A carbon-oxygen double bond in a certain organic molecule absorbs radiation that has a frequency of \(6.0 \times 10^{13} \mathrm{~s}^{-1}\). a. What is the wavelength of this radiation? b. To what region of the spectrum does this radiation belong? c. What is the energy of this radiation per photon? per mole of photons? d. A carbon-oxygen bond in a different molecule absorbs radiation with frequency equal to \(5.4 \times 10^{13} \mathrm{~s}^{-1}\). Is this radiation more or less energetic?

Imagining light as a wave with peaks and valleys can be helpful when exploring the concept of wavelength. The wavelength (symbol \( \lambda \)) of light is the physical distance from one peak of the wave to the next, and it's measured in meters. In infrared radiation in chemistry, calculating the wavelength involves the speed of light (\( c \) – approximately \( 3.0 \times 10^8 \ \text{m/s} \)) and the frequency of the radiation (\( u \) – measured in cycles per second or Hertz). The simple relationship \( \lambda = \frac{c}{u} \) allows chemists to find the wavelength of light absorbed by a molecule.
To clarify, if the frequency of radiation is \( 6.0 \times 10^{13} \ \text{s}^{-1} \), then the wavelength is \( \lambda = \frac{3.0 \times 10^8 \text{m/s}}{6.0 \times 10^{13} \text{s}^{-1}} \), which calculates to \( 5.0 \times 10^{-6} \text{m} \). Remember, this wavelength tells us the type of light and gives insight into the energy levels involved in chemical reactions.

The electromagnetic spectrum is like a vast ocean of different types of radiation waves, each with its own energy level. In infrared radiation in chemistry, it's essential to know which part of the spectrum the observed radiation falls into. Infrared radiation lies between visible light and microwave regions, typically in the range of 700 nm to 1 mm in wavelength.
When a molecule absorbs radiation with a wavelength calculated as \( 5.0 \times 10^{-6} \text{m} \), we can pinpoint its location in the infrared region of the spectrum. By identifying this, chemists can deduce the type of molecular vibrations and bonds that may be responsible for such absorption, leading to deeper understanding of molecular structure and behavior.

Moving beyond the visible, each photon of light carries a quantum of energy, and calculating this energy can reveal much about the chemical interactions taking place. The energy of a photon (\( E \)) is given by the equation \( E = h \times u \), where \( h \) is Planck's constant (approximately \( 6.626 \times 10^{-34} \ \text{Js} \)), and \( u \) is the frequency of the photon. Therefore, if a photon's frequency is \( 6.0 \times 10^{13} \ \text{s}^{-1} \), its energy would be calculated as:\( E = (6.626 \times 10^{-34} \text{Js}) \times (6.0 \times 10^{13} \text{s}^{-1}) \), resulting in \( E = 3.976 \times 10^{-20} \text{J} \) per photon.
Furthermore, to understand the energy involved in a mole of these photons—Avogadro's number (\( N_A \)) comes into play. By multiplying the single photon's energy by \( N_A \), chemists can determine the total energy per mole, which is significant when considering reaction in bulk matter.

When comparing radiation from different sources or processes, energy content offers a clear way to contrast their respective influences on chemical bonds. More energetic radiation will typically induce greater changes at the molecular level. Using the photon energy equation already discussed (\( E = h \times u \)), we determine which radiation source is more potent by comparing their photon energies. For instance, radiation with a frequency of \( 6.0 \times 10^{13} \text{s}^{-1} \) would have more energy per photon than radiation with a frequency of \( 5.4 \times 10^{13} \text{s}^{-1} \). This difference implies that the bond in the molecule absorbing the higher frequency radiation is undergoing greater energy changes, potentially leading to significant chemical reactions.