Question

The diameter of a rubidium atom is \(4.95 \AA\). We will consider two different ways of placing the atoms on a surface. In arrangement \(\mathrm{A}\), all the atoms are lined up with one another. Arrangement \(\mathrm{B}\) is called a close-packed arrangement because the atoms sit in the "depressions" formed by the previous row of atoms:

Short answer

In arrangement A, the rubidium atoms are lined up along a straight line, and the distance between the neighboring atoms is \(4.95 \AA\). In arrangement B, the rubidium atoms form a more compact hexagonal lattice pattern, with a distance between neighboring atoms of approximately \(4.29 \AA\). Therefore, arrangement B is more compact compared to arrangement A and can be considered as the close-packed arrangement for rubidium atoms on a surface.

Step-by-step solution

Understand Arrangement A (atoms lined up)

In arrangement A, the rubidium atoms are simply lined up along a straight line. Since the atoms are positioned in a linear fashion, it is straightforward to calculate the distance between atoms in this arrangement using their diameters.

Distance calculation for Arrangement A

The distance between the centers of the adjacent atoms is equal to the diameter of a single rubidium atom. So the distance between neighboring atoms in Arrangement A is \(4.95 \AA\).

Understand Arrangement B (close-packed)

In arrangement B, the rubidium atoms are situated such that each atom is positioned into the depression created by the previous row of atoms. This creates a more compact arrangement, and eventually forms a hexagonal lattice pattern. To determine the distance between the centers of the atoms in this case, we need to identify the length of the sides of an equilateral triangle formed by three adjacent atoms.

Distance calculation for Arrangement B

In order to find this distance, we can use the fact that an equilateral triangle can be split into two 30-60-90 triangles. The side length (s) of an equilateral triangle equals the diameter of the atom (\(4.95 \AA\)). When split into two 30-60-90 triangles, the longer leg (distance between neighboring atoms) will have length equal to \(s * \frac{\sqrt{3}}{2}\).

Compute the distance for Arrangement B

Using the diameter of a rubidium atom, the distance between neighboring atoms in Arrangement B is \(4.95 * \frac{\sqrt{3}}{2} \approx 4.29 \AA\).

Compare distances in both arrangements.

Comparing the distances between neighboring atoms in both arrangements, we have: - Arrangement A: \(4.95 \AA\) - Arrangement B: \(4.29 \AA\) Thus, arrangement B is more compact compared to arrangement A, as it has lesser distance between neighboring atoms. Therefore, arrangement B can be considered as the close-packed arrangement for rubidium atoms on a surface.

Key concepts

Rubidium Atom

Rubidium is a chemical element with the symbol Rb and atomic number 37. It belongs to the alkali metal group in the periodic table. Rubidium atoms are relatively large, with a diameter of approximately \(4.95 \AA\). This size influences how rubidium atoms can be arranged on a surface.

When rubidium atoms are placed in a single line, similar to pearls on a necklace, each rubidium atom takes up space equal to its diameter. This arrangement is straightforward and serves as a basis to explore more efficient packing techniques, like the close-packed arrangement. Understanding rubidium's size and properties is essential to predict how it interacts in different atomic structures.

Close-packed Arrangement

A close-packed arrangement is a method to fit atoms or spheres together as closely as possible. This leads to a more efficient use of space, reducing the gaps between the particles.

Typically, there are types of close-packed structures: hexagonal close-packed (HCP) and face-centered cubic (FCC), but in a two-dimensional context, we often focus on the formation of a hexagonal lattice.

This arrangement maximizes the number of nearest neighbors, allowing atoms to sit in the depressions of the preceding row. The close-packed structure results in minimized interatomic distances compared to simply lining them up. This is crucial for understanding materials science and how different elements stack together in solid forms.

Hexagonal Lattice Pattern

The hexagonal lattice pattern emerges when atoms are arranged in close-packed structures. This pattern forms because each row of atoms is offset so that each fits into the gaps of the previous one, creating triangles.

An interesting property of this hexagonal pattern is that if you connect the centers of three adjacent atoms, they form an equilateral triangle. Each side of these triangles equals the diameter of one atom. To find the precise distance between the centers of these atoms, a bit of geometry is used: consider an equilateral triangle divided into two 30-60-90 right triangles.

The relationship between the side lengths of these right triangles helps calculate tighter spacing. With rubidium, this calculated distance results in a more compact structure compared to a simple line-up, confirming the efficiency of the hexagonal lattice.