Question

The diameter of a neutral helium atom is about \(1 \times 10^{2} \mathrm{pm} .\) Suppose that we could line up helium atoms side by side in contact with one another. Approximately how many atoms would it take to make the distance 1 in from end to end?

Short answer

254,000 helium atoms are needed.

Step-by-step solution

Convert Inch to Picometers

Start by converting the distance from inches to picometers (pm). Since 1 inch is equal to approximately 2.54 cm and there are 10 million picometers in a centimetre, the conversion is:\[ 1 \text{ in} = 2.54 \text{ cm} \times 10^7 \text{ pm/cm} = 2.54 \times 10^7 \text{ pm} \]

Diameter of One Helium Atom

We know the diameter of a helium atom is given as \( 1 \times 10^2 \text{ pm} \). This means each helium atom has a diameter of 100 picometers.

Calculate Number of Atoms

To find the number of helium atoms needed to span 1 inch, divide the total distance in picometers by the diameter of one helium atom:\[ \text{Number of atoms} = \frac{2.54 \times 10^7 \text{ pm}}{1 \times 10^2 \text{ pm/atom}} \] Simplifying this expression gives:\[ \text{Number of atoms} = 2.54 \times 10^5 \]

Conclusion

Approximately 254,000 helium atoms lined up side by side are required to span a distance of 1 inch.

Key concepts

Helium Atom

Helium atoms are fascinating and simple structures, and they make up one of the most basic elements, helium, in the periodic table. Each helium atom consists of two protons, two neutrons, and two electrons. These components make helium the second lightest element after hydrogen. Helium is a noble gas, meaning it is non-reactive due to its full outer electron shell, which gives it stability. This noble gas is commonly found in natural gas deposits and is used in applications such as inflating balloons or in cryogenics due to its low boiling point.
Understanding the properties of a helium atom, such as its diameter, can be essential for various scientific calculations, including those related to atomic structure and material science. On average, the diameter of a helium atom is about 100 picometers (pm), a unit that is very helpful in measuring atomic-scale distances.

Diameter Measurement

Measuring the diameter of atoms, like helium, involves understanding dimensions at a very small scale. The diameter of an atom is the straight line passing from one side of the atom to the other through its center. In the case of helium, the measurement of its diameter is typically done in picometers (pm), which is a unit tailored for such small distances.
Picometers are part of the metric system and are equal to one trillionth of a meter ( 1 ext{ pm} = 10^{-12} ext{ m} ). Because atoms are so incredibly small, using units like picometers helps scientists and students manage these sizes effectively in calculations. Knowing the size of helium atoms allows us to estimate how they would align if placed side by side, vital in exercises where estimation of atomic distances in real-world scales is needed.

Unit Conversion

Unit conversion is an important skill in scientific calculations, especially when dealing with various measurement units. In the example exercise, the conversion is from inches to picometers. Such a conversion is necessary when comparing different measurement units that are more or less familiar.
To convert inches into picometers, you first convert inches to centimeters, as 1 inch equals approximately 2.54 centimeters. Then, since there are 10 million picometers in a centimeter, this becomes:

• 2.54 cm × 10⁷ pm/cm = 2.54 × 10⁷ pm

This straightforward conversion ensures we use the same units consistently when calculating or comparing dimensions. Consistent units are key in ensuring accuracy in both academic exercises and practical applications across scientific fields.

Distance Calculation

Calculating distances using the properties of atoms, like the helium atom, can be intriguing. It involves understanding how atoms would arrange themselves when placed side by side in a line. In the given solution, the task is to calculate how many helium atoms would span a distance of one inch.
Once you've done the unit conversion from inches to picometers, you have the total distance in a manageable unit. Then, you divide this distance by the diameter of a single helium atom:

• Number of atoms = \( \frac{2.54 imes 10^7 ext{ pm}}{1 imes 10^2 ext{ pm/atom}} \)

This calculation indicates that \(2.54 \times 10^5\) or approximately 254,000 helium atoms would be needed to reach the desired distance of 1 inch. By understanding this process, you can get a better visualization of atomic sizes in relation to common everyday distances like an inch, showcasing the vast difference in scales when moving from the atomic to the macroscopic world.