Problem 40 - Chemistry: Structure and Properties

Question

Calculate the wavelength of each frequency of electromagnetic radiation: a. 100.2 MHz (typical frequency for FM radio broadcasting) b. 1070 kHz (typical frequency for AM radio broadcasting) (assume four significant figures) c. 835.6 MHz (common frequency used for cell phone communication)

Step-by-Step Solution

Answer

The wavelengths are approximately \r\( 2.99 \text{ m} \r\) for 100.2 MHz, \r\( 280.4 \text{ m} \r\) for 1070 kHz, and \r\( 0.359 \text{ m} \r\) for 835.6 MHz.

Step 1: Understand the Relationship between Wavelength and Frequency

The speed of light (c) is the product of the wavelength (\r\( \lambda \r\)) and the frequency (f) of the electromagnetic radiation. This is described by the equation \r\( c = \lambda \cdot f \r\), where c is the speed of light in a vacuum (approximately \r\( 3 \times 10^8 \) meters per second).

Step 2: Convert MHz to Hz

The frequency is given in megahertz (MHz) and kilohertz (kHz) and must be converted to hertz (Hz) by multiplying the MHz value by \r\( 10^6 \r\) and the kHz value by \r\( 10^3 \r\).

Step 3: Calculate the Wavelength for 100.2 MHz

Convert the frequency from MHz to Hz: \r\( 100.2 \text{ MHz} = 100.2 \times 10^6 \text{ Hz} \r\). Then use the speed of light to find the wavelength: \r\( \lambda = \frac{c}{f} = \frac{3 \times 10^8 \text{ m/s}}{100.2 \times 10^6 \text{ Hz}} \r\).

Step 4: Calculate the Wavelength for 1070 kHz

Convert the frequency from kHz to Hz: \r\( 1070 \text{ kHz} = 1070 \times 10^3 \text{ Hz} \r\). Then calculate the wavelength: \r\( \lambda = \frac{c}{f} = \frac{3 \times 10^8 \text{ m/s}}{1070 \times 10^3 \text{ Hz}} \r\), ensuring that the result has four significant figures.

Step 5: Calculate the Wavelength for 835.6 MHz

Convert the frequency from MHz to Hz: \r\( 835.6 \text{ MHz} = 835.6 \times 10^6 \text{ Hz} \r\). Calculate the wavelength using the speed of light: \r\( \lambda = \frac{c}{f} = \frac{3 \times 10^8 \text{ m/s}}{835.6 \times 10^6 \text{ Hz}} \r\).

Key Concepts

Speed of Light

The speed of light, denoted as 'c', is a fundamental constant in physics, representing the swiftest speed at which energy, matter, and information can travel. In the vacuum of space, it has a value of approximately \requal to \( 3 \times 10^8 \) meters per second or about 300,000 kilometers per second.

This constant is not only a speed limit but also a crucial element in the study of electromagnetic radiation. Since light is an electromagnetic wave, its velocity in a vacuum remains constant regardless of the observer's frame of reference. This consistency allows scientists to use it as a foundation for measurements and equations in various areas including physics, astronomy, and communication technologies. Understanding the constancy of the speed of light is essential when calculating the wavelengths of different frequencies of electromagnetic radiation.

Frequency and Wavelength Relationship

Electromagnetic radiation comprises waves that carry energy through space. Each wave has a frequency (f) and a wavelength (\( \lambda \)), which are inversely related. The frequency represents the number of waves that pass a point in one second and is measured in Hertz (Hz). The wavelength, on the other hand, is the distance between two consecutive peaks or troughs of a wave.

The equation \( c = \lambda \cdot f \) expresses the direct relationship between wavelength and frequency, where increasing the frequency results in a reduction in wavelength and vice versa. Recognizing this relationship helps to understand why different electromagnetic waves (like radio waves, microwaves, and X-rays) have both very different frequencies and wavelengths but nonetheless travel at the same speed – the speed of light – when in a vacuum.

Electromagnetic Spectrum

The electromagnetic spectrum encompasses all types of electromagnetic radiation, ranging from very low-frequency radio waves to very high-frequency gamma rays. Each category within the spectrum—including radio, microwave, infrared, visible light, ultraviolet, X-rays, and gamma rays—has its unique frequency and wavelength range.

From long to short wavelengths, the spectrum is ordered as follows: radio waves, microwaves, infrared, visible light, ultraviolet, X-rays, and gamma rays. Understanding the electromagnetic spectrum is crucial for technologies like radio broadcasting, where certain frequency bands are dedicated to FM and AM signals, or cell phone communication that operates at specific frequencies. Each application uses a particular band of the spectrum to effectively transmit information without undue interference from other types of electromagnetic waves.

Units Conversion

In physics and engineering, precise unit conversion is a fundamental skill. When computing the wavelengths of electromagnetic radiation, it is often necessary to convert the given frequency into a consistent unit such as hertz (Hz).

For instance, a frequency given in megahertz (MHz) must be converted to Hz by multiplying by \( 10^6 \), and a frequency in kilohertz (kHz) is converted by multiplying by \( 10^3 \). Through these conversions, we can apply the speed of light and frequency relationship to find the wavelength of electromagnetic waves for different frequencies in a standard unit. Being diligent with unit conversions is necessary to ensure accuracy and to prevent significant errors in calculations, which is especially critical when dealing with high precision requirements in science and technology.