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

Verified step by step guidance

1. Identify the energy required to break a carbon-carbon bond, which is given as 348 kJ/mol.
2. Convert the energy from kJ/mol to J/photon. Use the conversion factor: 1 kJ = 1000 J and Avogadro's number (6.022 x 10^23 mol^-1) to find the energy per photon.
3. Use the equation E = h\nu, where E is the energy per photon, h is Planck's constant (6.626 x 10^-34 J·s), and \nu is the frequency of the radiation. Rearrange the equation to solve for \nu: \nu = E/h.
4. Use the relationship between frequency and wavelength: c = \lambda\nu, where c is the speed of light (3.00 x 10^8 m/s) and \lambda is the wavelength. Rearrange to solve for \lambda: \lambda = c/\nu.
5. Calculate the longest wavelength \lambda using the values obtained from the previous steps. This wavelength corresponds to the minimum energy required to break the carbon-carbon bond.

Key Concepts

Energy and Wavelength Relationship

The energy of electromagnetic radiation is inversely related to its wavelength, described by the equation E = hc/λ, where E is energy, h is Planck's constant, c is the speed of light, and λ is the wavelength. This means that shorter wavelengths correspond to higher energy, which is crucial for understanding how radiation can break chemical bonds.

Bond Energy

Bond energy is the amount of energy required to break a bond between two atoms. For carbon–carbon bonds, this energy is approximately 348 kJ/mol. Understanding bond energy is essential for determining the minimum energy required for radiation to effectively break these bonds.

Calculating Wavelength from Energy

To find the longest wavelength of radiation that can break a carbon–carbon bond, one can rearrange the energy-wavelength relationship equation to λ = hc/E. By substituting the bond energy into this equation, one can calculate the corresponding wavelength, which indicates the threshold for bond-breaking radiation.