Problem 27 - Chemistry: The Central Science in SI Units, Global Edition

Question

The average bond enthalpies of the \(\mathrm{C}-\mathrm{C}\) and \(\mathrm{C}-\mathrm{H}\) bonds are \(348 \mathrm{~kJ} / \mathrm{mol}\) and \(413 \mathrm{~kJ} / \mathrm{mol}\), respectively. (a) What is the maximum wavelength that a photon can possess and still have sufficient energy to break the \(\mathrm{C}-\mathrm{H}\) and \(\mathrm{C}-\mathrm{C}\) bonds, respectively? (b) Given the fact that \(\mathrm{O}_{2}, \mathrm{~N}_{2},\) and \(\mathrm{O}\) in the upper atmosphere absorb most of the light with wavelengths shorter than \(240 \mathrm{nm}\), would you expect the photodissociation of \(\mathrm{C}-\mathrm{C}\) and \(\mathrm{C}-\mathrm{H}\) bonds to be significant in the lower atmosphere?

Step-by-Step Solution

Answer

The maximum wavelength for C-C and C-H bonds are 343 nm and 289 nm, respectively. Since these values are greater than 240 nm, photodissociation of C-C and C-H bonds can be significant in the lower atmosphere.

Step 1: Planck's Equation

Planck's equation is given by: \[E = h \nu\] Where E is the energy of the photon, h is Planck's constant (6.626 x 10^(-34) J.s), and ν is the frequency of the photon. The frequency of a photon is related to its wavelength (λ) by the speed of light (c): \[\nu = \frac{c}{\lambda}\] Where c = 3 x 10^8 m/s. Substituting this equation into Planck's equation, we get: \[E = \frac{hc}{\lambda}\]

Step 2: Calculate wavelength for C-C bond

Given the bond enthalpy of the C-C bond is 348 kJ/mol, we first need to convert this to energy per photon: \[\frac{348\,\text{kJ}}{\text{mol}} \times \frac{1000\,\text{J}}{1\,\text{kJ}} \times \frac{1 \text{mol}}{6.022 \times 10^{23}\,\text{atoms}} = 5.78 \times 10^{-19}\,\text{J}\] Now we can use Planck's equation to find the maximum wavelength: \[\lambda_{\mathrm{C-C}} = \frac{hc}{E_{\mathrm{C-C}}} = \frac{(6.626 \times 10^{-34}\,\text{J.s})(3 \times 10^8\,\text{m/s})}{5.78 \times 10^{-19}\,\text{J}} = 343\,\text{nm}\]

Step 3: Calculate wavelength for C-H bond

Similarly, do the same for the C-H bond with a bond enthalpy of 413 kJ/mol: \[\frac{413\,\text{kJ}}{\text{mol}} \times \frac{1000\,\text{J}}{1\,\text{kJ}} \times \frac{1 \text{mol}}{6.022 \times 10^{23}\,\text{atoms}} = 6.86 \times 10^{-19}\,\text{J}\] And find the maximum wavelength: \[\lambda_{\mathrm{C-H}} = \frac{hc}{E_{\mathrm{C-H}}} = \frac{(6.626 \times 10^{-34}\,\text{J.s})(3 \times 10^8\,\text{m/s})}{6.86 \times 10^{-19}\,\text{J}} = 289\,\text{nm}\]

Step 4: Compare the wavelengths to atmospheric absorption

We are given that wavelengths shorter than 240 nm are absorbed in the upper atmosphere. The calculated wavelengths for the C-C and C-H bonds were 343 nm and 289 nm, respectively. Both of these values are greater than 240 nm.

Step 5: Determine the significance of photodissociation

Since the maximum wavelengths required to break C-C and C-H bonds are greater than 240 nm, the photons with these wavelengths can pass through the upper atmosphere without being absorbed. This means that photodissociation of C-C and C-H bonds can be significant in the lower atmosphere.

Key Concepts

Photon Energy

Photon energy is a measure of the energy carried by a single photon. This energy is crucial in many physical processes, including chemical reactions that involve breaking bonds.
The energy of a photon (\(E\)) is directly proportional to its frequency (\(u\)) and is determined using Planck's equation:

• \[E = h u\]

where \(h = 6.626 \times 10^{-34} \text{ J.s}\) is Planck's constant. This equation shows that higher frequency photons have more energy.
This is a key principle for understanding how photons can break chemical bonds during processes like photodissociation.

Wavelength Calculation

Calculating the wavelength of a photon is an essential step in understanding its properties, particularly its energy. The speed of light (\(c\)) connects the frequency (\(u\)) of a photon to its wavelength (\(\lambda\)). The relationship is given by:

• \[u = \frac{c}{\lambda}\]
• where \(c = 3 \times 10^8 \text{ m/s}\) is a constant representing the speed of light in a vacuum.

To find the wavelength, we can rearrange Planck's equation:

• \[E = \frac{hc}{\lambda}\]

From this, the wavelength can be calculated as:

• \[\lambda = \frac{hc}{E}\]

Knowing the wavelength helps in predicting whether certain photons can cause photodissociation by matching their energy to that needed for breaking chemical bonds.

Photodissociation

Photodissociation is the process where a chemical bond is broken by the absorption of a photon of light. Photons with sufficient energy can overcome the bond energy holding atoms together.
For the dissociation of \(\mathrm{C}-\mathrm{H}\) or \(\mathrm{C}-\mathrm{C}\) bonds, a photon needs to provide energy equal to or greater than the bond enthalpy.
The concept is crucial in atmospheric chemistry and industrial applications, as it describes how molecules can be broken down by light. In terms of atmospheric exposure, knowing the exact photon energy and wavelength needed allows us to predict and control such processes effectively.

Atmospheric Absorption

Atmospheric absorption refers to the ability of gases in the atmosphere to absorb specific wavelengths of light. Gases like \(\mathrm{O}_{2}\),\(\mathrm{N}_{2}\), and \(\mathrm{O}\) absorb most light below a 240 nm wavelength. This is essential for protecting life on Earth by filtering harmful ultraviolet (UV) radiation.
Wavelengths higher than 240 nm, like those required to break \(\mathrm{C}-\mathrm{H}\) and \(\mathrm{C}-\mathrm{C}\) bonds (343 nm and 289 nm, respectively), are not absorbed as efficiently, allowing them to reach lower atmospheric layers. Understanding atmospheric absorption aids in predicting the impact of photodissociation on the environment.

Planck's Equation

Planck's equation forms the foundation for understanding photon energy by linking it with frequency and wavelength. It is expressed as:

• \[E = hu\]

By knowing the speed of light and substituting \(u = \frac{c}{\lambda}\), Planck's equation can be rewritten to find photon energy using wavelength:

• \[E = \frac{hc}{\lambda}\]

This is pivotal for calculations involving energy levels and quantum mechanics, and it helps predict the behavior of photons in various chemical processes. Understanding Planck's equation allows for precise control and prediction of light-induced reactions, crucial in fields like photochemistry and atmospheric science.