Question

The diameter of a hydrogen atom is \(212 \mathrm{pm}\). Find the length in kilometers of a row of \(6.02 \times 10^{23}\) hydrogen atoms. The diameter of a ping pong ball is \(4.0 \mathrm{~cm} .\) Find the length in kilometers of a row of \(6.02 \times 10^{23}\) ping pong balls.

Short answer

The length of a row of hydrogen atoms is \(127.424 \times 10^{11}\) km, and the length of a row of ping pong balls is \(2408 \times 10^{17}\) km.

Step-by-step solution

Convert Diameter of a Hydrogen Atom to Meters

First, convert the diameter of a single hydrogen atom from picometers to meters. There are \(1 \times 10^{12}\) picometers in a meter, which means you divide the diameter in picometers by \(1 \times 10^{12}\) to get the diameter in meters. So, the diameter of a hydrogen atom in meters is \(212 \times 10^{-12}\) meters.

Calculate the Total Length of Hydrogen Atoms

Multiply the diameter of a single hydrogen atom in meters by the number of hydrogen atoms to get the total length in meters. You have \(212 \times 10^{-12} \times 6.02 \times 10^{23}\) meters.

Convert the Length from Meters to Kilometers

As there are \(1 \times 10^{3}\) meters in a kilometer, divide the total length in meters by \(1 \times 10^{3}\) to find the length in kilometers. Perform the calculation to find the length of a row of hydrogen atoms in kilometers.

Convert Diameter of a Ping Pong Ball to Meters

First, convert the diameter of the ping pong ball from centimeters to meters by dividing by 100, since there are 100 centimeters in a meter. So, the diameter in meters is \(4.0 \times 10^{-2}\) meters.

Calculate the Total Length of Ping Pong Balls

Multiply the diameter of a single ping pong ball in meters by the number of ping pong balls to find the total length in meters. You have \(4.0 \times 10^{-2} \times 6.02 \times 10^{23}\) meters.

Convert the Length from Meters to Kilometers

Divide the total length in meters by \(1 \times 10^{3}\) to convert the length from meters to kilometers for the ping pong balls row. Perform the calculation to find the length in kilometers.

Key concepts

Avogadro's number

Understanding Avogadro's number is crucial when dealing with the microscopic world of atoms and molecules. Essentially, it provides a link between the macroscopic and atomic scales. Avogadro's number, denoted as \(6.02 \times 10^{23}\), represents the number of units, be they atoms, molecules, or ions, in one mole of any substance. This vast number allows chemists to convert between the number of particles and the amount of substance in moles, a bridge between the atomic scale and measurable quantities.

For instance, in our exercise, we consider a row of \(6.02 \times 10^{23}\) hydrogen atoms. This quantity, by definition, is one mole of hydrogen atoms. In a similar fashion, one mole of ping pong balls, if such a concept were applied to larger objects, would also contain this many ping pong balls. Avogadro's number enables us to perform calculations that exhibit the staggering difference in scale between everyday objects and atomic particles.

Unit Conversion

Unit conversion is a powerful tool in the realm of science and mathematics, allowing us to translate different units of measurement into a common language that is easier to understand and work with. In our exercise, we have converted measurements from picometers to meters and centimeters to meters. The key point of unit conversion is to keep the numerical value of a measurement consistent regardless of the unit system used.

For example, there are \(1 \times 10^{12}\) picometers in a meter, so to convert the diameter of a hydrogen atom from picometers to meters, you divide by \(1 \times 10^{12}\). This simplification streamlines the process to reach the final goal, which is finding the length of a hypothetical row of atoms or objects in a more conventional unit of length, like kilometers. The concept of unit conversion also emphasizes attention to detail since overlooking a conversion factor can lead to significant errors in calculation.

Dimensional Analysis

Dimensional analysis, also known as factor-label method or the unit factor method, is a systematic way of checking the consistency of units in calculations. It is an essential process that ensures the system's units cancel out properly, leaving the desired unit.

At its core, dimensional analysis is the practice of multiplying by conversion factors that describe the relationship between units and that are equal to one—for example, \(1 \text{ km} = 1000 \text{ m}\). When we multiplied the diameter of the hydrogen atom by Avogadro's number and then converted it to kilometers, each step maintained the integrity of dimensional units.

Using dimensional analysis ensures that our final answer is in terms of the correct units. It acts as a validation tool, helping to avoid mistakes that can occur when dealing with complex unit transformations, like those needed for our exercise of aligning picometers with kilometers.