Question

The first LEDs were made from GaAs, which has a band gap of \(1.43 \mathrm{eV}\). What wavelength of light would be emitted from an LED made from GaAs? What region of the electromagnetic spectrum does this light correspond to: UV, Visible, or IR?

Short answer

The GaAs LED emits light with a wavelength of 866 nm, corresponding to the infrared region of the electromagnetic spectrum.

Step-by-step solution

Convert energy to Joules

First, we have to convert the energy from electron volts (eV) to Joules (J) using the conversion factor 1 eV = \(1.602 \times 10^{-19}\) J. Thus: \(E = 1.43 \ \text{eV} \times 1.602 \times 10^{-19} \frac{\text{J}}{\text{eV}} = 2.29 \times 10^{-19} \ \text{J}\)

Rearrange the formula to solve for wavelength

Now, we can rearrange the formula: \(\lambda = \dfrac{hc}{E}\)

Plug in values to find the wavelength

We know the values of h and c, so we can plug them in: \(\lambda = \dfrac{(6.626 \times 10^{-34} \ \text{J} \cdot \text{s})(2.998 \times 10^8 \ \frac{\text{m}}{\text{s}})}{2.29 \times 10^{-19} \ \text{J}}\)

Calculate the wavelength

Now, we can calculate the wavelength: \(\lambda = \dfrac{(6.626 \times 10^{-34} \ \text{J} \cdot \text{s})(2.998 \times 10^8 \ \frac{\text{m}}{\text{s}})}{2.29 \times 10^{-19} \ \text{J}} \approx 8.66 \times 10^{-7} \ \text{m}\)

Convert wavelength to nanometers

For convenience, we can convert the wavelength to nanometers (nm): \(\lambda = 8.66 \times 10^{-7} \ \text{m} \times \dfrac{10^9 \ \text{nm}}{1 \ \text{m}} = 866 \ \text{nm}\)

Identify the region of the electromagnetic spectrum

With the wavelength calculated, we can identify the region of the electromagnetic spectrum it falls into: - UV: 10 nm - 400 nm - Visible: 400 nm - 700 nm - IR: 700 nm - 1,000,000 nm The wavelength of 866 nm falls in the infrared (IR) region. Hence, the GaAs LED emits light with a wavelength of 866 nm, corresponding to the infrared region of the electromagnetic spectrum.