Problem 24 - Chemistry: An Atoms-Focused Approach

Question

Garage Door Openers The remote control units for garage door openers transmit electromagnetic radiation. Before 2005 they operated on a frequency of \(390 \mathrm{MHz}\), but since 2005, the operating wavelength has been 952 mm. Which radiation has the lower frequency, the pre-2005 or the post- 2005 devices?

Step-by-Step Solution

Answer

Answer: Post-2005 devices have a lower frequency.

Step 1: Calculate post-2005 device frequency

To start, we will calculate the frequency of post-2005 devices using the provided wavelength. First, we need to convert the given wavelength from mm to meters. Wavelength (λ) in meters = 952 mm * (1 meter / 1000 mm) = 0.952 m Now, we can use the relationship between frequency, wavelength, and the speed of light to find the frequency of the post-2005 devices. Recall the speed of light is approximately 3 * 10^8 meters per second. f = c / λ f = (3 * 10^8 m/s) / 0.952 m f ≈ 315.126 MHz

Step 2: Compare the frequencies of pre-2005 and post-2005 devices

Now that we have the frequency of post-2005 devices (approximately 315.126 MHz), we should compare it with the frequency of pre-2005 devices (390 MHz). Pre-2005 frequency: 390 MHz Post-2005 frequency: ≈ 315.126 MHz Since 315.126 MHz < 390 MHz, the post-2005 devices have a lower frequency than pre-2005 devices.

Key Concepts

Frequency Calculation

Frequency is a measure of how often an event occurs over a specific period of time. When it comes to electromagnetic radiation, frequency refers to the number of cycles that occur per second in a wave. This is measured in hertz (Hz). In our specific scenario, understanding frequency is important to determine how electromagnetic waves are transmitted by garage door openers.
To calculate the frequency of electromagnetic radiation, we employ the formula:

• Frequency (\(f\)) = Speed of Light (\(c\)) / Wavelength (\(\lambda\)).

If you have the speed of light and the wavelength of the transmitted wave, you can find the frequency by dividing the speed of light by the wavelength.

• Remember: The speed of light is approximately \(3 \times 10^8\) m/s.

So, when calculating the frequency of post-2005 garage opener devices, we converted the wavelength from millimeters to meters to use in our frequency formula. This conversion resulted in a frequency of about 315.126 megahertz (MHz).
Comparing frequencies allows us to understand which technology operates at a higher or lower frequency, a key factor in determining their behavior and compatibility.

Wavelength Conversion

Wavelength is the distance between consecutive points of a wave that are in phase, such as from crest to crest or trough to trough. It plays a crucial role in understanding electromagnetic radiation and can be expressed in various units, such as meters or millimeters.
In real-world applications, like our garage door opener exercise, sometimes the wavelength is given in millimeters instead of meters.
For standard calculations, wavelengths must be in meters (m). To convert the wavelength from millimeters to meters, a simple formula is used:

• Wavelength in meters = Wavelength in millimeters * (1 meter / 1000 millimeters)

In this context, the given wavelength was 952 mm, which converted to 0.952 meters. This conversion is a crucial step before applying the frequency formula.

Speed of Light

The speed of light is a fundamental constant crucial for calculating electromagnetic properties, including frequency and wavelength. It is one of the cornerstones of physics. In a vacuum, the speed of light is approximately \(3 \times 10^8\) meters per second (m/s).
This constant speed plays a role in various calculations involving electromagnetic waves, such as predicting the frequency of devices based on their wavelengths.

• When using the formula \(f = c / \lambda\), the speed of light (\(c\)) allows us to solve for frequency (\(f\)) given the wavelength (\(\lambda\)).

In scenarios incorporating electromagnetic technologies like the garage door openers, this constant provides a bridge between different measurements, linking wavelength and frequency. It enables us to compare different technologies or devices operating at different frequencies, as seen in the comparison of pre- and post-2005 garage door openers.