Problem 6 - Chemistry and Chemical Reactivity

Question

If a gold atom has a radius of \(145 \mathrm{pm},\) and you could string gold atoms like beads on a thread, how many atoms would you need to have a necklace \(36 \mathrm{cm}\) long?

Step-by-Step Solution

Answer

You need approximately \(1.241 \times 10^9\) gold atoms.

Step 1: Understanding Atom and Necklace Length

To solve this problem, we need to understand that the atom of gold has a radius of 145 picometers (pm), and we want to determine how many such atoms would form a length of 36 centimeters (cm).

Step 2: Conversion of Picometers to Centimeters

First, let's convert the radius from picometers to centimeters. Recall that 1 pm is equal to \(1 \times 10^{-12}\) meters. To work in centimeters, we also convert from meters to centimeters by recognizing that 1 meter is 100 centimeters. Therefore, \(1 \text{ pm} = 1 \times 10^{-12} \text{ m} = 1 \times 10^{-10} \text{ cm}\).

Step 3: Calculate Diameter of a Gold Atom

A gold atom's diameter is twice its radius. Given \(145\ \mathrm{pm}\), the diameter equals \(2 \times 145\ \mathrm{pm} = 290\ \mathrm{pm}\). Converting this to centimeters: \(290 \times 10^{-12} \, \text{m}\) which is \(290 \times 10^{-10} \, \text{cm}\) or \(2.9 \times 10^{-8} \, \text{cm}\).

Step 4: Determine Number of Atoms

To find out how many gold atoms are needed, divide the total length of the necklace by the diameter of one gold atom. So, \(\frac{36 \text{ cm}}{2.9 \times 10^{-8} \text{ cm}}\). This gives \(1.24137 \times 10^9\) atoms approximately.

Key Concepts

Understanding the Gold Atom

Gold atoms are fundamental in understanding the structure of the element gold. The size of a gold atom is minuscule, measured in picometers (pm). Specifically, each gold atom has a radius of 145 pm. To put things into perspective, a picometer is one trillionth of a meter, making it incredibly tiny.
Understanding the atomic radius is key when discussing atomic structures and properties. The atomic radius helps in understanding how atoms come together and form different materials.
Gold is a dense and malleable metal, valued for both its aesthetic beauty and functional properties. Its small atomic size plays a role in these properties, influencing everything from conductivity to how gold bonds with other elements.

Picometer to Centimeter Conversion

When dealing with atomic measurements, like the radius of a gold atom, we often use picometers. However, for practical calculations, converting these measurements to more familiar units like centimeters can be helpful.
A picometer is an extremely tiny unit of length, with 1 pm equal to \(1 \times 10^{-12}\) meters. To convert picometers to centimeters, you'd use the conversion:

• 1 pm = \(1 \times 10^{-12}\) meters
• 1 meter = 100 centimeters

Thus, 1 pm equals \(1 \times 10^{-10}\) centimeters. This conversion is crucial when calculating the size of atoms in larger, more understandable terms.

Atomic Scale Measurements

Atomic scale measurements help us understand the nature and behavior of materials at the tiniest levels. These measurements often use units like picometers or nanometers to describe atomic sizes and distances between atoms.
Traditional units like meters or centimeters are too large to effectively measure atomic dimensions. For example, the diameter of a gold atom is 290 pm. Converting this into centimeters gives about \(2.9 \times 10^{-8}\) cm. This emphasizes how minuscule and precise these atomic scales are.
Understanding these concepts is vital for fields like chemistry and material science, where knowing the behavior of atoms can lead to the development of new materials and technologies.

Necklace Length Calculation

To calculate the number of gold atoms needed to form a necklace, you need to consider the length of the necklace and the size of each atom.
Given the necklace's total length of 36 centimeters and the diameter of a gold atom as \(2.9 \times 10^{-8}\) cm, it becomes a simple division problem. You determine how many atomic diameters fit into the 36 cm length:

• Calculate the number of gold atoms needed: \( \frac{36 \, \text{cm}}{2.9 \times 10^{-8} \, \text{cm}} \)

This calculation yields approximately \(1.24137 \times 10^9\) atoms. This enormous number reflects the incredibly tiny size of an individual atom and illustrates how countless atoms are needed even for a modestly sized object like a necklace.