Problem 1 - Chemistry The Central Science In Si Units

Question

The speed of sound in dry air at \(20^{\circ} \mathrm{C}\) is \(343 \mathrm{~m} / \mathrm{s}\) and the lowest frequency sound wave that the human ear can detect is approximately \(20 \mathrm{~Hz}\). (a) What is the wavelength of such a sound wave? (b) What would be the frequency of electromagnetic radiation with the same wavelength? (c) What type of electromagnetic radiation would that correspond to? [Section 6.1]

Step-by-Step Solution

Answer

(a) 17.15 m; (b) 17.5 MHz; (c) FM radio waves.

Step 1: Understand the relation between speed, frequency, and wavelength

To find the wavelength of a sound wave in air, use the equation \( v = f \cdot \lambda \), where \( v \) is the speed of sound, \( f \) is the frequency, and \( \lambda \) is the wavelength. We are given \( v = 343 \, \text{m/s} \) and \( f = 20 \, \text{Hz} \).

Step 2: Calculate the wavelength of the sound wave

Rearrange the formula to solve for \( \lambda \): \( \lambda = \frac{v}{f} \). Substitute the values to find \( \lambda \): \( \lambda = \frac{343}{20} = 17.15 \, \text{m} \).

Step 3: Use wavelength to find the frequency of electromagnetic radiation

The speed of light \( c \) is approximately \( 3 \times 10^8 \, \text{m/s} \). Use the same formula \( c = f \cdot \lambda \) to find the frequency of electromagnetic radiation: \( f = \frac{c}{\lambda} \). Substitute \( \lambda = 17.15 \, \text{m} \): \( f = \frac{3 \times 10^8}{17.15} \approx 1.75 \times 10^7 \, \text{Hz} \).

Step 4: Identify the type of electromagnetic radiation

The frequency \( 1.75 \times 10^7 \, \text{Hz} \) (or \(17.5 \, \text{MHz}\)) corresponds to the Very High Frequency (VHF) band, which is typically used for FM radio broadcasting.

Key Concepts

Speed of Sound in Air

When we talk about the speed of sound in air, we're discussing how fast sound waves travel through the air. For dry air at a temperature of 20 degrees Celsius, the speed of sound is approximately 343 meters per second. This speed can vary depending on factors like temperature, humidity, and air pressure.

• Temperature: Sound travels faster at higher temperatures because the molecules move more quickly.
• Humidity: Higher humidity generally increases the speed of sound, as water vapor is less dense than dry air.
• Pressure: Has little direct effect on the speed of sound, given the proportional change between density and pressure at constant temperature.

Sound waves are mechanical waves, which means they require a medium, like air, to travel through. The relationship between the speed of a sound wave ( \( v \) ), its frequency ( \( f \) ), and its wavelength ( \( \lambda \) ) is expressed by the equation: \( v = f \cdot \lambda \) . Understanding this relationship is crucial for calculating the wavelength of sound waves.

Electromagnetic Radiation

Electromagnetic radiation encompasses a wide range of wave types that do not require a medium to travel through, unlike sound. These waves move through the vacuum of space at the speed of light, approximately 3 x 10^8 meters per second. Electromagnetic radiation varies vastly in wavelength and frequency, ranging from radio waves to gamma rays.

• Wavelength: The distance between successive crests of a wave.
• Frequency: The number of waves that pass a point in one second, measured in Hertz (Hz).
• Speed: For electromagnetic waves, speed is constant in vacuum: \( c = 3 \times 10^8 \, \text{m/s} \).

Knowing the speed and wavelength of electromagnetic waves allows us to calculate their frequency using the equation: \( c = f \cdot \lambda \). This relationship is essential for understanding how different types of electromagnetic waves behave, such as infrared, visible light, and radio waves. The variety of these waves has practical applications in communications, medicine, and more.

Frequency of Sound Waves

The frequency of a sound wave is related to how low or high a sound is perceived by the human ear. Frequency is measured in Hertz (Hz), and it determines the pitch of the sound. For instance, a low-frequency sound has a low pitch, while a high-frequency sound has a high pitch. The human ear can typically hear frequencies ranging from about 20 Hz to 20,000 Hz.

When calculating the frequency of a sound wave, the same formula used for electromagnetic waves applies: \( v = f \cdot \lambda \). By rearranging this equation, you can determine frequency if you know the speed and wavelength: \( f = \frac{v}{\lambda} \).

• The frequency range of sounds is measured as audible frequency, with any sound below 20 Hz referred to as infrasound, and any sound above 20,000 Hz referred to as ultrasound.
• Application areas include music production, sonar technology, and medical imaging, among others.

Understanding frequency is crucial for various fields, including acoustics and audio engineering, allowing us to take advantage of sound in music, communication, and technology.