Which one of the following is the correct relation between Ȧ and nm?
1 nm = 10 Ȧ
Understanding Angstrom and Nanometer Relation
The question asks for the correct relationship between the Angstrom (Ȧ) and the nanometer (nm). Both are units of length commonly used in science, particularly for very small scales like atomic sizes, molecular distances, and wavelengths of light.
Defining Angstrom (Ȧ) and Nanometer (nm)
- Angstrom (Ȧ): The Angstrom is a non-SI unit of length equal to \(10^{-10}\) meters. It is often used in crystallography, spectroscopy, and other fields to express atomic radii, bond lengths, and lattice constants.
- Nanometer (nm): The nanometer is an SI unit of length equal to \(10^{-9}\) meters. It is widely used in nanotechnology, optics, and semiconductor physics, often for wavelengths of visible light (\(400-700\) nm) and feature sizes in microelectronics.
Deriving the Relationship
To find the relationship between Angstrom and nanometer, we can use their definitions in terms of meters:
- \(1 \, \text{Ȧ} = 10^{-10} \, \text{m}\)
- \(1 \, \text{nm} = 10^{-9} \, \text{m}\)
We want to express \(1 \, \text{nm}\) in terms of Angstroms. We can start with the definition of the nanometer:
\(1 \, \text{nm} = 10^{-9} \, \text{m}\)
From the definition of the Angstrom, we can express 1 meter in terms of Angstroms:
\(1 \, \text{Ȧ} = 10^{-10} \, \text{m}\)
Dividing both sides by \(10^{-10}\):
\(\frac{1 \, \text{Ȧ}}{10^{-10}} = \text{m}\)
Using the property of exponents \(\frac{1}{a^{-n}} = a^n\):
\(1 \, \text{m} = 10^{10} \, \text{Ȧ}\)
Now substitute this expression for meters into the nanometer definition:
\(1 \, \text{nm} = 10^{-9} \, \times \, (10^{10} \, \text{Ȧ})\)
Using the property of exponents \(a^m \times a^n = a^{m+n}\):
\(1 \, \text{nm} = 10^{(-9 + 10)} \, \text{Ȧ}\)
\(1 \, \text{nm} = 10^{1} \, \text{Ȧ}\)
\(1 \, \text{nm} = 10 \, \text{Ȧ}\)
Thus, one nanometer is equal to ten Angstroms.
Comparing with Options
Let's compare our derived relationship \(1 \, \text{nm} = 10 \, \text{Ȧ}\) with the given options:
| Option | Relation | Comparison |
|---|---|---|
| 1 | \(1 \, \text{nm} = 10^{-1} \, \text{Ȧ}\) | Does not match \(1 \, \text{nm} = 10 \, \text{Ȧ}\) |
| 2 | \(1 \, \text{nm} = 10 \, \text{Ȧ}\) | Matches our derived relationship |
| 3 | \(1 \, \text{nm} = 1 \, \text{Ȧ}\) | Does not match \(1 \, \text{nm} = 10 \, \text{Ȧ}\) |
| 4 | \(1 \, \text{nm} = 10^{-2} \, \text{Ȧ}\) | Does not match \(1 \, \text{nm} = 10 \, \text{Ȧ}\) |
Option 2 correctly states the relationship between nanometers and Angstroms.
Revision Table: Unit Conversions
| Unit | Symbol | Equivalent in Meters |
|---|---|---|
| Angstrom | Ȧ | \(10^{-10} \, \text{m}\) |
| Nanometer | nm | \(10^{-9} \, \text{m}\) |
| Micrometer (micron) | μm | \(10^{-6} \, \text{m}\) |
| Millimeter | mm | \(10^{-3} \, \text{m}\) |
| Centimeter | cm | \(10^{-2} \, \text{m}\) |
| Kilometer | km | \(10^{3} \, \text{m}\) |
Additional Information: Importance of Small Units
Units like Angstrom and nanometer are crucial for describing phenomena at the nanoscale. For example:
- Typical atomic radii are around \(1-2\) Angstroms.
- Covalent bond lengths are typically between \(1-2\) Angstroms.
- The diameter of a DNA double helix is about \(2\) nanometers.
- Features on modern computer chips are measured in tens of nanometers.
- The wavelengths of visible light range from about \(400\) nm (violet) to \(700\) nm (red).
Understanding these unit conversions is fundamental for working with measurements in fields like chemistry, physics, biology, and engineering at the atomic and molecular levels.
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