Problem 3 - Chemistry: Principles and Reactions

Question

A line in the spectrum of neon has a wavelength of \(837.8 \mathrm{~nm}\). (a) In what spectral range does the absorption occur? (b) Calculate the frequency of this absorption. (c) What is the energy in kilojoules per mole?

Step-by-Step Solution

Answer

Answer: The neon absorption line occurs in the infrared spectral range. The frequency of the absorption is 3.58 x 10^14 Hz, and the energy of the absorption is 143.25 kJ/mol.

Step 1: Determine the spectral range of the absorption

To determine the spectral range, we need to examine the wavelength of the absorbed light. The given wavelength is 837.8 nm. The spectral ranges are as follows: - Ultraviolet: wavelengths below 400 nm - Visible: wavelengths between 400 nm and 700 nm - Infrared: wavelengths above 700 nm Since the wavelength of 837.8 nm is above 700 nm, the absorption occurs in the infrared spectral range.

Step 2: Calculate the frequency of the absorption

To calculate the frequency of the absorbed light, we use the formula: Frequency (f) = Speed of light (c) / Wavelength (λ) where c = 3.0 x 10^8 m/s, and the wavelength (λ) = 837.8 nm = 837.8 x 10^-9 m f = (3.0 x 10^8 m/s) / (837.8 x 10^-9 m) f = 3.58 x 10^14 Hz The frequency of this absorption is 3.58 x 10^14 Hz.

Step 3: Calculate the energy of the absorption

To compute the energy per photon, we can use the equation: Energy (E) = Planck's constant (h) x Frequency (f) where Planck's constant (h) = 6.63 x 10^-34 Js E = (6.63 x 10^-34 Js) x (3.58 x 10^14 Hz) E = 2.38 x 10^-19 J

Step 4: Convert energy to kilojoules per mole

To convert the energy per photon to kilojoules per mole, we need to multiply it by Avogadro's number (6.022 x 10^23/mol) and divide it by 1000 to convert Joules to kilojoules: Energy per mole = Energy per photon x Avogadro's number Energy per mole = (2.38 x 10^-19 J) x (6.022 x 10^23/mol) Energy per mole = 143,246 J/mol Now convert this value to kilojoules per mole. Energy per mole = 143,246 J/mol / 1000 Energy per mole = 143.25 kJ/mol To summarize, (a) The absorption occurs in the infrared spectral range. (b) The frequency of the absorption is 3.58 x 10^14 Hz. (c) The energy of the absorption is 143.25 kJ/mol.

Key Concepts

Infrared Spectroscopy

Infrared spectroscopy is a widely used analytical technique that explores the interaction of infrared light with matter. Each molecule has specific bonds and molecular motions that absorb infrared light at characteristic wavelengths, resulting in a unique spectrum. Imagine it as the molecular equivalent of a fingerprint. This method is especially useful for identifying organic compounds because it provides detailed information about the molecular structure.

Infrared light falls just beyond the visible spectrum and has a wavelength range typically from 700 nm to 1 mm. When compounds absorb infrared light, molecular vibrations such as stretching, bending, and twisting are induced. The wavelength of the absorbed light corresponds to the energy required to excite these vibrations. Because the absorption in our example occurs at 837.8 nm, which is above 700 nm, it falls within the infrared range. This is particularly significant in the field of chemistry where the analysis of molecular vibrations can reveal the presence of certain functional groups and even the overall structure of a molecule.

Wavelength and Frequency Calculation

Wavelength and frequency are fundamental properties of waves, including light. The wavelength, represented by the Greek letter lambda (), is the distance between two successive peaks of a wave, and the frequency, denoted as f, is the number of waves that pass a certain point per unit time. In the context of light, wavelength determines the color we see or, in the case of infrared, the spectral signature detected by instruments.

Light, including infrared, travels at a constant speed in a vacuum, known as the speed of light (). The relationship between wavelength, frequency, and the speed of light is given by the equation: f = / . This fundamental relationship allows scientists to calculate the frequency of light if they know its wavelength. In our scenario with neon light of a wavelength of 837.8 nm, we convert the wavelength to meters and apply the equation to find a frequency of 3.58 x 10^14 Hz. Understanding this relationship is crucial for scientists as it connects the observable property (wavelength) with an intrinsic property (frequency) of the electromagnetic wave.

Energy per Mole Calculation

The energy associated with a photon of light is directly proportional to its frequency, a relationship described by the equation (E) = Planck's constant () x Frequency (). Planck's constant () is a fundamental physical constant that describes the quantization of energy. This relationship shows that as the frequency increases, so does the energy.

Chemists often express energy in terms of moles, which is a standard unit for the amount of a substance. Avogadro's number (6.022 x 10^23/mol) links the microscopic scale of atoms and molecules with the macroscopic world by defining the number of particles in one mole of substance. When calculating the energy per mole, we multiply the energy of a single photon by Avogadro's number. For the 837.8 nm infrared light absorbed by neon, this results in an energy of 143.25 kJ/mol. This calculation is indispensable in chemistry because it allows scientists to predict the amount of energy involved in reactions and processes on a molar scale, tying together the properties of individual photons with the practicalities of the laboratory and industry.