Problem 17 - Physics for Scientists and Engineers with Modern Physics

Question

A typical atom has a diameter of about \(1.0 \times 10^{-10} \mathrm{~m}\). (a) What is this in inches? (b) Approximately how many atoms are there along a \(1.0-\mathrm{cm}\) line?

Step-by-Step Solution

Answer

(a) The diameter is approximately \(3.93701 \times 10^{-9}\) inches. (b) There are approximately \(1.0 \times 10^{8}\) atoms along a 1-cm line.

Step 1: Convert Meters to Inches

First, we need to convert the diameter of an atom from meters to inches. We know that 1 meter equals 39.3701 inches. So we multiply:\[ 1.0 \times 10^{-10} \text{ m} \times 39.3701 \text{ in/m} = 3.93701 \times 10^{-9} \text{ inches} \]

Step 2: Convert Length of Line to Meters

Next, convert the length of the line from centimeters to meters because we already have the diameter of an atom in meters. Since 1 cm equals 0.01 meters:\[ 1.0 \text{ cm} = 0.01 \text{ m} \]

Step 3: Calculate Number of Atoms Along the Line

To find out how many atoms fit along a line of 0.01 meters, divide the length of the line by the diameter of an atom:\[ \frac{0.01 \text{ m}}{1.0 \times 10^{-10} \text{ m}} = 1.0 \times 10^{8} \]

Key Concepts

Unit Conversion

Unit conversion is the process of converting a value from one unit of measurement to another. This is a crucial skill in many fields including physics, especially when working with different measurement systems like metric and imperial.
When we have a measurement in meters and we wish to convert it to inches, we use the conversion factor that 1 meter equals 39.3701 inches. For example, to convert 1.0 \( \times 10^{-10} \) meters to inches, we would use the formula:

• Multiply the value in meters by the number of inches per meter.

This gives us: \[ 1.0 \times 10^{-10} \text{ m} \times 39.3701 \text{ in/m} = 3.93701 \times 10^{-9} \text{ inches} \] Learning to convert units across systems helps to ensure accuracy and consistency in scientific calculations.

Atomic Diameter

The atomic diameter refers to the approximate size of an atom. It's a fascinating aspect of atomic physics that highlights how tiny atoms are in comparison to everyday measurements.
Typically, the diameter of an atom is around 1.0 \( \times 10^{-10} \) meters. This scale is incredibly tiny and challenges our perception of size. To translate this into more familiar terms, that's roughly 0.0000000001 meters wide.
Understanding atomic diameter helps us compare the relative sizes of different atoms and make calculations regarding their interactions and placements in materials science. The small size of atoms allows for a high number of them to fit within any given space, influencing material properties.

Number of Atoms

The concept of the number of atoms along a line involves understanding how to calculate how many times a smaller unit can fit into a larger unit. This is heavily relied upon in physics to comprehend atomic arrangements.
When given a line of a certain length and an atomic diameter, we can determine how many atoms can line up along that length by dividing the total length by the diameter of a single atom:

• Convert the line length into the same unit as the atomic diameter. Here, 1 cm = 0.01 meters.
• Use the formula: \( \frac{\text{line length in meters}}{\text{atomic diameter in meters}} \)

For example, for a 1.0 cm line:\[ \frac{0.01 \text{ m}}{1.0 \times 10^{-10} \text{ m}} = 1.0 \times 10^{8} \] This means approximately 100 million atoms, showing just how densely packed atoms can be even in a short distance.