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?

Short answer

Answer: The spectral range is infrared, the frequency is 3.58 x 10^14 Hz, and the energy is 142.89 kJ/mol.

Step-by-step solution

Identify the spectral range

To identify the spectral range of the absorption, we must check where 837.8 nm falls within the electromagnetic spectrum's wavelength ranges. For visible light, the wavelength ranges are approximately: - Infrared: 700 nm to 1 mm, - Red: 620 nm to 750 nm, - Orange: 590 nm to 620 nm, - Yellow: 570 nm to 590 nm, - Green: 495 nm to 570 nm, - Blue: 450 nm to 495 nm, - Violet: 380 nm to 450 nm. As 837.8 nm is greater than 700 nm, it falls in the infrared range.

Calculate the frequency

To find the frequency of the absorption, we must use the formula: Frequency (v) = speed of light (c) / wavelength (λ) The speed of light (c) is approximately 3.00 x 10^8 meters per second (m/s). First, we need to convert the wavelength from nm to meters: 837.8 nm = 837.8 x 10^-9 meters Now, we can calculate the frequency: v = (3.00 x 10^8 m/s) / (837.8 x 10^-9 m) v ≈ 3.58 x 10^14 Hz

Calculate the energy per mole

To find the energy of one photon, we can use Planck's constant (h = 6.626 x 10^-34 Js) and the previously calculated frequency: Energy (E) = Planck's constant (h) * frequency (v) E = (6.626 x 10^-34 Js) * (3.58 x 10^14 Hz) E ≈ 2.373 x 10^-19 J Now, we can convert the energy to kilojoules per mole. We know that 1 mole of photons contains Avogadro's number (N_A = 6.022 x 10^23) of photons, and 1 Joule = 0.001 Kilojoules: E (kJ/mol) = (2.373 x 10^-19 J/photon) * (6.022 x 10^23 photons/mol) * (0.001 kJ/J) E ≈ 142.89 kJ/mol The final answers are: (a) Infrared (b) 3.58 x 10^14 Hz (c) 142.89 kJ/mol