Problem 96 - Chemistry The Central Science

Question

CdS has a band gap of 2.4 eV. If large crystals of 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 CdS crystals with a band gap of 2.4 eV is green, with a wavelength of 517 nm. Appropriately sized CdS quantum dots can emit blue light, as their band gap energy can be tuned higher for blue light emission. However, they are unlikely to emit red light, as the band gap energy would need to decrease, which is not possible with CdS.

Step 1: Convert band gap energy to wavelength

To convert the given band gap energy of CdS (2.4 eV) to wavelength, we can use the formula: \(λ = \frac{hc}{E}\), where λ represents the wavelength, h is Planck's constant (6.626 x 10^-34 Js), c is the speed of light (3 x 10^8 m/s), and E is the given energy (2.4 eV). Note that 1 eV = 1.602 x 10^-19 J. First, let's convert the energy from eV to J: \(E = 2.4 eV * 1.602 * 10^{-19} J/eV = 3.8448 * 10^{-19} J\). Now we can calculate the wavelength: \(λ = \frac{(6.626 * 10^{-34} Js) * (3 * 10^8 m/s)}{3.8448 * 10^{-19} J} = 5.17 * 10^{-7} m\), which can be expressed as 517 nm.

Step 2: Determine the color of emitted light

Based on the calculated wavelength of 517 nm, we can now determine the color of the emitted light. The wavelength range for visible light is around 380 nm to 750 nm. The colors are distributed within this range as follows: - Violet: 380 - 450 nm - Blue: 450 - 495 nm - Green: 495 - 570 nm - Yellow: 570 - 590 nm - Orange: 590 - 620 nm - Red: 620 - 750 nm Since the wavelength of the emitted light is 517 nm, it falls within the range for green light.

Step 3: Discuss the ability of CdS quantum dots to emit blue and red light

Quantum dots have size-dependent optical properties, meaning that their band gap energy, and therefore the wavelength of emitted light, can be tuned based on their size. (a) For blue light emission, the wavelength should be in the range of 450 - 495 nm. To check if appropriately sized CdS quantum dots can emit blue light, we will need to check if the wavelength can be achieved by adjusting the band gap energy. In this case, since our given band gap of 2.4 eV produces green light, it is possible for the band gap energy to be higher for blue light emission. Thus, appropriately sized CdS quantum dots can emit blue light. (b) For red light emission, the wavelength should be in the range of 620 - 750 nm. However, as the size of the quantum dots increases, their band gap energy will decrease. Since the given band gap energy of 2.4 eV already corresponds to green light, it is unlikely that appropriately sized CdS quantum dots can emit red light, as the band gap energy would need to decrease further, which is not possible with CdS.

Key Concepts

Understanding Band Gap Energy

In the realm of semiconductor physics, band gap energy is a fundamental concept that refers to the energy difference between the top of the valence band and the bottom of the conduction band. Electrons need to gain this precise energy amount to jump from the valence band, where they are bound to atoms, to the conduction band, where they are free to conduct electricity.

For a material like cadmium sulfide (CdS), the band gap energy is 2.4 electronvolts (eV). When light with energy equal to or greater than this band gap strikes the material, electrons are excited and move across the band gap. Upon returning to their ground states, these electrons release energy in the form of light—or, more specifically, photons. This photon energy corresponds to the band gap energy and thus determines the color of the light emitted.

Through a conversion process using Planck's constant (h) and the speed of light (c), we can relate this energy to a wavelength using the formula: \[ \lambda = \frac{hc}{E} \]
In the case of CdS with a band gap of 2.4 eV, the emitted light corresponds to a wavelength in the visible spectrum, showcasing the direct connection between band gap energy and photon wavelength.

Quantum Dot Size-Dependent Properties

Quantum dots are tiny particles that have remarkable optical and electronic properties that are intricately linked to their size. These nanocrystals can be as small as a few nanometers, a size that instills unique quantum mechanical behaviors.

Their size-dependent properties are especially notable when it comes to their optical characteristics. Altering the size of a quantum dot changes its band gap energy, which in turn affects the color of light it emits when illuminated—you can think of it as tuning a radio to different frequencies to change stations. Smaller quantum dots have larger band gaps and emit light with shorter wavelengths (towards the violet end of the spectrum), whereas larger dots have smaller band gaps and emit light with longer wavelengths (towards the red end).

For CdS quantum dots, this means that by carefully controlling their size, it is possible to 'dial in' the band gap energy to a value that will emit blue light. Conversely, increasing the size of the quantum dots typically leads to a decrease in band gap energy but does not necessarily allow for the emission of red light, given the inherent limits of the material's band gap range.

The Visible Light Spectrum

The visible light spectrum encompasses all the wavelengths of light that the human eye can perceive. Ranging approximately from 380 nm to 750 nm, it includes all the colors of the rainbow, with violet on one end and red on the other.

The wavelength of light determines its color:

• Violet: 380 - 450 nm
• Blue: 450 - 495 nm
• Green: 495 - 570 nm
• Yellow: 570 - 590 nm
• Orange: 590 - 620 nm
• Red: 620 - 750 nm

When a material like CdS is excited by ultraviolet light, it emits photons in a wavelength that typically falls into the green portion of this spectrum, as the 2.4 eV band gap of CdS translates to a wavelength of approximately 517 nm. Understanding where each color falls within the spectrum is crucial when predicting or analyzing the emitted light color from any light-emitting source, including quantum dots.

Thus, the visible light spectrum is not only key to understanding the colors we see around us but also central in determining which quantum dot sizes are needed to achieve specific colors in technological applications, such as LED displays and biological imaging.