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: $$ \mathrm{AgCl} \stackrel{h v}{\longrightarrow} \mathrm{Ag}+\mathrm{Cl} $$ The silver metal that is formed causes the lenses to darken. The enthalpy 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?

Short answer

The maximum wavelength of light that can cause the reaction in photogray lenses is approximately \(386 \mathrm{nm}\).

Step-by-step solution

Write down the formula for the energy of a photon

We can use the energy-wavelength relationship, given by the formula: $$E = h \cdot \nu$$ where E is the energy of the photon, h is the Planck constant (\(6.626 \times 10^{-34}\) J·s), and \(\nu\) is the frequency of the light. As we are given the energy, we will rewrite this to make use of the speed of light (c) and wavelength (λ) instead of frequency: $$E = \frac{h \cdot c}{\lambda}$$

Convert the enthalpy change to energy per photon

The enthalpy change given is in \(\mathrm{kJ/mol}\), but we need the energy in \(\mathrm{J}\) for each photon. We will use Avogadro's number (\(N_A = 6.022 \times 10^{23} \mathrm{mol}^{-1}\)) to convert the energy. $$E_{\mathrm{photon}} = \frac{\Delta H}{N_A}$$ where \(\Delta H = 3.10 \times 10^{2} \mathrm{kJ/mol}\) $$E_{\mathrm{photon}} = \frac{3.10 \times 10^2 \times 10^3}{6.022 \times 10^{23}}$$

Calculate the energy per photon

Calculate the energy per photon using the values from the previous step: $$E_{\mathrm{photon}} = \frac{3.10 \times 10^2 \times 10^3}{6.022 \times 10^{23}} = 5.15 \times 10^{-19} \mathrm{J}$$

Solve for the maximum wavelength of light

Use the energy-wavelength formula and energy per photon to calculate the maximum wavelength: $$\lambda = \frac{h \cdot c}{E_{\mathrm{photon}}}$$ Plug in the values for h, c, and \(E_{\mathrm{photon}}\): $$\lambda = \frac{6.626 \times 10^{-34} \cdot 2.998 \times 10^8}{5.15 \times 10^{-19}}$$

Calculate the final result

Calculate the maximum wavelength: $$\lambda = \frac{6.626 \times 10^{-34} \cdot 2.998 \times 10^8}{5.15 \times 10^{-19}} = 3.86 \times 10^{-7} \mathrm{m}$$ Converting to nanometers (nm), we get: $$\lambda = 3.86 \times 10^{-7} \mathrm{m} \times \frac{1}{10^{-9}} = 386 \mathrm{nm}$$ The maximum wavelength of light that can cause the reaction in photogray lenses is approximately \(386 \mathrm{nm}\).

Key concepts

Enthalpy Change

The concept of enthalpy change is crucial in understanding chemical reactions such as the one involving silver chloride (AgCl). Enthalpy change, denoted by \( \Delta H \), represents the heat absorbed or released during a chemical reaction at constant pressure. It gives us insights into the energy requirements or releases associated with the reaction. In this case, the reaction where AgCl is split into silver (Ag) and chlorine (Cl) is endothermic, meaning it absorbs energy.- Given: \( \Delta H = 3.10 \times 10^2 \, \mathrm{kJ/mol} \)When dealing with photonic processes, this energy requirement must be met by the energy supplied by the photons. Hence, understanding and calculating the enthalpy change helps determine the characteristics of the light necessary to trigger the reaction.

Energy of a Photon

The energy of a photon is a key concept in photochemistry and is essential for understanding how light interacts with matter. A photon is a particle representing a quantum of light, carrying energy proportional to its frequency. The energy of a photon can be calculated using the equation:\[ E = h \cdot u \]where:- \( E \) is the energy of the photon,- \( h \) is the Planck constant \( (6.626 \times 10^{-34} \, \mathrm{J \cdot s}) \), and- \( u \) is the frequency of the light.However, since we often work with wavelength rather than frequency, the relationship between energy and wavelength becomes essential, as described in the next section.

Energy-Wavelength Relationship

The energy-wavelength relationship provides an important link between the energy possessed by a photon and its corresponding wavelength. Since energy and wavelength are linked through the speed of light, we can rearrange the photon's energy formula to connect these quantities:\[ E = \frac{h \cdot c}{\lambda} \]where:- \( E \) is the energy of the photon,- \( h \) is the Planck constant,- \( c \) is the speed of light \( (2.998 \times 10^8 \, \mathrm{m/s}) \), and- \( \lambda \) is the wavelength of the light.This formula reveals that the energy of a photon is inversely proportional to its wavelength. Thus, shorter wavelengths (higher frequency light) carry more energy than longer wavelengths. For reactions requiring a certain amount of energy, this relationship allows us to calculate the specific wavelength of light necessary.

Maximum Wavelength Calculation

To determine the maximum wavelength of light that can cause a reaction, we must apply our understanding of enthalpy change and the energy of photons in the context of the energy-wavelength relationship. First, we convert the enthalpy change to energy per photon using Avogadro's number \( N_A \):\[ E_{\mathrm{photon}} = \frac{\Delta H}{N_A} \]For the reaction \( \mathrm{AgCl} \rightarrow \mathrm{Ag} + \mathrm{Cl} \), this converts to:\[ E_{\mathrm{photon}} = \frac{3.10 \times 10^2 \times 10^3}{6.022 \times 10^{23}} = 5.15 \times 10^{-19} \, \mathrm{J} \]Next, we use the energy-wavelength formula to solve for the maximum wavelength \( \lambda \):\[ \lambda = \frac{h \cdot c}{E_{\mathrm{photon}}} \]Plugging in the constants and calculated energy:\[ \lambda = \frac{6.626 \times 10^{-34} \cdot 2.998 \times 10^8}{5.15 \times 10^{-19}} = 3.86 \times 10^{-7} \, \mathrm{m} \]Converting this to nanometers gives us approximately 386 nm. This wavelength represents the threshold above which photons do not possess enough energy to trigger the reaction. Understanding this threshold can be crucial in designing lenses or materials that respond to specific light conditions.