Question

A penny has a thickness of approximately 1.0 \(\mathrm{mm}\) . If you stacked Avogadro's number of pennies one on top of the other on Earth's surface, how far would the stack extend (in km)? For comparison, the sun is about 150 million km from Earth and the nearest star, Proxima Centauri, is about 40 trillion km from Earth.

Short answer

The stack of pennies would extend approximately \(6.022 \times 10^{20}\) kilometers. This is over 4 trillion times the distance to the sun and about 15 times the distance to Proxima Centauri.

Step-by-step solution

Understand Avogadro's Number

Avogadro's number is a fundamental constant used in chemistry, representing the number of constituent particles (usually atoms or molecules) in one mole of a substance. Avogadro's number is approximately \(6.022 \times 10^{23}\).

Convert Penny Thickness to Meters

The thickness of a penny is given in millimeters. To make it compatible with Avogadro's number and to ultimately calculate the height in kilometers, convert the thickness to meters by dividing by 1000. Therefore, the thickness of a penny in meters is \(1.0 \, \text{mm} = 0.001 \, \text{m}\).

Calculate the Height of the Stack

To find the height of the stack in meters, multiply Avogadro's number by the thickness of one penny in meters: \(6.022 \times 10^{23} \times 0.001 \, \text{m}\).

Convert the Height to Kilometers

Since there are 1000 meters in 1 kilometer, divide the height in meters by 1000 to convert it to kilometers.

Perform the Calculation

By performing the multiplication and division, the height of the stack is \((6.022 \times 10^{23}) \times (0.001 \, \text{m}) / 1000 = 6.022 \times 10^{20} \, \text{km}\).

Comparison with Astronomical Distances

For comparison purposes, note that the sun is about 150 million km (\(1.5 \times 10^{8}\) km) from Earth, making the stack \(\frac{6.022 \times 10^{20}}{1.5 \times 10^{8}} = 4.0147 \times 10^{12}\) times the distance to the sun. The nearest star, Proxima Centauri, is about 40 trillion km from Earth (\(4 \times 10^{13}\) km), so the stack would be \(\frac{6.022 \times 10^{20}}{4 \times 10^{13}} = 15.055\) times the distance to Proxima Centauri.

Key concepts

Mole Concept

Understanding the mole is crucial in chemistry, as it is the bridge between the atomic scale and the one we can measure and observe. The mole is defined as the amount of substance that contains as many entities (atoms, molecules, ions, or other particles) as there are atoms in 12 grams of carbon-12. This number is Avogadro's number, approximately \(6.022 \times 10^{23}\).
In the context of our penny stacking exercise, we consider each penny as a 'particle' and use Avogadro's number to imagine if we had a 'mole' of pennies. The mole concept enables us to measure substances at the atomic level and relate it to macroscopic quantities, which otherwise would be hard to comprehend due to the incredibly small size of atoms and molecules.

Atomic Theory

Atomic theory provides the foundational principles for understanding the chemical behavior of matter. It was first proposed in the early 19th century by John Dalton and has been developed over the years. According to this theory, all matter is made up of tiny, indestructible units called atoms. Each element has unique atoms that differ in size, mass, and properties from those of other elements.
In our exercise, atomic theory is indirectly applied through Avogadro's number—the same number that relates the atoms in a mole of substance. If we think of pennies as 'atoms', stacking Avogadro's number of pennies helps us visualize the vast quantities that atomic theory and chemistry deal with.

Stoichiometry

Stoichiometry involves the calculation of reactants and products in chemical reactions. It is based on the conservation of mass and the concept of the mole. Using stoichiometry, chemists can predict the amounts of substances consumed and produced in a given chemical reaction.
In our penny example, while it's not a chemical reaction, the exercise utilizes similar principles of stoichiometry—multiplying a known number (Avogadro's number) by a fixed proportion (thickness of a penny) to determine an overall magnitude (height of the stack). This helps understand the scale and precision required in stoichiometric calculations.

Astronomical Distances Comparison

The concept of astronomical distances introduces us to the vastness of space. For instance, the sun is approximately 150 million km from Earth, while the nearest star, Proxima Centauri, is about 40 trillion km away. These enormous distances are difficult to imagine.
To put these distances into perspective, we used Avogadro's number to extend a hypothetical stack of pennies from Earth. With a calculated height of \(6.022 \times 10^{20}\) kilometers, this stack dramatically surpasses the distance to the sun and even Proxima Centauri, providing a vivid illustration of both the immeasurable scale of the universe and the enormity of Avogadro's number.