Problem 23 - Chemistry Central Science

Question

The semiconductor GaP has a band gap of \(2.2 \mathrm{eV}\). Green LEDs are made from pure GaP. What wavelength of light would be emitted from an LED made from GaP?

Step-by-Step Solution

Answer

The wavelength of light emitted from a GaP-based green LED is approximately \(565 \, nm\).

Step 1: Identify the given values

We are given the band gap energy of GaP semiconductor as \(2.2 \mathrm{eV}\).

Step 2: Convert energy to joules

To work with the wavelength formula, we need the energy value in joules. To convert electron volts (eV) to joules (J), use the following conversion factor: \(1\, electronvolt (eV) = 1.6 \times 10^{-19}\,Joules\). So, we have: \(E = 2.2 \, eV \times \frac{1.6 \times 10^{-19} \, Joules}{1 \, eV}\) \(E = 3.52 \times 10^{-19} \, Joules\)

Step 3: Apply the photon energy-wavelength formula

The energy of a photon is related to its wavelength by the following formula: \(E = h \frac{c}{\lambda}\) Where, \(E\) is the energy of the photon, \(h\) is the Planck's constant (\(6.63 \times 10^{-34} \, Js\)), \(c\) is the speed of light (\(3 \times 10^8 \, m/s\)), \(\lambda\) is the wavelength of the photon.

Step 4: Solve for the wavelength

Rearrange the photon energy-wavelength formula to solve for the wavelength: \(\lambda = h \frac{c}{E}\) Now, plug in the values of \(E\), \(h\), and \(c\) into the equation to get \(\lambda\): \(\lambda = \frac{(6.63 \times 10^{-34} \, Js)(3 \times 10^8 \, m/s)}{3.52 \times 10^{-19} \, Joules}\)

Step 5: Calculate the wavelength

Solve for the wavelength, \(\lambda\): \(\lambda = \frac{(6.63 \times 10^{-34} \, Js)(3 \times 10^8 \, m/s)}{3.52 \times 10^{-19} \, Joules} = 5.65 \times 10^{-7} \, m\)

Step 6: Convert wavelength to nanometers

Finally, to express the wavelength in nanometers, convert meters to nanometers by using the conversion factor (\(1\,m = 1 \times 10^{9}\,nm\)): \(\lambda = 5.65 \times 10^{-7} \, m \times \frac{1 \times 10^{9} \, nm}{1 \, m}\) \(\lambda = 565 \, nm\) So, the wavelength of light emitted from a GaP-based green LED is approximately \(565 \, nm\).