Chapter 7: Problem 146

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?

Short Answer

Expert verified

a. The wavelength of the radiation is \(5.0 \times 10^{-6} m\). b. The radiation belongs to the infrared region of the electromagnetic spectrum. c. The energy of this radiation is \(3.98 \times 10^{-20} J\) per photon and \(2.40 \times 10^{4} J/mol\) per mole of photons. d. The radiation absorbed with a frequency of \(5.4 \times 10^{13} s^{-1}\) is less energetic than the radiation with a frequency of \(6.0 \times 10^{13} s^{-1}\).

Step by step solution

a. Wavelength calculation

To find the wavelength, use the formula \(c = \lambda \nu\). Solve for \(\lambda\): \(\lambda = \frac{c}{\nu}\) Plug in the given values: \(\lambda = \frac{3.0 \times 10^8 m/s}{6.0 \times 10^{13} s^{-1}}\) Calculate the result: \(\lambda = 5.0 \times 10^{-6} m\)

b. Region of the electromagnetic spectrum

Since the wavelength is \(5.0 \times 10^{-6} m\), the radiation belongs to the infrared region of the electromagnetic spectrum.

c. Energy of radiation calculation

To find the energy of the radiation per photon, use the formula \(E = h\nu\). Plug in the given values: \(E = (6.626 \times 10^{-34} Js)(6.0 \times 10^{13} s^{-1})\) Calculate the result: \(E = 3.98 \times 10^{-20} J\) Now, to find the energy per mole of photons, use the formula \(E_{mole} = N_A * E\). Plug in the given values: \(E_{mole} = (6.022 \times 10^{23} mol^{-1})(3.98 \times 10^{-20} J)\) Calculate the result: \(E_{mole} = 2.40 \times 10^{4} J/mol\)

d. Comparing the energy of the different radiation

The exercise asks if a different molecule with a radiation frequency of \(5.4 \times 10^{13} s^{-1}\) is more or less energetic. To find out, we must compare the energy of the first molecule to the energy of this new molecule. First, we will calculate the energy of this new molecule. \(E_{new} = (6.626 \times 10^{-34} Js)(5.4 \times 10^{13} s^{-1})\) Calculate the result: \(E_{new} = 3.58 \times 10^{-20} J\) Now, compare the energy of the first molecule and the new molecule: Since \(3.58 \times 10^{-20} J < 3.98 \times 10^{-20} J\), the radiation absorbed by the new molecule is less energetic than the first molecule.

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Short Answer

Step by step solution

a. Wavelength calculation

b. Region of the electromagnetic spectrum

c. Energy of radiation calculation

d. Comparing the energy of the different radiation

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