Ch.7 - Quantum-Mechanical Model of the Atom

Problem 73

Chapter 7, Problem 73

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

Identify the energy required to break a carbon-carbon bond, which is given as 348 kJ/mol.

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.

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.

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.

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.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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Frequency-Wavelength Relationship

Related Practice

Textbook Question

Calculate the frequency of the light emitted when an electron in a hydrogen atom makes each transition: a. n = 4 → n = 3 b. n = 5 → n = 1 c. n = 5 → n = 4 d. n = 6 → n = 5

An electron in the n = 7 level of the hydrogen atom relaxes to a lower-energy level, emitting light of 397 nm. What is the value of n for the level to which the electron relaxed?

An electron in a hydrogen atom relaxes to the n = 4 level, emitting light of 114 THz. What is the value of n for the level in which the electron originated?

The human eye contains a molecule called 11-cis-retinal that changes shape when struck with light of sufficient energy. The change in shape triggers a series of events that results in an electrical signal being sent to the brain that results in vision. The minimum energy required to change the conformation of 11-cis-retinal within the eye is about 164 kJ/mol. Calculate the longest wavelength visible to the human eye.

An argon ion laser puts out 5.0 W of continuous power at a wavelength of 532 nm. The diameter of the laser beam is 5.5 mm. If the laser is pointed toward a pinhole with a diameter of 1.2 mm, how many photons travel through the pinhole per second? Assume that the light intensity is equally distributed throughout the entire cross-sectional area of the beam. (1 W = 1 J/s)

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Open Question

A green leaf has a surface area of 2.50 cm². If solar radiation is 1000 W/m², how many photons strike the leaf every second? Assume three significant figures and an average wavelength of 504 nm for solar radiation.