Question

The diameter of a hydrogen atom is 212 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 6.02 x 10^23 hydrogen atoms is approximately 127,424 kilometers. The length of a row of 6.02 x 10^23 ping pong balls is approximately 24,080,000 kilometers.

Step-by-step solution

Convert Diameter of Hydrogen Atom to Meters

Convert the given diameter of a hydrogen atom from picometers (pm) to meters (m). There are 1 trillion (1 x 10^12) picometers in 1 meter. Thus, the conversion is: Diameter in meters = 212 pm x (1 meter / 1 x 10^12 pm).

Calculate Length of Hydrogen Atoms in Meters

Multiply the diameter of a single hydrogen atom in meters by the number of atoms (Avogadro's number) to find the total length in meters. Length in meters = Diameter in meters x 6.02 x 10^23.

Convert Length of Hydrogen Atoms to Kilometers

Since there are 1,000 meters in a kilometer, divide the length in meters by 1,000 to find the length in kilometers.

Convert Diameter of Ping Pong Ball to Meters

Convert the given diameter of a ping pong ball from centimeters (cm) to meters (m). There are 100 centimeters in 1 meter. Diameter in meters = 4.0 cm x (1 meter / 100 cm).

Calculate Length of Ping Pong Balls in Meters

Multiply the diameter of a single ping pong ball in meters by the number of balls (Avogadro's number) to find the total length in meters. Length in meters = Diameter in meters x 6.02 x 10^23.

Convert Length of Ping Pong Balls to Kilometers

Divide the length in meters by 1,000 to find the length in kilometers.

Key concepts

Atomic Diameter Conversion

Understanding how to convert the diameter of an atom into various units is crucial for length calculation in chemistry.

As in our original example with the hydrogen atom, which measures 212 picometers in diameter, we must convert this figure to a more manageable unit such as meters for further calculations.

Picometers (pm) are a unit of length in the metric system, where 1 picometer equals one trillionth of a meter (\(1 \text{pm} = 10^{-12} \text{m}\)).

To perform this conversion, simply multiply the atomic diameter by the conversion factor: \
\text{Diameter in meters} = \text{atomic diameter in pm} \times \frac{1 \text{m}}{10^{12} \text{pm}} \
This process of conversion is crucial to understand as it lays the groundwork for more complex calculations in chemistry.

Length Calculation in Chemistry

Calculating the length of a sequence of atoms or molecules involves understanding Avogadro's number. Avogadro's number, \(6.02 \times 10^{23}\), represents the number of units in one mole of a substance.

When considering a row of hydrogen atoms, the length calculation would be the product of the diameter of one hydrogen atom and Avogadro's number. Assuming we have converted the atomic diameter to meters, the equation becomes: \
\text{Total length in meters} = \text{Diameter in meters} \times 6.02 \times 10^{23} \
Such calculations are essential in helping chemists and students visualize and understand the scale of atoms and molecules. Length calculations of this sort provide vital context for the minuscule dimensions of the chemical world.

Unit Conversion in Chemistry

Chemistry often requires converting measurements into different units to solve a problem effectively.

In the case of the hydrogen atoms and ping pong ball example, length calculations initially result in meters, but we may want the answer in kilometers. Since 1 kilometer equals 1,000 meters (\(1 \text{km} = 1,000 \text{m}\)), you need to divide the total length in meters by 1,000 to convert to kilometers: \
\text{Length in kilometers} = \frac{\text{Length in meters}}{1,000} \
Unit conversion is a fundamental skill that enables students to solve problems across different scales, from the atomic to the macroscopic level. Whether it is converting meters to kilometers or picometers to meters, recognizing how to switch between units is a crucial aspect of scientific literacy.