If the band-gap energy of a gallium phosphide (GaP) semiconductor is 222 kJ/mol, calculate the wavelength of light emitted in a GaP light-emitting diode (LED). (LO 12.11) (a) 186 nm (b) 245 nm (c) 539 nm (d) 854 nm

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Convert the band-gap energy from kJ/mol to J/photon by dividing by Avogadro's number (6.022 x 10^{23} mol^{-1}).

Use the energy-wavelength relationship given by the equation E = \frac{hc}{\lambda}, where E is the energy of a photon, 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.

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

Substitute the values for h, c, and the energy per photon (calculated in step 1) into the equation to find \lambda.

Convert the wavelength from meters to nanometers by multiplying by 10^9 nm/m.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 conduction. In the context of light-emitting diodes (LEDs), the band-gap energy corresponds to the energy of the photons emitted when electrons recombine with holes.

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 by a semiconductor based on its band-gap energy. By rearranging the equation, we can find the wavelength corresponding to the energy of the emitted photons.

Units of Energy Conversion

In this problem, it is essential to convert the band-gap energy from kJ/mol to joules per photon for accurate calculations. Since 1 kJ/mol equals approximately 6.022 x 10^23 J (Avogadro's number), dividing the band-gap energy by this number gives the energy per individual photon. This conversion is crucial for applying the photon energy and wavelength relationship correctly.

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