The diameter of an oxygen atom is 1.32 angstroms. What is this in meters? If you placed oxygen atoms side by side in a line with a length of ten inches, how many atoms are in the line. Assume that oxygen atoms are hard spheres.
The correct answer and explanation is:
Answer:
1. Diameter in meters:
1 angstrom (Å) = 1×10−101 \times 10^{-10} meters
So, diameter of an oxygen atom = 1.32 A˚=1.32×10−10 m1.32 \, \text{Å} = 1.32 \times 10^{-10} \, \text{m}
2. Length of line in meters:
10 inches = 10 × 0.0254 = 0.254 meters
3. Number of atoms in 0.254 meters: Number of atoms=0.254 m1.32×10−10 m≈1.92×109 atoms\text{Number of atoms} = \frac{0.254 \, \text{m}}{1.32 \times 10^{-10} \, \text{m}} \approx 1.92 \times 10^9 \, \text{atoms}
Explanation (300 words):
To convert the size of an atom from angstroms to meters, we start by understanding the metric relationship: 1 angstrom (Å) is 1×10−101 \times 10^{-10} meters. Since the diameter of an oxygen atom is given as 1.32 Å, its equivalent in meters is 1.32×10−101.32 \times 10^{-10} m. This unit conversion allows us to work with standard SI units when calculating how many atoms fit into a specified length.
Next, we convert the total length from inches to meters. One inch equals 0.0254 meters, so ten inches equal 10×0.0254=0.25410 \times 0.0254 = 0.254 meters. Now that both the length and the diameter of the atom are in meters, we can determine how many oxygen atoms, modeled as hard spheres (meaning they are packed without overlapping), can be placed end-to-end in this line.
To find the number of atoms, we divide the total line length by the diameter of a single oxygen atom: 0.2541.32×10−10≈1.92×109\frac{0.254}{1.32 \times 10^{-10}} \approx 1.92 \times 10^9
This means that approximately 1.92 billion oxygen atoms can fit in a straight line that is ten inches long.
This thought experiment illustrates the incredibly small scale of atoms. Despite a ten-inch line appearing relatively short to us, it can accommodate nearly 2 billion oxygen atoms in a row. This helps convey the scale difference between everyday macroscopic objects and the atomic world, and highlights why nanotechnology and molecular-scale engineering require highly precise instruments.