Question

How many 10-nm-diameter nanotubes, laid side by side, would it take to make a line \(0.10 \mathrm{~mm}\) in width?

Short answer

To make a line of 0.10 mm in width with 10-nm-diameter nanotubes, you need 10,000 nanotubes laid side by side.

Step-by-step solution

Convert millimeters to nanometers

First, convert the width from millimeters to nanometers to match the units for the diameter of the nanotubes. Since 1 millimeter is equal to 1,000,000 nanometers, you can convert 0.10 mm to nanometers by multiplying by 1,000,000.

Calculate the number of nanotubes

Divide the total width in nanometers by the diameter of one nanotube in nanometers to find out how many nanotubes can fit side by side. This will give you the total number of nanotubes needed to make a line of the given width in nanometers.

Perform the division

Perform the division calculated in the previous step to get the exact number of nanotubes required.

Key concepts

Unit Conversion

Understanding unit conversion is vital in many scientific computations, particularly in chemistry where measurements need to be precise. Unit conversion allows us to translate a quantity expressed in one kind of unit to another for comparison, calculation, or to adhere to specific measurement standards. In the context of our exercise, we have a measurement in millimeters that we need to convert to nanometers.

Here's a simple way to approach unit conversion: First identify how many of the smaller units are contained in one of the larger units. In our case, there are 1,000,000 nanometers in one millimeter. This factor, known as a conversion factor, is then used to multiply by the measurement we wish to convert. This critical step ensures all subsequent calculations are comparing 'apples to apples', so to speak.

Nanometer

The nanometer is a unit of length in the metric system, equivalent to one billionth of a meter (\(1 nm = 10^{-9} m\)). In the realm of chemistry and materials science, the nanometer is often used because it is a convenient size for expressing the dimensions of atoms, molecules, and nanostructures like nanotubes.

Understanding the scale of a nanometer can be challenging because it's not visible to the naked eye. A helpful comparison is that a human hair is approximately 80,000 to 100,000 nanometers wide, which puts into perspective the size of the nanotubes we're working with in this exercise - only 10 nanometers in diameter, significantly smaller than even a single strand of hair.

Geometry in Chemistry

The field of chemistry often involves geometric concepts, especially when discussing molecular shapes, bond angles, and intermolecular forces. When considering the arrangement of nanotubes, we apply geometry to determine how many of these cylindrical structures can fit side by side.

In our exercise, the diameter of the nanotubes is essential in calculating how many tubes will fit across a specified width. If one imagines these tubes as circles lying flat side by side, we can use simple geometric understanding to deduce that the diameter corresponds to the width each nanotube will occupy.

Problem-Solving in Chemistry

Chemistry problems often require a structured approach to arrive at the correct solution. The step-by-step method applied in this exercise is a classic example of problem-solving in chemistry. Initially, we ensure that all units align, followed by applying relevant formulas or principles - in this case, dividing the total width by the diameter of an individual nanotube to determine how many fit within a given space, resembling an approach similar to finding how many times a number can be divided by another.

Problem-solving also involves a logical progression of steps, anticipating what information is needed down the line. By converting measurements to the same units and employing division, we can arrive at an accurate and meaningful answer without errors or misinterpretation.