Problem 14 - Chemistry The Central Science In Si Units

Question

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

Step-by-Step Solution

Answer

(a) The frequency is approximately \(7.5 \times 10^{13} \) Hz. (b) This radiation occurs in the infrared region.

Step 1: Understand the Problem

We have been given the wavelength of radiation absorbed by carbon dioxide, which is 4.0 micrometers, and we need to find its frequency. Frequency can be found using the equation \( f = \frac{c}{\lambda} \), where \( c \) is the speed of light and \( \lambda \) is the wavelength.

Step 2: Convert Wavelength from Micrometers to Meters

First, convert the wavelength from micrometers to meters for correct calculations. The conversion is \( 1 \mu m = 10^{-6} m \). So, \( 4.0 \mu m = 4.0 \times 10^{-6} m \).

Step 3: Apply the Speed of Light Equation

Now, use the equation \( f = \frac{c}{\lambda} \) to find the frequency, where \( c = 3.00 \times 10^8 \, m/s \) is the speed of light. Substitute the values: \( f = \frac{3.00 \times 10^8}{4.0 \times 10^{-6}} \).

Step 4: Calculate the Frequency

Perform the calculation: \( f = \frac{3.00 \times 10^8}{4.0 \times 10^{-6}} = 7.5 \times 10^{13} \) Hz. Thus, the frequency of the \(4.0 \mu m \) radiation is \(7.5 \times 10^{13} \) Hz.

Step 5: Determine the Region of the Spectrum

Radiation with wavelengths in the range of approximately 1 micrometer to 100 micrometers falls in the infrared region of the electromagnetic spectrum.

Key Concepts

Wavelength Conversion

To calculate the frequency of electromagnetic radiation, one often needs to convert the wavelength into compatible units.
This conversion process involves translating the wavelength from micrometers (\(\mu m\)) to meters (m) since most scientific equations, including the speed of light equation, utilize meters.
Here’s a step-by-step guide to understand this conversion easily.Every micrometer is equivalent to\(10^{-6}\ m\). Therefore, converting the given wavelength involves multiplying the numerical value of micrometers by\(10^{-6}\). So, for our original problem where the wavelength is\(4.0\ \mu m\), it converts into:

• \(4.0 \times 10^{-6}\ m\)

This conversion makes the wavelength compatible with the speed of light equation \(f = \frac{c}{\lambda}\), facilitating easy calculations.

Frequency Calculation

After converting the wavelength into the requisite units, you can determine the frequency of the radiation.
Frequency, denoted as \(f\), represents how many wavelengths pass a point in a second.
It is essential in understanding how radiation interacts with matter, such as atmospheric gases.The equation to find frequency is: \(f = \frac{c}{\lambda}\), where \(c\) denotes the speed of light \( (3.00 \times 10^8\ m/s)\). To find the frequency of the given radiation:

• Substitute the speed of light \((3.00 \times 10^8\ m/s)\) and the converted wavelength \((4.0 \times 10^{-6}\ m)\) into the equation.
• The equation thus becomes: \[f = \frac{3.00 \times 10^8}{4.0 \times 10^{-6}}\]
• Upon solving, this yields: \(f = 7.5 \times 10^{13}\ Hz\)

Frequency is a vital characteristic in identifying the part of the electromagnetic spectrum to which the radiation belongs.

Infrared Radiation

Infrared radiation is part of the electromagnetic spectrum closely associated with heat.
It lies beyond the visible spectrum and has wavelengths typically ranging from around 1 micrometer (\(\mu m\)) to 100 micrometers.In our context, the calculated frequency (\(7.5 \times 10^{13}\ Hz\)) falls within this extended range.
Infrared radiation is vital in environmental studies since gases like carbon dioxide absorb infrared wavelengths.
This absorption affects global temperature and climate patterns.IR radiation is categorized into three segments:

• Near-infrared (0.7 to 1.4 \(\mu m\))
• Mid-infrared (1.4 to 3 \(\mu m\))
• Far-infrared (3 to 1000 \(\mu m\))

The wavelengths we studied fall in the far-infrared segment.
Understanding these nuances assists scientists in monitoring and predicting global warming trends.