Question

Advances in the field of nanotechnology have made it possible to construct chains of single metal atoms linked one to the next. Physicists are particularly interested in the ability of such chains to conduct electricity with little resistance. Estimate how many gold atoms would be required to make such a chain long enough to wear as a necklace. How many would be required to make a chain that encircled the Earth? If 1 mole of a substance is equivalent to roughly \(6.022 \cdot 10^{23}\) atoms, how many moles of gold are required for each necklace?

Short answer

Additionally, how many moles of gold are required for each? Answer: For the 45 cm necklace, around 1.56 x 10^9 gold atoms and 2.59 x 10^-15 moles of gold are required. For the chain that encircles the Earth, approximately 1.39 x 10^17 gold atoms and 2.31 x 10^-7 moles of gold are needed.

Step-by-step solution

1. Determine the length of the gold chain for each scenario

For the first scenario, let's assume a necklace has a length of 45 cm (18 inches). Therefore, the chain's length for a necklace would be 0.45 meters. To determine the length of a chain that encircles the Earth, we need to use the Earth's circumference, which is approximately 40,000 km or 40,000,000 meters.

2. Find the distance between two gold atoms

Gold has a crystalline structure with a face-centered cubic (FCC) unit cell. The distance between two adjacent gold atoms, also known as the atomic radius, can be calculated using the lattice parameter (the edge length) of the FCC unit cell. The lattice parameter of gold is 4.08 angstroms. In FCC, the distance between the two adjacent atoms is calculated as \(a \cdot \sqrt{2}/2\) where a is the lattice parameter. Converting the lattice parameter to meters: \(4.08 \cdot 10^{-10} m\), the distance between two gold atoms would be approximately \(2.88 \cdot 10^{-10} m\).

3. Calculate the number of gold atoms required

To find the number of gold atoms required for each chain, divide the length of the chain by the distance between two gold atoms: For the necklace: \(\frac{0.45}{2.88 \cdot 10^{-10}} = 1.56 \cdot 10^{9}\) gold atoms. For the chain that encircles the Earth: \(\frac{4.0 \cdot 10^{7}}{2.88 \cdot 10^{-10}} = 1.39 \cdot 10^{17}\) gold atoms.

4. Calculate the number of moles of gold

To determine the number of moles required for each chain, divide the number of gold atoms by Avogadro's number, which is \(6.022 \cdot 10^{23}\) atoms per mole: For the necklace: \(\frac{1.56 \cdot 10^{9}}{6.022 \cdot 10^{23}} = 2.59 \cdot 10^{-15}\) moles. For the chain that encircles the Earth: \(\frac{1.39 \cdot 10^{17}}{6.022 \cdot 10^{23}} = 2.31 \cdot 10^{-7}\) moles.

Key concepts

Electric Conductivity

Imagine if electricity could flow through a material as effortlessly as water through a pipe. This is the essence of electric conductivity, a fundamental property that describes how well a material can transmit electric current.

For example, the nanotechnology gold chains mentioned in the exercise are of interest due to their potential to conduct electricity with minimal resistance. The atoms in these chains are meticulously organized, allowing electrons to move freely across the structure, minimizing energy loss and overheating.

Electric conductivity is measured in siemens per meter (S/m). The higher the conductivity, the better the material is at transmitting electric current. Gold is known for its excellent conductivity, making it an ideal material for creating efficient electrical connectors in nano-scale devices.

Atomic Radius Calculation

The size of an atom is characterized by its atomic radius, a measure from the center of the nucleus to the outer boundary of the electron cloud. Knowing the atomic radius is key to calculating the length of chains like the hypothetical gold necklace.

The step-by-step solution provided uses the lattice parameter to determine the atomic radius. In a more direct approach, the radius can be calculated by examining the geometry of gold's face-centered cubic (FCC) lattice. By knowing the relation between the lattice parameter and the radius, we can establish that for FCC, the atomic radius is defined by the equation \( r = \frac{a \cdot \sqrt{2}}{4} \), where \( a \) is the lattice parameter.

Such calculations are integral to designing materials at the nanoscale, where precision is indispensable.

Lattice Parameter

Diving deeper into the structure of a solid, the lattice parameter becomes an important concept. It is the physical dimension of unit cells in a crystal lattice—the repeatable pattern that atoms are arranged in a solid.

In the context of the exercise, gold's lattice parameter is given as 4.08 angstroms, which is equivalent to 4.08 x 10^-10 meters. With a crystalline structure (FCC), each gold atom is shared among the corners of the unit cell with the neighbors, and this parameter helps determine the precise arrangement of atoms within the lattice.

Understanding the lattice parameter helps materials scientists and nanotechnology experts predict material properties like density, thermal expansion, and, yes, electric conductivity.

Avogadro's Number

When measuring substances at the atomic level, we rely on Avogadro's number—a constant that tells us the number of constituent particles, usually atoms or molecules, in one mole of a given substance. This colossal number is roughly 6.022 x 10²³.

In our exercise, Avogadro's number provides a bridge between the macroscopic world, where we wear necklaces and live on Earth, and the atomic world, with its incomprehensibly vast numbers of gold atoms. By using Avogadro's number, we can estimate the amount of substance, in moles, needed to construct the nanoscale gold chains.

This constant is a cornerstone of chemistry and material science, vital for converting between atoms/molecules and moles, aiding in the understanding of reactions, and the design of materials at the molecular level.