Problem 149 - Chemistry An Atoms First Approach

Question

Photogray lenses incorporate small amounts of silver chloride in the glass of the lens. When light hits the AgCl particles, the following reaction occurs: $$ \operatorname{AgCl} \stackrel{h v}{\longrightarrow} \mathrm{Ag}+\mathrm{Cl} $$ The silver metal that is formed causes the lenses to darken. The energy change for this reaction is \(3.10 \times 10^{2} \mathrm{kJ} / \mathrm{mol} .\) Assuming all this energy must be supplied by light, what is the maximum wavelength of light that can cause this reaction?

Step-by-Step Solution

Answer

The maximum wavelength of light that can cause the reaction in photogray lenses is 385 nm.

Step 1: Convert the energy change per mole to energy per photon

We are given the energy change for the reaction as \(3.10 \times 10^{2} \mathrm{kJ/mol}\). We will now convert it to energy per photon by using Avogadro's constant, which is \(N_A = 6.022\times10^{23}\, \text{particles/mol}\). Energy per photon: \(E = \frac{3.10 \times 10^{2} \mathrm{kJ/mol}}{6.022\times10^{23}\, \text{particles/mol}}\) Now, convert the energy from kJ to J: \(E = \frac{3.10 \times 10^{2} \mathrm{ kJ/mol} \times 1000 \mathrm{\, J/ kJ}}{6.022\times10^{23}\, \text{particles/mol}}\) E = \(5.15 \times 10^{-19} \mathrm{J}\)

Step 2: Calculate the maximum wavelength of light

Now, we will use the equation \(E = \frac{h\cdot c}{\lambda}\) to find the maximum wavelength of light. Here, E is the energy per photon, h is Planck's constant (\(6.63\times10^{-34}\, \text{Js}\)), and c is the speed of light (\(3.00\times10^{8}\, \text{m/s}\)). Wavelength: \(\lambda = \frac{h \cdot c}{E}\) \(\lambda = \frac{6.63\times10^{-34}\, \text{Js} \cdot 3.00\times10^{8}\, \text{m/s}}{5.15 \times 10^{-19} \mathrm{J}}\) \(\lambda = 3.85 \times 10^{-7}\) m To convert the wavelength to nanometers, multiply by \(1\times10^{9}\, \text{nm/m}\): \(\lambda = 3.85 \times 10^{-7} \mathrm{m} \times 1\times10^{9}\, \text{nm/m}\) \(\lambda = 385 \mathrm{nm}\) So, the maximum wavelength of light that can cause the reaction is 385 nm.

Key Concepts

Silver Chloride

Silver chloride, known chemically as AgCl, is a compound widely used in the manufacturing of photogray lenses. When light impacts these lenses, a chemical reaction occurs involving silver chloride. Here, silver chloride is decomposed into silver (Ag) and chlorine atoms. This reaction is fundamental in the lens darkening process. Such lenses contain minute particles of silver chloride embedded within the glass structure. The exposure to light causes the AgCl particles to break down into elemental silver and chloride ions. The elemental silver formed is what contributes to the darkening effect on lenses. This property makes silver chloride essential for developing transitional lenses that adapt to varying light conditions.

Photon Energy

Photon energy is a crucial concept when it comes to understanding the reaction facilitated by silver chloride in photogray lenses. Photons are particles of light that carry energy related to their wavelength.

• Energy Per Photon: The amount of energy a single photon holds can be calculated and is a significant factor in its ability to drive chemical reactions like the one involving silver chloride.
• Conversion: The exercise outlines that energy needs to be converted from energy per mole, which is given, to energy per photon for precise calculations. Using Avogadro's constant, this conversion allows us to understand how much energy a specific number of photons can supply in a reaction.

Calculating the energy per photon involves not only understanding the basic energy transformation, but also how this energy can influence chemical bonds, leading to specific reactions in substances like silver chloride.

Wavelength Calculation

Wavelength calculation is essential in determining what kind of light can activate reactions in photogray lenses. A maximum wavelength calculation helps us understand what light wavelengths will cause silver chloride to dissociate effectively:

• Formula: We use the formula \(E = \frac{h \cdot c}{\lambda}\) to find the wavelength, where \(E\) stands for energy, \(h\) is Planck’s constant, and \(c\) is the speed of light.
• Calculations: By rearranging the formula to \(\lambda = \frac{h \cdot c}{E}\), we can calculate the wavelength that provides the exact energy needed for the chemical reaction.

This calculated wavelength reveals the capability of certain wavelengths of light to facilitate reactions, such as those needed in photogray lens reactions, allowing the lenses to transition properly in sunlight.

Planck's Constant

Planck’s constant, denoted as \(h\), represents a fundamental role in the relationship between energy and frequency in photons. Its value is approximately \(6.63 \times 10^{-34} \text{Js}\). This constant is pivotal when calculating the energy per photon in various chemical reactions.

• Frequency and Energy: Planck's constant is used in the equation \(E = h \cdot f\) to show that energy is directly proportional to frequency.
• Usage in Calculations: In photogray lens reactions, \(h\) enables us to find the wavelength of light required to supply enough energy for the reaction by relating energy to wavelength through the speed of light \(c\).

Understanding Planck's constant helps explain the quantized nature of energy levels in reactions, which is key to determining how photogray lenses respond to different light conditions.