Problem 23 - Chemistry: The 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 emitted wavelength of light from a GaP LED can be calculated using the energy-wavelength relationship: \(E = h \times f\) and the frequency-wavelength relationship: \(c = f \times \lambda\). Firstly, convert the given band gap energy of 2.2 eV to Joules: 2.2 eV = 2.2 × \(1.60218 \times 10^{-19}\) J. Then, calculate the frequency of the emitted light: \(f = \dfrac{E}{h}\), where h is Planck's constant (\(6.626 \times 10^{-34}\) J·s). Finally, calculate the emitted wavelength: \(\lambda = \dfrac{c}{f}\), where c is the speed of light in vacuum (\(3 \times 10^8\) m/s). To express the wavelength in nanometers (nm), multiply the result by \(10^9\).

Step 1: Given Values

The band gap energy of GaP is given as 2.2 eV.

Step 2: Convert energy from eV to Joules

First, we need to convert the energy value from electron volts (eV) to Joules (J) by using the conversion factor: 1 eV = \(1.60218 \times 10^{-19}\) J So, 2.2 eV = 2.2 × \(1.60218 \times 10^{-19}\) J

Step 3: Find the energy-frequency relation

We can use the energy-frequency relation, which is derived from Planck's equation: E = h × f Where E represents energy, f represents frequency, and h is Planck's constant (\(6.626 \times 10^{-34}\) J·s).

Step 4: Calculate the frequency of emitted light

Now, we can calculate the frequency of emitted light using the energy-frequency relation: f = E / h Where E is the energy in Joules, and h is Planck's constant (\(6.626 \times 10^{-34}\) J·s). f = (2.2 × \(1.60218 \times 10^{-19}\) J) / (\(6.626 \times 10^{-34}\) J·s)

Step 5: Finding the frequency-wavelength relation

We can relate the frequency and wavelength using the speed of light (c) in vacuum: c = f × λ Where c is the speed of light in vacuum (\(3 \times 10^8\) m/s), f is frequency, and λ is wavelength.

Step 6: Calculate the emitted wavelength

Finally, we can rearrange the frequency-wavelength equation to find the wavelength of emitted light: λ = c / f λ = (\(3 \times 10^8\) m/s) / ((2.2 × \(1.60218 \times 10^{-19}\) J) / (\(6.626 \times 10^{-34}\) J·s)) Now, calculate the emitted wavelength, which will be in meters. To express the wavelength in nanometers (nm), multiply the result by \(10^9\).