Problem 34 - Chemistry: Principles and Reactions

Question

Convert \(6743 \mathrm{~nm}\) to (a) \(\AA\) (b) inches (c) miles

Step-by-Step Solution

Answer

Answer: The equivalent lengths of 6743 nm are approximately 67,430 Å, 2.65 x 10^-7 inches, and 4.19 x 10^-12 miles.

Step 1: Convert nanometers to angstroms

To convert the given length in nanometers to angstroms, multiply the length by the conversion factor of 10. In this case, 6743 nm multiplied by 10 gives us: \[6743 \times 10 = 67430 ~\AA\]

Step 2: Convert nanometers to inches

To convert the given length in nanometers to inches, divide the length by the conversion factor of 25.4 million (25,400,000). In this case, 6743 nm divided by 25,400,000 gives us: \[\frac{6743}{25,400,000} \approx 2.65 \times 10^{-7} ~\text{inches}\]

Step 3: Convert nanometers to miles

To convert the given length in nanometers to miles, divide the length by the conversion factor of 1.609 * 10^12. In this case, 6743 nm divided by 1.609 * 10^12 gives us: \[\frac{6743}{1.609 \times 10^{12}} \approx 4.19 \times 10^{-12} ~\text{miles}\]

Key Concepts

Understanding Nanometers to Angstroms

Nanometers (nm) and angstroms (\(\AA\)) are both units used to measure extremely small lengths. They are particularly important in fields like physics and chemistry, when dealing with atomic and molecular scales. One nanometer is equal to 10 angstroms. So, to convert from nanometers to angstroms, you simply multiply by 10.

Here's how it works: if you have 6743 nm, you calculate it in angstroms as follows:

• Multiply 6743 by 10
• Result: 67430 \(\AA\)

This straightforward multiplication step makes the conversion simple and rapid.

Converting Nanometers to Inches

Converting from nanometers to inches might seem tricky, but it becomes simple with the right conversion factor. An inch is quite larger compared to a nanometer, so the conversion requires dividing by a large number. To convert nanometers to inches, we use the factor of 25.4 million (or 25,400,000). This is based on the number of nanometers per inch.

Let's see this with our example of 6743 nm:

• Divide 6743 nm by 25,400,000
• Result: approximately \(2.65 \times 10^{-7}\) inches

This conversion highlights how small a nanometer is compared to an inch.

Converting Nanometers to Miles

Miles are even larger units compared to nanometers. When converting nanometers to miles, we deal with an enormous conversion factor. One mile is equivalent to roughly 1.609 trillion (1.609 x 10¹²) nanometers. This highlights the significant difference in scale.

Using the example of 6743 nm, you perform the conversion as follows:

• Divide 6743 nm by \(1.609 \times 10^{12}\)
• Result: approximately \(4.19 \times 10^{-12}\) miles

This conversion shows how minuscule a nanometer is when considered in the context of miles.