Problem 14 - Chemistry: The Central Science in SI Units, Global Edition

Question

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?

Step-by-Step Solution

Answer

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 1: 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: \(4.0\, \mu m = 4.0 \times 10^{-6}\, m\)

Step 2: 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) \(ν = \frac{3.00 \times 10^8\, m/s}{4.0 \times 10^{-6}\, m}\)

Step 3: 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.

Step 4: 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.

Key Concepts

Wavelength

Wavelength is a key property of electromagnetic waves, describing the distance between one peak of a wave to the next. In the electromagnetic spectrum, wavelengths vary greatly, from long radio waves to short gamma rays. Understanding wavelength is crucial when analyzing the properties of light and other forms of electromagnetic radiation. Typically, we measure wavelength in meters (m), but smaller units like nanometers (nm) and micrometers (μm) are often used for convenience in specific contexts.

When dealing with conversions, it's helpful to remember:

• 1 meter = 1,000,000 micrometers (μm)
• 1 meter = 1,000,000,000 nanometers (nm)

Converting units is often the first step when solving wavelength-related problems, as it simplifies calculations and ensures consistency in our equations.

Frequency Calculation

Frequency refers to how often the crest of a wave passes a fixed point in one second. It's measured in Hertz (Hz) and is an essential concept for understanding the behavior of waves in the electromagnetic spectrum. The relationship between frequency and wavelength is inverse; as one increases, the other decreases.

This relationship is expressed mathematically by the formula:\[u = \frac{c}{\lambda}\]where:

• \( u \) is the frequency
• \( c \) is the speed of light (approximately \( 3.00 \times 10^8 \ \text{m/s}\))
• \( \lambda \) is the wavelength in meters

Using this formula allows you to calculate the frequency if the wavelength is known, and vice versa, highlighting a fundamental property of waves.

Infrared Radiation

Infrared radiation is a type of electromagnetic radiation that lies between visible light and microwaves in the electromagnetic spectrum. It is not visible to the naked eye but can be felt as heat. The wavelength range of infrared radiation is typically from about 700 nanometers (nm) to 1 millimeter (mm).

Infrared radiation plays a significant role in various applications:

• Remote controls, using infrared light to send signals to devices.
• Infrared cameras, which detect heat emitted by objects.
• For medical and military uses, like in thermal imaging.

Understanding infrared radiation is essential, as it has vast implications and uses across different industries, from healthcare to consumer electronics.

Micrometers to Meters Conversion

Converting from micrometers to meters is a straightforward but significant process when working with electromagnetic waves. It involves changing the measurement unit to fit standard formulas and understandings in physics.

The conversion is simple:

• There are 1,000,000 micrometers in a single meter
• To convert micrometers to meters, multiply the micrometer value by \( 10^{-6} \).

For example, converting 4.0 micrometers (μm) to meters results in:\[ 4.0 \, \mu\text{m} = 4.0 \times 10^{-6} \text{ m} \]Such conversions help maintain consistency in scientific calculations, ensuring accurate results. This is especially important when calculating properties like the frequency of electromagnetic radiation, which requires measurement consistency for precise computation.