Problem 5 - General Chemistry: Principles and Modern Applications

Question

Without doing detailed calculations, determine which of the following wavelengths represents light of the highest frequency: (a) \(6.7 \times 10^{-4} \mathrm{cm} ;\) (b) \(1.23 \mathrm{mm}\) (c) \(80 \mathrm{nm} ;\) (d) \(6.72 \mu \mathrm{m}\)

Step-by-Step Solution

Answer

The shortest wavelength corresponds to the highest frequency. In this case, the wavelength of \(80 nm\) is the shortest, so light with this wavelength has the highest frequency.

Step 1: Understand Units

Identify the units used and convert all into the same unit for easy comparison. Here, the smallest unit nanometer (nm) could be chosen as the standard.

Step 2: Unit Conversion

Convert all the units into nanometers. (a) To convert centimeters to nanometers, multiply by \(1 \times 10^{7}\). So, \(6.7 \times 10^{-4} cm = 6.7 \times 10^{3} nm. (b) To convert millimeters to nanometers, multiply by \(1 \times 10^{6} \). So, \(1.23 mm = 1.23 \times 10^{6} nm. (c) Is already in the unit nm. (d) To convert micrometers to nanometers, multiply by \(1 \times 10^{3} \). So, \(6.72 \mu m = 6.72 \times 10^{3} nm.\)

Step 3: Comparision

Now that all the wavelengths are in the same unit, compare them to find out which one is the shortest. The shortest wavelength will represent light of the highest frequency.

Key Concepts

Frequency of Light

Light, also known as electromagnetic radiation, has various properties. One key aspect is its frequency, which is the number of wave cycles that pass through a point per second. The frequency of light is inversely proportional to its wavelength. This means that as the wavelength gets shorter, the frequency becomes higher. This inverse relationship can be mathematically expressed as:\[ c = \lambda u \]where \( c \) is the speed of light (approximately \( 3 \times 10^8 \) meters per second), \( \lambda \) is the wavelength, and \( u \) is the frequency.Understanding the frequency of light is crucial when comparing different types of light waves, such as those in the exercise given. Higher frequency light, like ultraviolet, has shorter wavelengths compared to lower frequency light, such as infrared.So when you compare different wavelengths to determine which has the highest frequency, remember that you're essentially looking for the smallest wavelength. This simple rule—short wavelength equals high frequency—makes it easier to understand the visible light spectrum and beyond.

Unit Conversion

Unit conversion can seem like a daunting task, but it's actually a straightforward process once you understand the basic steps. Different scientific fields use various units, depending on what they measure. For wavelength, common units include nanometers \( (nm) \), micrometers \( (\mu m) \), millimeters \( (mm) \), and centimeters \( (cm) \).
To compare wavelengths accurately, you must first convert them all to the same unit. Converting involves using a conversion factor that represents the relationship between two units. For example:

• 1 centimeter is equal to 10⁷ nanometers.
• 1 millimeter is equal to 10⁶ nanometers.
• 1 micrometer is equal to 10³ nanometers.

Here's how you convert:

• Multiply the number in its current unit by its respective factor to convert to nanometers.
• After this, you'll have all measurements in nanometers, making your data easy to compare.

Mastering unit conversion gives you the tools to tackle many scientific problems by allowing you to streamline and unify your data.

Light Wavelength Comparison

When comparing light wavelengths, the goal is to identify which wavelength is the shortest—because shorter wavelengths correspond to higher frequencies. The process of comparison involves a few simple steps: First, as demonstrated in the original problem, convert all wavelengths into the same unit. This unification of units is crucial to avoiding errors when comparing magnitudes.
Once in the same unit, the next step is to simply line up the numbers and see which one is the smallest. The smaller the number, the shorter the wavelength, and hence, the higher the frequency. In practical terms:

• Check each converted measurement and identify the smallest value.
• The smallest wavelength translates to the highest frequency.

This method of comparison is frequently used in scientific studies, such as physics and engineering, to understand how different light waves interact within various environments, including natural and technological contexts.