Problem 47 - Chemistry Matter and Change

Question

What is the frequency of electromagnetic radiation with a wavelength of \(3.33 \times 10^{-8} \mathrm{m}\) ? What type of electromagnetic radiation is this?

Step-by-Step Solution

Answer

The frequency is \(9.01 \times 10^{15} \text{ Hz}\), and it is ultraviolet radiation.

Step 1: Identify the formula

To find the frequency of electromagnetic radiation, use the formula \( c = \lambda \cdot f \), where \( c \) is the speed of light (approximately \( 3.00 \times 10^8 \text{ m/s} \)), \( \lambda \) is the wavelength, and \( f \) is the frequency. Rearrange the formula to solve for frequency: \( f = \frac{c}{\lambda} \).

Step 2: Substitute given values

Substitute the given wavelength \( \lambda = 3.33 \times 10^{-8} \text{ m} \) into the formula along with the speed of light \( c = 3.00 \times 10^8 \text{ m/s} \). This gives \( f = \frac{3.00 \times 10^8 \text{ m/s}}{3.33 \times 10^{-8} \text{ m}} \).

Step 3: Calculate the frequency

Perform the division to find the frequency: \( f = 9.01 \times 10^{15} \text{ Hz} \).

Step 4: Determine the type of electromagnetic radiation

Consult the electromagnetic spectrum to determine the type of radiation based on frequency. A frequency of \( 9.01 \times 10^{15} \text{ Hz} \) is in the ultraviolet range of the electromagnetic spectrum.

Key Concepts

Wavelength

Wavelength is a fundamental property of waves, including electromagnetic waves. It refers to the distance between consecutive crests or troughs in a wave. Essentially, it's how "long" a wave is from one peak to the next. In terms of electromagnetic waves, such as light, wavelength is often measured in meters.

• Shorter wavelengths are closer together, which means a higher frequency.
• Longer wavelengths are further apart, resulting in lower frequencies.

To visualize wavelength, imagine a wave on the ocean. If the crests (highest points) are close together, this corresponds to a short wavelength. If they are far apart, the wavelength is long. For electromagnetic radiation, measuring wavelength helps identify the type of radiation, as different types have characteristic wavelength ranges. For example, radio waves have longer wavelengths than visible light.

Speed of Light

The speed of light is a constant in physics, often represented by the symbol "c." Light travels at an impressive speed of approximately 3.00 x 10⁸ meters per second in a vacuum. This speed is crucial in calculating the frequency or wavelength of electromagnetic radiation through the formula:
\[c = \lambda \cdot f\]Where:

• \(c\) is the speed of light,
• \(\lambda\) is the wavelength,
• \(f\) is the frequency.

The speed of light ties together the relationship between frequency and wavelength, helping us understand and measure electromagnetic phenomena. It is the universal speed limit; nothing can travel faster than light in a vacuum. That makes it an important tool to cross-verify calculations regarding electromagnetic waves.

Ultraviolet Radiation

Ultraviolet (UV) radiation is a type of electromagnetic radiation with a frequency higher than that of visible light, but lower than X-rays. Its name reflects the fact that its frequency places it beyond the violet end of the visible light spectrum.

• UV radiation is mostly invisible to the human eye, but certain insects, like bees, can see it.
• Sunlight is a natural source of ultraviolet radiation.
• It is well-known for causing sunburns, but it also plays a role in producing vitamin D in the skin.

Understanding UV radiation is crucial for both protection against excessive exposure, which can lead to skin cancer, and harnessing its benefits, such as sterilization and medical applications.

Electromagnetic Spectrum

The electromagnetic spectrum is the range of all types of electromagnetic radiation. Waves are categorized based on their wavelengths and frequencies. The spectrum includes everything from very long waves, like those used for radio transmissions, to very short waves, like gamma rays.
Key regions within the electromagnetic spectrum include:

• Radio Waves: Long wavelengths, low frequencies. Used in broadcasting.
• Microwaves: Shorter than radio waves, used in microwave ovens.
• Infrared: Felt as heat, with wavelengths longer than visible light.
• Visible Light: The small part of the spectrum visible to the human eye.
• Ultraviolet: Shorter wavelengths than visible light.
• X-rays: Even shorter wavelengths, used in medical imaging.
• Gamma Rays: Shortest wavelengths, highest frequencies, emitted by nuclear reactions.

Exploring the electromagnetic spectrum allows scientists to understand the properties and uses of different types of radiation. By identifying where a given frequency falls on the spectrum, we can describe what kind of electromagnetic radiation it is, as exemplified in determining that a specific frequency is ultraviolet radiation.