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Chapter 6: Problem 14

Carbon dioxide in the atmosphere absorbs energy in the $4.0-4.5 \mu \mathrm{m}\( range of the spectrum. (a) Calculate the frequency of the \)4.0 \mu \mathrm{m}\( radiation. \)(\mathbf{b})$ In what region of the electromagnetic spectrum does this radiation occur?

Short Answer

Expert verified
The frequency of the 4.0 μm radiation is \(7.5 \times 10^{13} Hz\), and this radiation falls within the infrared region of the electromagnetic spectrum.

Step by step solution

01

Convert Wavelength to Meters

To use the formula to calculate frequency, we first need to convert the given wavelength from micrometers to meters. There are 10^6 micrometers in 1 meter, so we can convert 4.0 μm to meters: \(4.0\, \mu m = 4.0 \times 10^{-6}\, m\)
02

Calculate Frequency

Now we can use the formula to calculate the frequency of the 4.0 μm radiation. The speed of light is approximately 3.00 x 10^8 m/s. Plug in the values into the formula: Frequency (ν) = Speed of light (c) / Wavelength (λ) \(ν = \frac{3.00 \times 10^8\, m/s}{4.0 \times 10^{-6}\, m}\)
03

Simplify the Expression

Now, simplify the expression: \(ν = \frac{3.00}{4.0} \times \frac{10^8}{10^{-6}}\) \(ν = 0.75 \times 10^{14}\, Hz\) The frequency of the 4.0 μm radiation is 7.5 x 10^13 Hz.
04

Determine the Electromagnetic Spectrum Region

To determine the region of the electromagnetic spectrum, compare the given wavelength (4.0 μm) to the known ranges of the spectrum. The electromagnetic spectrum can be divided as follows: - Radio waves: > 1 mm - Microwaves: 1 mm - 700 nm - Infrared: 700 nm - 400 nm - Visible light: 400 nm - 10 nm - Ultraviolet: 10 nm - 0.01 nm - X-rays: 0.01 nm - 1 pm - Gamma rays: < 1 pm As we can see, 4.0 μm falls within the infrared part of the spectrum. So the 4.0 μm radiation is in the infrared region of the electromagnetic spectrum.

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