A photovoltaic cell contains a p–n junction that converts solar light to electricity. (a) Silicon semiconductors with a band-gap energy of 107 kJ/mol are commonly used to make photovoltaic cells. Calculate the wavelength that corresponds to the band-gap energy in silicon.

Verified step by step guidance

1. Convert the band-gap energy from kJ/mol to Joules per photon. Since 1 kJ = 1000 Joules and Avogadro's number (\(6.022 \times 10^{23} \text{ mol}^{-1}\)) is the number of particles per mole, use the formula: \(\text{Energy (J)} = \frac{\text{Energy (kJ/mol)}}{1000 \times \text{Avogadro's number}}\).
2. Use the Planck's equation to relate the energy of a photon to its wavelength: \(E = \frac{hc}{\lambda}\), where \(E\) is the energy of the photon, \(h\) is Planck's constant (\(6.626 \times 10^{-34} \text{ Js}\)), \(c\) is the speed of light (\(3.00 \times 10^8 \text{ m/s}\)), and \(\lambda\) is the wavelength in meters.
3. Rearrange the Planck's equation to solve for the wavelength \(\lambda\): \(\lambda = \frac{hc}{E}\).
4. Substitute the values of \(h\), \(c\), and the energy calculated in step 1 into the rearranged equation to find the wavelength.
5. Ensure the units are consistent and convert the wavelength from meters to nanometers if necessary, since wavelengths are often expressed in nanometers in the context of light (1 m = 1,000,000,000 nm).

Key Concepts

Band-Gap Energy

Band-gap energy is the energy difference between the valence band and the conduction band in a semiconductor. It determines the energy required for an electron to jump from the valence band to the conduction band, allowing for electrical conductivity. In photovoltaic cells, the band-gap energy is crucial as it dictates the wavelengths of light that can be absorbed and converted into electricity.

Wavelength and Energy Relationship

The relationship between wavelength and energy is 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 equation shows that energy is inversely proportional to wavelength; as energy increases, wavelength decreases. This principle is essential for calculating the wavelength corresponding to a specific band-gap energy in semiconductors.

Photovoltaic Effect

The photovoltaic effect is the process by which a material converts light energy into electrical energy. In a photovoltaic cell, when photons with sufficient energy strike the semiconductor, they can excite electrons across the band gap, generating an electric current. Understanding this effect is fundamental to grasping how photovoltaic cells operate and their efficiency in converting solar energy into usable electricity.