Problem 21 - Chemistry: An Atoms-Focused Approach

Question

Broadcast Frequencies FM radio stations broadcast in a band of frequencies between 88 and 108 megahertz (MHz). Calculate the wavelengths corresponding to the broadcast frequencies of the following radio stations: a. KRNU (Lincoln, NE), 90.3 MHz b. WBRU (Providence, RI), \(95.5 \mathrm{MHz}\) c. WYLD (New Orleans, LA), 98.5 MHz d. WAAF (Boston, MA), \(107.3 \mathrm{MHz}\)

Step-by-Step Solution

Answer

Question: Calculate the wavelengths of the FM radio stations KRNU (Lincoln, NE), 90.3 MHz; WBRU (Providence, RI), 95.5 MHz; WYLD (New Orleans, LA), 98.5 MHz; and WAAF (Boston, MA), 107.3 MHz. Answer: The wavelengths for the FM radio stations are: KRNU - 3.32 meters; WBRU - 3.14 meters; WYLD - 3.05 meters; WAAF - 2.80 meters.

Step 1: a. KRNU (Lincoln, NE), 90.3 MHz

1. Convert the frequency from MHz to Hz: \( f_1 = 90.3 MHz \times 10^6 = 9.03 \times 10^7 Hz\) 2. Use the formula to calculate wavelength: \( \lambda_1 = \frac{3 \times 10^8 m/s}{9.03 \times 10^7 Hz} = 3.32 m\) The wavelength for KRNU radio station is 3.32 meters.

Step 2: b. WBRU (Providence, RI), 95.5 MHz

1. Convert the frequency from MHz to Hz: \( f_2 = 95.5 MHz \times 10^6 = 9.55 \times 10^7 Hz\) 2. Use the formula to calculate wavelength: \( \lambda_2 = \frac{3 \times 10^8 m/s}{9.55 \times 10^7 Hz} = 3.14m\) The wavelength for WBRU radio station is 3.14 meters.

Step 3: c. WYLD (New Orleans, LA), 98.5 MHz

1. Convert the frequency from MHz to Hz: \( f_3 = 98.5 MHz \times 10^6 = 9.85 \times 10^7 Hz\) 2. Use the formula to calculate wavelength: \( \lambda_3 = \frac{3 \times 10^8 m/s}{9.85 \times 10^7 Hz} = 3.05 m\) The wavelength for WYLD radio station is 3.05 meters.

Step 4: d. WAAF (Boston, MA), 107.3 MHz

1. Convert the frequency from MHz to Hz: \( f_4 = 107.3 MHz \times 10^6 = 1.073 \times 10^8 Hz\) 2. Use the formula to calculate wavelength: \( \lambda_4 = \frac{3 \times 10^8 m/s}{1.073 \times 10^8 Hz} = 2.80 m\) The wavelength for WAAF radio station is 2.80 meters.

Key Concepts

Radio Frequencies

Radio frequencies refer to the range of electromagnetic waves that lie within the radio spectrum, specifically the frequencies from 3 kHz to 300 GHz. These frequencies are used in various communication systems such as radio broadcasting, television, satellites, and more.
Understanding radio frequencies is essential for tuning into different stations. These frequencies, measured in hertz (Hz), define how many cycles per second the wave oscillates.

• Hertz (Hz): The unit of frequency measurement, representing one cycle per second.
• Kilohertz (kHz): 1,000 Hz.
• Megahertz (MHz): 1,000,000 Hz or 10⁶ Hz.
• Gigahertz (GHz): 1,000,000,000 Hz or 10⁹ Hz.

Thus, by converting between different units (such as MHz to Hz), we can better understand and work with these frequencies in calculations and real-world applications.

FM Radio Band

The FM radio band is a range of frequencies in the electromagnetic spectrum used for frequency modulation (FM) broadcasting. The standard FM broadcast band covers 88 to 108 MHz.
This band is used for delivering high-fidelity audio content over distances. FM broadcasting is preferred for music and voice because it is less susceptible to interference compared to AM.

• Frequency Modulation (FM): Alters the frequency of the carrier signal according to the audio information, providing better sound quality.
• Broadcast Band: The specific range of frequencies allocated for FM radio stations to broadcast.
• High-Fidelity Audio: Superior sound reproduction capability, critical for music broadcasting.

Stations within this band are assigned specific frequencies to avoid overlap and signal interference, ensuring clear audio transmission.

Frequency to Wavelength Conversion

Converting frequency to wavelength involves understanding the relationship between these two properties in electromagnetic waves. The wavelength (\(\lambda\)) is the distance between two consecutive peaks of a wave, and frequency (\(f\)) is how often the wave cycles in one second.
The formula used for this conversion is:

• \[\lambda = \frac{c}{f}\]

Where:

• \(\lambda\) is the wavelength in meters.
• \(c\) is the speed of light in a vacuum, approximately \(3 \times 10^8\ m/s\).
• \(f\) is the frequency in hertz.

To convert from MHz to Hz:

• Multiply by \(10^6\)

This step is crucial because it ensures you're working with the correct units when applying the formula, thus providing accurate wavelength values.

Electromagnetic Waves

Electromagnetic waves are waves of the electromagnetic field, propagating through space, carrying electromagnetic radiant energy. These waves include a range of frequencies, from very low frequencies used in radio, extending to the very high frequencies of gamma rays.
Key characteristics of electromagnetic waves include:

• Speed: Travel at the speed of light in a vacuum, \(3 \times 10^8\ m/s\).
• Wavelength: The distance between successive crests of a wave.
• Frequency: The number of wave cycles per second.

Within the radio band of the electromagnetic spectrum, different frequencies are allocated for various forms of communication, such as:

• FM Radio: High-fidelity music and speech broadcasting.
• AM Radio: Amplitude modulation for talk radio, sports, etc.
• Television Broadcasting: Uses different parts of the spectrum for audio and visual data transmission.

Understanding these waves' properties helps in engineering applications like broadcasting, wifi, and even medical technologies, as they rely on these fundamental principles.