Problem 73 - Chemistry A Molecular Approach

Question

Ultraviolet radiation and radlation of shorter wavelengths can damage biological molecules because these kinds of radlation carry enough energy to break bonds within the molecules. A typical carbon-carbon bond fequires 348 kl / mol to beeak. What is the longest wavelength of radiation with enough energy to break carbon-carbon bonds?

Step-by-Step Solution

Answer

Calculate the required wavelength using the equation \(\lambda = \frac{(6.626 \times 10^{-34} J \cdot s) \times (3.00 \times 10^8 m/s)}{348,000 J/mol}\), then divide by Avogadro's number to get the wavelength per photon. The longest wavelength of radiation that has enough energy to break carbon-carbon bonds will be the result.

Step 1: Determine the energy required to break the bond

Find the total energy needed to break the carbon-carbon bond. This energy is given as 348 kJ/mol. We need to convert this energy into joules (J), since 1 kilojoule (kJ) is equal to 1000 joules (J).

Step 2: Convert kilojoules to joules

Convert the energy from kilojoules per mole to joules by multiplying by 1000. So, the energy in joules is 348 kJ/mol * 1000 J/kJ = 348,000 J/mol.

Step 3: Use the energy equation of a photon

Relate the energy (E) needed to break the bond to the energy of a photon using the equation E = h * f, or E = (h * c) / \(\lambda\), where h is Planck's constant (6.626 x 10^-34 J·s), c is the speed of light (3.00 x 10^8 m/s), and \(\lambda\) is the wavelength in meters.

Step 4: Calculate the wavelength (lambda)

Solve for the wavelength \(\lambda\) using the equation from Step 3, \(\lambda = \frac{h*c}{E}\). Plug in the values for h, c, and the energy in J (from Step 2) to calculate the wavelength.

Step 5: Solve for the wavelength

Insert the known values into the equation to find the wavelength: \(\lambda = \frac{(6.626 \times 10^{-34} J \cdot s) \times (3.00 \times 10^8 m/s)}{348,000 J/mol}\). Note that the answer will be in meters per mole.

Step 6: Convert the wavelength for a single photon

Since energy given is per mole of photons, and we know that one mole contains Avogadro's number of particles (6.022 x 10^23 photons/mol), divide the wavelength by Avogadro's number to find the wavelength of a single photon.

Key Concepts

Chemical Bond Energy

Understanding the concept of chemical bond energy is essential when considering how molecules interact and react to different forms of radiation. Chemical bond energy refers to the amount of energy required to break a bond between two atoms within a molecule. It's a crucial factor in determining the stability of a compound and how it might change under certain conditions, such as exposure to ultraviolet radiation.

In the context of our problem, knowing that it takes 348 kJ/mol to break a carbon-carbon bond allows us to calculate the minimum energy a photon must carry to induce this bond breakage. This scenario is highly relevant in the study of how radiation can lead to the degradation of biological molecules, a topic of importance in fields like molecular biology and material science.

Photon Energy Equation

To connect the wavelength of radiation to its ability to break chemical bonds, we use the photon energy equation. This fundamental relation in physics is given by the equation E = (h * c) / \(\lambda\), where E is the photon energy, h is Planck's constant, c is the speed of light, and \(\lambda\) is the photon's wavelength.

The equation illustrates that the energy carried by a photon is inversely proportional to its wavelength — shorter wavelengths mean higher energy photons. The concept is pivotal when evaluating the potential of different kinds of radiation to cause molecular changes, such as the breaking of chemical bonds in biological molecules.

Planck's Constant

Planck's constant is a fundamental quantity in quantum mechanics that plays a vital role in the photon energy equation. Its value is approximately 6.626 x 10^-34 J·s (joule seconds), and it represents the proportionality constant between the energy of a photon and its frequency.

This constant is named after Max Planck, the physicist who proposed that energy is quantized and can be released or absorbed in discrete units called 'quanta'. Planck's constant enables us to calculate the energy carried by a photon, which is essential in understanding phenomena such as the photoelectric effect, and also in solving problems like determining the wavelength of radiation needed to break a specific chemical bond.

Avogadro's Number

Avogadro's number, approximately 6.022 x 10^23 particles/mol, is another fundamental constant in chemistry and physics, defining the number of constituent particles, usually atoms or molecules, in one mole of a given substance.

When dealing with photons and bond energy, Avogadro's number enables us to transition from macroscopic amounts of substances (in moles) to the microscopic scale (individual particles or photons). This transition is crucial for converting the energy required to break bonds for a mole of bonds to the energy of a single photon. Therefore, when we measure quantities like the wavelength necessary to break chemical bonds, we often use Avogadro's number to scale our calculations to a per-molecule or per-photon basis.