Problem 14 - Chemistry The Central Science

Question

(a) What is the relationship between the wavelength and the frequency of radiant energy? (b) Ozone in the upper atmosphere absorbs energy in the \(210-230-\mathrm{nm}\) range of the spectrum. In what region of the electromagnetic spectrum does this radiation occur?

Step-by-Step Solution

Answer

The relationship between the wavelength (\(\lambda\)) and frequency (\(\nu\)) of radiant energy is given by the equation \(\lambda = \frac{c}{\nu}\), where \(c\) is the speed of light. For the wavelength range 210-230 nm, the radiation occurs in the Ultraviolet (UV) Rays region of the electromagnetic spectrum, as it falls within the range of 10 nm to 400 nm.

Step 1: Relationship between wavelength and frequency of radiant energy

The relationship between wavelength (λ) and frequency (v) of radiant energy can be described through the equation: \[c = \lambda \nu\] Here, \(c\) represents the speed of light (\(3.0 \times 10^8 \mathrm{m/s}\)), \(\lambda\) is the wavelength of the radiant energy, and \(\nu\) is the frequency of the radiant energy.

Step 2: Answering part (a) of the exercise

Based on the equation above, the relationship between wavelength and frequency of radiant energy is: \[\lambda = \frac{c}{\nu}\] where \(\lambda\) is the wavelength, \(c\) is the speed of light, and \(\nu\) is the frequency.

Step 3: Identifying the region in the electromagnetic spectrum

The electromagnetic spectrum is divided into several regions based on the wavelength range. We need to find the region in which radiation with a wavelength of 210-230 nm occurs. Here are the ranges for some common regions in the electromagnetic spectrum: 1. Gamma Rays: less than 0.01 nm 2. X-Rays: 0.01 nm to 10 nm 3. Ultraviolet (UV) Rays: 10 nm to 400 nm 4. Visible Light: 400 nm to 700 nm 5. Infrared (IR) Rays: 700 nm to 1 mm 6. Microwaves: 1 mm to 0.3 m 7. Radio Waves: greater than 0.3 m

Step 4: Answering part (b) of the exercise

Based on the given range 210-230 nm, we can identify the region in the electromagnetic spectrum: Since 210-230 nm falls within the range of Ultraviolet (UV) Rays (10 nm to 400 nm), the radiation occurs in the Ultraviolet region of the electromagnetic spectrum.

Key Concepts

Wavelength and Frequency Relationship

Understanding the relationship between wavelength and frequency of radiant energy is foundational in grasping electrodynamics. Essentially, these two properties are inversely proportional to each other. This means that as the wavelength of a wave increases, its frequency decreases and vice versa.

The equation that beautifully encapsulates this relationship is: c = \(\lambdau\)Where:\

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• \(c\) represents the speed of light, a constant valued at approximately \(3.0 \times 10^8 \mathrm{m/s}\)
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• \(\lambda\) symbolizes the wavelength of the radiant energy
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• \(u\) denotes the frequency at which the wave oscillates
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If you're tackling a problem and are given either wavelength or frequency, you can use this equation to find the other. An important takeaway is the fact that all electromagnetic waves travel at the speed of light in a vacuum, allowing this equation to hold true across the electromagnetic spectrum.

Speed of Light

The speed of light, denoted as \(c\), is a fundamental constant of nature. It's the speed at which all electromagnetic radiation travels in a vacuum and is valued at roughly \(3.0 \times 10^8 \mathrm{m/s}\). This speed is significant because it acts as the cosmic speed limit; nothing can travel faster than light in a vacuum.

The constancy of the speed of light is crucial in various laws of physics, including Einstein's theory of relativity. It underlies the calculations and understanding of phenomena across the entire electromagnetic spectrum, from radio waves to gamma rays. It's also the 'c' we use in the wavelength-frequency relationship, revealing how interconnected the properties of the electromagnetic spectrum truly are.

Ultraviolet Radiation

Ultraviolet (UV) radiation is a type of electromagnetic radiation with wavelengths shorter than visible light but longer than X-rays, specifically in the range of 10 nm to 400 nm. It is known for its higher energy levels compared to visible light, which accounts for both its usefulness and its danger.

When discussing UV radiation, we often refer to its ability to affect chemical reactions and biological processes. For example, it is instrumental in the production of vitamin D in humans but also can cause skin damage and increase the risk of cancer with excessive exposure. In the exercise, the absorption of energy by ozone in the range of 210-230 nm would place this radiation within the UV spectrum. This is a crucial aspect to consider when discussing anything from environmental science to health implications associated with exposure to UV light.