Problem 17 - Chemistry: The Central Science in SI Units, Global Edition

Question

The dissociation energy of a carbon-iodine bond is typically about \(240 \mathrm{~kJ} / \mathrm{mol} .(\mathbf{a})\) What is the maximum wavelength of photons that can cause \(\mathrm{C}-\mathrm{I}\) bond dissociation? (b) Which kind of electromagnetic radiation-ultraviolet, visible, or infrared- does the wavelength you calculated in part (a) correspond to?

Step-by-Step Solution

Answer

The maximum wavelength of photons that can cause C-I bond dissociation is \(5.0 \times 10^{-7} \mathrm{m}\) or 500 nm, which corresponds to visible electromagnetic radiation.

Step 1: Recall the Planck's equation for energy and wavelength

First, let's recall the Planck's equation that relates the energy (E) of a photon with its wavelength (\(\lambda\)): \(E = \frac{hc}{\lambda}\) where h is the Planck's constant (h = \(6.626 \times 10^{-34}\) J∙s) and c is the speed of light (c = \(3.0 \times 10^8\) m/s).

Step 2: Calculate the energy in joules per molecule

The dissociation energy is given in kJ/mol. We need to convert this value to J/molecule. The Avogadro's number helps us to do this conversion: \(\textrm{Energy per molecule (J/molecule)} = \frac{\textrm{Energy per mole (J/mol)}}{\textrm{Avogadro's number (molecule/mol)}}\) The dissociation energy of C-I bond is 240 kJ/mol, first convert it to joules: \(240 \mathrm{\: kJ/mol} \times \frac{1000\: \mathrm{J}}{1 \:\mathrm{kJ}} = 2.4 \times 10^5 \mathrm{\:J/mol}\) Now, use the Avogadro's number (6.022 × 10²³ molecules/mol) to find the energy per molecule: \(E = \frac{2.4 \times 10^5 \:\mathrm{J/mol}}{6.022 \times 10^{23} \:\mathrm{molecules/mol}} = 3.99 \times 10^{-19} \:\mathrm{J/molecule}\)

Step 3: Calculate the maximum wavelength of photons

Using the Planck's equation and the energy calculated in step 2, we can find the maximum wavelength that can break the carbon-iodine bond: \(\lambda = \frac{hc}{E}\) \(\lambda = \frac{(6.626 \times 10^{-34} \:\mathrm{J\: s})(3.0 \times 10^8 \:\mathrm{m/s})}{3.99 \times 10^{-19} \:\mathrm{J/molecule}} = 5.0 \times 10^{-7} \:\mathrm{m}\)

Step 4: Determine the electromagnetic radiation type

Let's find which category of electromagnetic radiation the calculated wavelength falls under: - Ultraviolet radiation: wavelength < 400 nm - Visible radiation: 400 nm < wavelength < 700 nm - Infrared radiation: wavelength > 700 nm With the calculated wavelength of 5.0 x 10⁻⁷ m (500 nm), it corresponds to visible electromagnetic radiation. #Conclusion#: The maximum wavelength of photons that can cause C-I bond dissociation is 5.0 × 10⁻⁷ m or 500 nm, which corresponds to the visible electromagnetic radiation.

Key Concepts

Planck's equation

In the realm of quantum mechanics, Planck's equation is pivotal when it comes to understanding the energy of photons. This famous equation expresses the relationship between the energy of a photon (E) and its wavelength (\lambda): \[E = \frac{hc}{\lambda}\]Here,

• \(h\) is Planck's constant, with a value of \(6.626 \times 10^{-34}\) Joule seconds (J∙s).
• \(c\) is the speed of light, approximately \(3.0 \times 10^8\) meters per second (m/s).

This equation highlights an inverse relationship between energy and wavelength: as the energy of a photon increases, its wavelength decreases, and vice versa. When tasked with photon energy calculation, we often rearrange this formula to find one quantity when given the others. For example, to determine the wavelength of a photon given its energy, you use: \[\lambda = \frac{hc}{E}\]This rearrangement allows us to explore various realms of the electromagnetic spectrum by identifying the wavelengths of light that correspond to specific energy levels.

Electromagnetic radiation

Electromagnetic radiation encompasses a broad range of wavelengths and frequencies in the electromagnetic spectrum. It encompasses various forms of light, including

• Ultraviolet (UV)
• Visible light
• Infrared (IR)

This type of radiation is essential in understanding how energy is transferred in the form of electromagnetic waves. Different kinds of electromagnetic radiation are characterized by their wavelengths:

• Ultraviolet radiation has wavelengths shorter than 400 nm.
• Visible radiation fits between 400 nm and 700 nm.
• Infrared radiation possesses wavelengths longer than 700 nm.

When dealing with exercises related to photon energy calculation, identifying the type of electromagnetic radiation based on its wavelength is crucial. Knowing that a calculated wavelength of 500 nm, like in our example, corresponds to visible light helps contextualize the type of radiation influencing a specific reaction, such as the dissociation of a carbon-iodine bond.

Carbon-iodine bond

The carbon-iodine bond ( C-I ) is a significant chemical link found in organic compounds. A critical aspect of this bond is its dissociation energy, which is the amount of energy required to break the bond - typically around 240 kJ/mol. To comprehend the mechanisms that cause bond dissociation, you must recognize how energy from photons plays a role. When electromagnetic radiation of a specific energy hits a carbon-iodine bond, it can supply enough energy to break the bond, a process termed photodissociation. In our exercise, calculating the maximum wavelength of photons capable of dissociating a C-I bond provides insights into the nature of the electromagnetic radiation involved. The calculated maximum wavelength (500 nm) indicates that visible light possesses the energy necessary to break the C-I bond. This concept is pivotal for applications in photochemistry and the analysis of reaction mechanisms involving electromagnetic radiation.