Problem 92 - Chemistry: The Central Science

Question

\mathrm{CdS}\( has a band gap of \)2.4 \mathrm{eV}\(. If large crystals of \)\mathrm{CdS}$ are illuminated with ultraviolet light, they emit light equal to the band gap energy. (a) What color is the emitted light? (b) Would appropriately sized CdS quantum dots be able to emit blue light? (c) What about red light?

Step-by-Step Solution

Answer

The emitted light from large crystals of CdS when illuminated with ultraviolet light is green, as the band gap energy of 2.4 eV corresponds to a wavelength of 517 nm, which falls within the green range of the visible light spectrum. Appropriately sized CdS quantum dots can emit blue light if they have a slightly larger band gap energy and red light if they have a slightly smaller band gap energy.

Step 1: Convert the band gap energy to wavelength

To find the wavelength of the emitted light, we can use the formula: \(\lambda = \frac{hc}{E}\) where \(\lambda\) is the wavelength, \(h\) is the Planck's constant \(h = 4.14\times 10^{-15}\mathrm{eV\,s}\), \(c\) is the speed of light \(c = 3\times 10^8 \mathrm{-m/s}\), and \(E\) is the band gap energy in eV. For CdS with a band gap of 2.4 eV, the wavelength is: \(\lambda = \frac{(4.14\times 10^{-15}\mathrm{eV\,s})(3\times 10^8 \mathrm{-m/s})}{2.4\mathrm{eV}}\)

Step 2: Calculate the wavelength

Now, calculating the wavelength, we have: \(\lambda = \frac{(4.14\times 10^{-15}\mathrm{eV\,s})(3\times 10^8 \mathrm{-m/s})}{2.4\mathrm{eV}} = 5.17 \times 10^{-7} \mathrm{m} = 517 \mathrm{nm}\)

Step 3: Determine the color of the emitted light

Based on the visible light spectrum: - Violet: 380nm - 450nm - Blue: 450nm - 495nm - Green: 495nm - 570nm - Yellow: 570nm - 590nm - Orange: 590nm - 620nm - Red: 620nm - 750nm The wavelength (517 nm) calculated in Step 2 falls within the green range. (a) The emitted light is green.

Step 4: Check the possibility of emitting blue and red light

(b) For CdS quantum dots to emit blue light, their band gap energy needs to be higher than that required for the blue wavelength range. Since the band gap energy of CdS is 2.4 eV, which corresponds to green light, it is possible for appropriately sized quantum dots to emit blue light by having a slightly larger band gap energy. (c) Similarly, if CdS quantum dots have a slightly smaller band gap energy, they can emit red light. It is, therefore, possible for appropriately sized CdS quantum dots to emit red light as well. In conclusion, the emitted light from large crystals of CdS when illuminated with ultraviolet light is green. Appropriately sized CdS quantum dots can emit both blue and red light.

Key Concepts

Understanding CdS Quantum Dots

CdS quantum dots are tiny semiconductor particles, each consisting of hundreds to thousands of atoms. As the name suggests, they're small enough to exhibit quantum mechanical properties. One of the fascinating aspects of CdS, or cadmium sulfide quantum dots, is their band gap energy. The band gap is critical because it determines the quantum dot's optical and electronic properties, including the color of light it can emit.

The size of the quantum dots plays a vital role in their behavior. When CdS quantum dots are very small, quantum confinement effects come into play. This means the electrons in the material are more restricted in their movement because of the small size of the crystals, leading to discrete energy levels. The band gap expands as the quantum dot size decreases, which results in a blue shift in the emitted light. In turn, larger dots will emit light that is red-shifted due to a smaller band gap.

Furthermore, by carefully controlling the size of these dots during synthesis, it is possible to tailor the band gap energy to our needs. Therefore, the quantum dots can be engineered to emit various colors, including blue and red, which is central to applications like LED displays and solar cells.

Wavelength of Emitted Light

The wavelength of emitted light from a material like a semiconductor correlates directly to its band gap energy. The higher the band gap energy, the shorter the wavelength of the emitted light, which is closer to the violet end of the visible spectrum. Conversely, a lower band gap energy corresponds to a longer wavelength, which is closer to the red end of the spectrum.

Using the formula \( \lambda = \frac{hc}{E} \), we can translate the band gap energy of a material to the specific wavelength of the light that the material emits upon excitation. With cadmium sulfide (CdS), a band gap energy of 2.4 eV translates to an emitted light wavelength of approximately 517 nm, which is in the green range of the visible light spectrum. This calculation is pivotal for understanding how the changes in quantum dot sizes alter the color of light they emit. Quantifying the emitted light’s wavelength allows engineers and scientists to design materials with precise optical properties for various applications.

The Visible Light Spectrum

The visible light spectrum is the portion of the electromagnetic spectrum that is visible to the human eye. It ranges from about 380 nm to about 750 nm in wavelength. The spectrum can be divided into colors that often blend smoothly into one another: violet, indigo, blue, green, yellow, orange, and red, progressing from shorter wavelengths to longer wavelengths.

The importance of the visible light spectrum lies in its relationship with materials like CdS quantum dots. Depending on their band gap energies, these materials can be engineered to absorb and emit light within specific regions of the visible spectrum. For example, with a wavelength of 517 nm, the light emitted by CdS falls into the green category. Nonetheless, if one could tweak the CdS quantum dots to have a band gap energy corresponding to wavelengths of around 450 nm or 650 nm, they would emit blue or red light, respectively. This versatility is why the manipulation of the visible light spectrum is critical in technology, from digital displays to lighting and beyond.