Problem 3 - Chemistry: Introducing Inorganic, Organic and Physical Chemistry

Question

What wavelength of radiation would provide sufficient energy to break a C-H bond in methane, where the bond dissociation energy is \(+439 \mathrm{kJ} \mathrm{mol}^{-1} ?(\text { Section } 10.2)\) \\[ \mathrm{CH}_{4} \rightarrow \mathrm{CH}_{3}^{*}+\mathrm{H}^{*} \\]

Step-by-Step Solution

Answer

The wavelength is approximately 272 nm.

Step 1: Understand the Bond Dissociation Energy

The bond dissociation energy, given as 439 kJ/mol, is the energy required to break one mole of C-H bonds in methane. We aim to find the wavelength of radiation needed to supply enough energy to break a single C-H bond.

Step 2: Convert Energy from kJ/mol to J/Photon

First, we convert the bond energy from kilojoules per mole to joules per photon. Since Avogadro's number is approximately \(6.022 \times 10^{23}\), there are \(6.022 \times 10^{23}\) photons in a mole.Thus, the energy per photon is calculated using the conversion:\[439 \, \text{kJ/mol} = 439,000 \, \text{J/mol}\]Energy per photon = \(\frac{439,000 \, \text{J}}{6.022 \times 10^{23} \text{ photons/mol}}\)

Step 3: Calculate the Energy per Photon

Now, compute the energy per photon:\[E = \frac{439,000 \, \text{J}}{6.022 \times 10^{23}} \approx 7.29 \times 10^{-19} \, \text{J/photon}\].This energy is what a single photon must have to break the C-H bond.

Step 4: Relate Energy to Wavelength Using Planck's Equation

Planck's equation \(E = \frac{hc}{\lambda}\) relates photon energy \(E\) to wavelength \(\lambda\) where \(h\) is Planck's constant \((6.626 \times 10^{-34} \, \text{J s})\) and \(c\) is the speed of light \((3.00 \times 10^8 \, \text{m/s})\). Rearrange to find:\[\lambda = \frac{hc}{E}\]\[\lambda = \frac{6.626 \times 10^{-34} \, \mathrm{J \cdot s} \times 3.00 \times 10^{8} \, \mathrm{m/s}}{7.29 \times 10^{-19} \, \mathrm{J}}\].

Step 5: Calculate the Wavelength

Now, execute the calculation:\[\lambda \approx \frac{1.986 \times 10^{-25} \, \text{J m}}{7.29 \times 10^{-19} \, \text{J}} \approx 2.72 \times 10^{-7} \, \text{m}\].Convert the wavelength from meters to nanometers (1 m = 1,000,000,000 nm):\[\lambda \approx 272 \text{ nm}\].

Key Concepts

Bond Dissociation Energy

Bond dissociation energy is a key concept in chemistry related to the strength of chemical bonds. It refers to the amount of energy required to break a specific bond in a molecule, resulting in the separation of atoms without altering other bonds. In the case of methane (14\(\text{CH}_4\)), its C-H bond has a dissociation energy of 439 kJ/mol. This value indicates the energy needed to break one mole of C-H bonds.

When considering bond dissociation energy, it's important to note that this energy is often measured under standard conditions, typically at 298 K and 1 atm pressure. Understanding this concept allows chemists to predict the stability of molecules and their reactivity under various conditions.

In practical terms, bond dissociation energy can be related to everyday processes and reactions, such as combustion or biochemical pathways, influencing how molecules change and interact.

Photon Energy Calculation

To calculate the energy of a single photon, especially in the context of breaking chemical bonds, converting units is crucial. Given that bond dissociation energies are typically provided in kilojoules per mole, we must find the energy per photon for effective calculations. Each mole contains Avogadro's number of entities, approximately \(6.022 \times 10^{23}\).

• First, convert the bond's energy from kilojoules to joules: 439 kJ/mol becomes 439,000 J/mol.
• Next, divide by Avogadro's number to determine the energy per photon, which results in about \(7.29 \times 10^{-19}\) J per photon.

This conversion is pivotal as it translates a macroscopic quantity (moles) into a microscopic one (photons), bridging our understanding of bulk material properties and individual molecular interactions.

Planck's Equation

Planck's equation is integral in understanding how energy relates to electromagnetic radiation, such as light. The formula, \(E = \frac{hc}{\lambda}\), connects a photon's energy \(E\) to its wavelength \(\lambda\). This equation is essential in finding the wavelength of light capable of breaking chemical bonds.

• Here, \(h\) is Planck's constant with a value of \(6.626 \times 10^{-34}\) J3s.
• \(c\) denotes the speed of light, approximately \(3.00 \times 10^8\) m/s.

Rearranging the equation to \(\lambda = \frac{hc}{E}\), we can calculate the wavelength associated with the energy from a photon that is sufficient to dissociate a bond. This direct relationship highlights the dependence of light's properties on its energy content.

Wavelength Calculation

Calculating the wavelength of radiation that can break a chemical bond involves using the energy per photon derived from Planck's equation. By substituting known values, we bridge the gap between energy and the wavelength of light.

• Given \(E = 7.29 \times 10^{-19}\) J/photon, the calculation becomes \(\lambda = \frac{6.626 \times 10^{-34} \, \text{J} \cdot \text{s} \times 3.00 \times 10^8 \, \text{m/s}}{7.29 \times 10^{-19} \, \text{J}}\).
• This math results in about \(2.72 \times 10^{-7}\) metres, or 272 nanometres.

Understanding wavelength is crucial in fields like spectroscopy, where it helps identify substances through their interaction with different wavelengths of light. This calculation exemplifies the practical application of theoretical concepts to real-world scientific problems.