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 of silver chloride to silver metal and chloride is 385 nm.

Step-by-step solution

Write down the given information and the constants needed

We are given the enthalpy change for the reaction: ΔH = \(3.10 \times 10^{2} \mathrm{kJ/mol}\) = \(3.10 \times 10^{5} \mathrm{J/mol}\) (converting from kJ to J). We also need the Planck's constant (h) which is \(6.626 \times 10^{-34} \mathrm{J} \cdot \mathrm{s}\) and the speed of light (c) which is \(3.00 \times 10^8 \mathrm{m/s}\).

Use the relationship between energy and wavelength

We need to find the energy per photon (E) that corresponds to ΔH. To find this, use the equation: \[E = \dfrac{h \cdot c}{\lambda}\] Where E is energy per photon, h is Planck's constant, c is the speed of light and λ is the wavelength of the light. We can rearrange this equation for λ: \[\lambda = \dfrac{h \cdot c}{E}\]

Determine the energy per photon that can initiate the reaction

We know the enthalpy change per mole (ΔH) of the reaction, but we need to find the energy per photon (E). To do this, use Avogadro's number (NA): \[E = \dfrac{\Delta H}{N_{A}}\] Where NA is Avogadro's number, approximately \(6.022 \times 10^{23} \mathrm{mol}^{-1}\). Substitute the values: \[E = \dfrac{3.10 \times 10^{5} \mathrm{J/mol}}{6.022 \times 10^{23} \mathrm{mol}^{-1}} = 5.15 \times 10^{-19} \mathrm{J}\]

Calculate the maximum wavelength of light

Now that we have the energy per photon, we can find the maximum wavelength of light that can initiate the reaction. Substitute the values into the rearranged equation for λ: \[\lambda = \dfrac{6.626 \times 10^{-34} \mathrm{J} \cdot \mathrm{s} \cdot 3.00 \times 10^8 \mathrm{m/s}}{5.15 \times 10^{-19} \mathrm{J}}\] \[\lambda = 3.85 \times 10^{-7} \mathrm{m}\] Since we need the wavelength in nanometers, we can convert it: \[\lambda = 3.85 \times 10^{-7} \mathrm{m} \cdot \dfrac{10^9 \mathrm{nm}}{1 \mathrm{m}} = 385 \mathrm{nm}\] Therefore, the maximum wavelength of light that can cause the reaction of silver chloride to silver metal and chloride is 385 nm.

Key concepts

Enthalpy Change

In photochemistry, the enthalpy change for a reaction is a crucial factor. It represents the energy change when reactants are transformed into products. In our exercise, the enthalpy change is given as \(3.10 \times 10^{2} \ \text{kJ/mol}\). This value tells us how much energy is needed per mole of silver chloride to initiate the reaction. When dealing with these calculations, it's essential to convert the energy into the proper units, which commonly involves converting from \(\text{kJ/mol}\) to \(\text{J/mol}\) since energy for photon calculations is typically in joules. Therefore, multiplying by 1000 gives us the converted enthalpy change as \(3.10 \times 10^{5} \ \text{J/mol}\). Understanding this conversion is vital for further steps in the calculation.

Wavelength Calculation

For photochemical reactions, light wavelength is related to the energy required to cause the reaction. This means calculating the maximum wavelength of light capable of initiating a specific reaction. The relationship between energy (E), Planck's constant (h), the speed of light (c), and wavelength (\lambda) is given by: \[E = \dfrac{h \cdot c}{\lambda}\] The equation shows that as energy increases, the wavelength decreases, meaning more energetic (or shorter wavelength) light is needed to carry out the reaction.We rearrange this equation to find wavelength: \[\lambda = \dfrac{h \cdot c}{E}\] By understanding this equation, we can determine the exact type of light (or its wavelength) needed for specific photochemical processes, confirming its fundamental role in such reactions.

Energy Per Photon

To understand what kind of light can cause a reaction, it's essential to determine the energy per photon. This is the actual amount of energy that each photon of light provides. For the reaction in our exercise, even though we have the enthalpy change per mole, we need to determine the energy available per individual photon to initiate the reaction.Here's how you calculate energy per photon:

• Start with the given enthalpy change (\(\Delta H\)) per mole of the substance.
• Use Avogadro's number (\(6.022 \times 10^{23} \ \text{mol}^{-1}\)) to convert this value to energy per photon.

The formula for this conversion is: \[E = \dfrac{\Delta H}{N_{A}}\] By inserting the given values, you derive how much energy each photon carries, which is crucial for understanding the light's effectiveness in causing a chemical transformation. This concept underpins why not all light can cause every reaction, emphasizing the need for precise energy matches between photons and chemical changes.