Question

A semiconductor is used to make an LED that emits red light at \(\lambda=630 \mathrm{nm}\) (a) Find the energy gap in eV of the semiconductor. (b) Find the energy in joules of a photon emitted. (c) Find the energy in joules for Avogadro constant number of these photons.

Short answer

(a) 1.97 eV; (b) \(3.155 \times 10^{-19}\) J; (c) \(1.9 \times 10^{5}\) J.

Step-by-step solution

Understanding the formula for energy from wavelength

We start with the relation between energy and wavelength, given by the equation \(E = \frac{hc}{\lambda}\), where \(h\) is Planck's constant \(6.626 \times 10^{-34} \text{ J s}\), \(c\) is the speed of light \(3 \times 10^8 \text{ m/s}\), and \(\lambda\) is the wavelength.

Convert wavelength to meters

Given \(\lambda = 630 \text{ nm}\). We convert this to meters by noting that \(1 \text{ nm} = 10^{-9} \text{ m}\). Thus, \(630 \text{ nm} = 630 \times 10^{-9} \text{ m}\).

Calculate the energy of a photon in Joules

Substitute the values \(h = 6.626 \times 10^{-34} \text{ J s}\), \(c = 3 \times 10^8 \text{ m/s}\), and \(\lambda = 630 \times 10^{-9} \text{ m}\) into the energy equation: \[ E = \frac{6.626 \times 10^{-34} \times 3 \times 10^8}{630 \times 10^{-9}} \approx 3.155 \times 10^{-19} \text{ J} \]

Convert energy from Joules to electronvolts

1 electronvolt (eV) is equivalent to \(1.602 \times 10^{-19} \text{ J}\). Thus, we convert the energy from Joules to eV: \[ E = \frac{3.155 \times 10^{-19} \text{ J}}{1.602 \times 10^{-19} \text{ J/eV}} \approx 1.97 \text{ eV} \]

Calculate energy for Avogadro's number of photons

Avogadro's number is \(6.022 \times 10^{23}\). To find the energy for Avogadro's number of photons, multiply the energy of one photon by Avogadro's number: \[ E_{\text{total}} = 3.155 \times 10^{-19} \text{ J/photon} \times 6.022 \times 10^{23} \approx 1.9 \times 10^{5} \text{ J} \]

Key concepts

Energy-Wavelength Relation

The energy-wavelength relation is crucial in understanding how light interacts with materials. It describes the direct relationship between the energy of a photon, which is a particle of light, and its wavelength. The formula used to express this relationship is given by the equation: \(E = \frac{hc}{\lambda}\)where:

• \(E\) is the energy of the photon.
• \(h\) is Planck's constant, valued at \(6.626 \times 10^{-34} \text{ J s}\).
• \(c\) is the speed of light, approximately \(3 \times 10^8 \text{ m/s}\).
• \(\lambda\) is the wavelength of light in meters.

This equation shows that energy and wavelength are inversely proportional. It means that as the wavelength decreases, the energy increases and vice versa. This principle is fundamental, especially in technologies like LEDs (Light Emitting Diodes), where controlling the wavelength of emitted light is essential. For instance, in the case of a red LED, with a wavelength of \(630 \text{ nm}\), understanding this relationship helps determine the energy gap and the efficiency of the light emitted.

Photon Energy

Photons are the smallest unit of light and carry energy that is crucial for various optical technologies. To find the energy of a photon, the formula used is derived from the energy-wavelength relation. For light with a known wavelength, you can directly calculate the energy of a single photon using:\(E = \frac{6.626 \times 10^{-34} \times 3 \times 10^8}{\lambda}\)For a red LED emitting light at \(\lambda = 630 \text{ nm} \) (which translates to \(630 \times 10^{-9} \text{ m}\)), the energy of a photon is about \(3.155 \times 10^{-19} \text{ J}\). To convert from energy in joules to electronvolts—another common unit of energy in physics—remember that \(1 \text{ eV} = 1.602 \times 10^{-19} \text{ J}\). Thus, when you convert \(3.155 \times 10^{-19} \text{ J}\) to electronvolts, you get approximately \(1.97 \text{ eV}\). This conversion is essential to relate quantum events at the atomic scale where energies are comparatively small.

Avogadro's Number

Avogadro's Number (\(6.022 \times 10^{23}\)) is a fundamental constant in chemistry and physics, representing the number of atoms, ions, or photons in one mole of a substance. This constant is crucial in quantifying the amount of basic entities in a given sample. When dealing with light, one photon might seem negligible in energy. However, when considering many photons, the total energy becomes significantly impactful. For instance, calculating the energy for Avogadro's number of photons, each with \(3.155 \times 10^{-19} \text{ J}\) energy, involves multiplying by Avogadro's Number:\[ E_{\text{total}} = 3.155 \times 10^{-19} \text{ J/photon} \times 6.022 \times 10^{23} \approx 1.9 \times 10^{5} \text{ J} \]This calculation illustrates the massive amount of energy conveyed when a large quantity of photons is considered, which is crucial for efficient energy utilization in various applications such as solar cells and other light-based technologies.