

Band Gap Energy

The band gap energy is the energy difference between the valence band and the conduction 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 requires more energy to excite electrons from the valence band to the conduction band, influencing the wavelengths of light it can absorb or emit.

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 indicates that higher energy photons correspond to shorter wavelengths, while lower energy photons correspond to longer wavelengths. Understanding this relationship is crucial for converting band gap energy into the corresponding wavelength of light.

Planck's Constant

Planck's constant (h) is a fundamental constant in quantum mechanics that relates the energy of a photon to its frequency. Its value is approximately 6.626 x 10^-34 J·s. This constant is essential for calculations involving photon energy and is used in the equation E = hf, where f is the frequency of the photon. It plays a critical role in determining the energy associated with electromagnetic radiation.