Problem 51 - General Chemistry: Principles and Modern Applications

Question

In \(\mathrm{ZnO}\), the band gap between the valence and conduction bands is \(290 \mathrm{kJmol}^{-1}\), and in \(\mathrm{CdS}\) it is \(250 \mathrm{kJmol}^{-1} .\) Show that CdS absorbs some visible light but ZnO does not. Explain the observed colors: \(\mathrm{ZnO}\) is white and \(\mathrm{CdS}\) is yellow.

Step-by-Step Solution

Answer

CdS absorbs blue light (479 nm) and appears yellow as it reflects other colors, whereas ZnO absorbs violet light (413 nm) but appears white as it reflects all other visible light.

Step 1: Calculate the Energy of the Incident Light

The energy (E) of a photon can be calculated using the formula\[E = hc/{\lambda}\] where h is Planck’s constant, c is the speed of light, and \({\lambda}\) is the wavelength of light. For visible light, the wavelength range lies between 400 nm to 700 nm.

Step 2: Convert the Band Gap to Energy in electron volts (eV)

1 eV is equal to 96.485 kJ/mol. Therefore, the band gaps in eV of ZnO and CdS are\[ZnO = 290/96.485 = 3.01 eV;\] \[CdS = 250/96.485 = 2.59 eV.\]

Step 3: Calculate the Wavelength that Corresponds to the Band Gap

Use the formula for energy from step 1 to calculate the wavelength that corresponds to the band gap for ZnO and CdS. The wavelength of light absorbed by a material is given by\[{\lambda} = hc/E\]Here, h is Planck’s constant, c is the speed of light, and E is energy. Substituting the values for ZnO, we find that\[\lambda_{ZnO} = (6.63 x 10^-34 Js x 3.0 x 10^8 ms^-1)/(3.01 x 1.6 x 10^-19 J) = 413 nm.\]Similarly, for CdS,\[\lambda_{CdS} = (6.63 x 10^-34 Js x 3.0 x 10^8 ms^-1)/(2.59 x 1.6 x 10^-19 J) = 479 nm.\]

Step 4: Relate the Absorbed Wavelength to the Observed Colors

Substances appear colored as they absorb certain wavelengths of visible light and transmit or reflect the rest. For ZnO, we calculated that it absorbs light with a wavelength of 413 nm which falls in the violet range. As a result, it would reflect all the other colors, including white light, making it appear white. For CdS, we calculated that it absorbs light with a wavelength of 479 nm which falls in the blue range. Consequently, its reflected light is subtractive of blue, tending towards yellow, hence its yellow appearance.

Key Concepts

Photons

Photons are the basic units of light and other forms of electromagnetic radiation. They carry energy that is proportional to the light's frequency. One interesting thing about photons is that they have no mass, yet they can exert force and transfer energy.
Photons travel at the speed of light, which is approximately 3 x 10^8 meters per second in a vacuum. This speed remains constant regardless of the photon's energy or wavelength.

• Photons are produced when electrons in an atom transition between energy levels.
• They can be absorbed or emitted by atoms, which changes the energy state of these atoms.
• The amount of energy that a photon carries is directly related to its frequency, as described by the equation: \[ E = h u \] where \( E \) is energy, \( h \) is Planck's constant (\(6.63 \times 10^{-34} \text{Js}\)), and \( u \) is the frequency.

Photons' interaction with atoms is key to understanding phenomena like color absorption, which is determined by which wavelengths are absorbed or reflected.

Visible Light

Visible light is the range of electromagnetic radiation that the human eye can detect. It spans wavelengths from about 400 nm to 700 nm. Each specific wavelength within this range is perceived as a different color to the human eye, from violet at the shorter wavelengths to red at the longer ones.

• Red light has longer wavelengths and lower energy.
• Violet light has shorter wavelengths and higher energy.
• This range of colors is what we typically see in rainbows.

The ability of a material to absorb visible light depends on its electronic structure. When light hits a material, photons with a specific energy are absorbed, which corresponds to the energy needed to move electrons across the band gap of the material. This is how different materials can appear in different colors. Structures with a larger band gap tend to absorb less visible light, thus appearing lighter or even white.

Wavelength

A wavelength is the distance between two consecutive peaks in a wave. In the context of light and electromagnetic waves, it determines the light's color. Wavelength is inversely proportional to frequency; as the wavelength increases, the frequency decreases.

• Wavelength is commonly measured in nanometers (nm) or meters (m).
• Shorter wavelengths correspond to higher energy photons.
• The wavelength of light absorbed depends on the electronic transitions available in a material.

For example, ZnO and CdS have different band gaps, which means they absorb light at different wavelengths, influencing their color. \( \mathrm{ZnO} \) absorbs at around 413 nm (violet), which is why it looks white overall as most visible light is not absorbed. Meanwhile, \( \mathrm{CdS} \) absorbs at around 479 nm (blue), hence it reflects and appears yellow.

Energy Conversion

Energy conversion in the realm of photons involves the transformation of light energy into other forms. This principle is important when materials absorb light, as photons provide the energy to move electrons to a higher energy state. The amount of energy a photon has depends on the wavelength of light, described by the equation \( E = \frac{hc}{\lambda} \), where \( h \) is Planck's constant and \( c \) is the speed of light.

• Conversion processes occur when a photon is absorbed, its energy goes into raising an electron to a higher energy level.
• Different materials require different amounts of energy (band gap energy) for such transitions.
• Only photons with energy equal to or greater than the material's band gap can initiate these transitions.

This principle pairs closely with concepts in semiconductors and photovoltaics, where light is converted into electricity. Differences in band gap energies between materials like \( \mathrm{ZnO} \) and \( \mathrm{CdS} \) affect their visible color.

Color Absorption

Color absorption is what determines the color that we perceive a material to be. It occurs when certain wavelengths of visible light are absorbed by a material, and the remaining wavelengths are reflected or transmitted. The absorbed light's energy is associated with electronic transitions within the material.
For example, when examining the band gaps of \( \mathrm{ZnO} \) and \( \mathrm{CdS} \):

• \( \mathrm{ZnO} \) has a larger band gap and absorbs in the violet range (413 nm), yet appears white because most visible light is not absorbed.
• \( \mathrm{CdS} \), with a smaller band gap, absorbs blue light (absorption at 479 nm), reflecting light that hints towards yellow.
• This subtraction of specific colors from the light contributes to the material's overall appearance.

Hence, the color a substance appears is the complementary color of the light it absorbs. This is a core principle in material science and helps explain the visible properties of materials.