Simple Cubic Unit Cell: Videos & Practice Problems

Simple Cubic Volume Example

To calculate the volume of a simple cubic unit cell composed of atoms with a radius of 2.5 angstroms, we start by recalling that the volume of a cube is given by the formula:

V = a³

where V is the volume and a is the edge length of the cube. For a simple cubic unit cell, the edge length a is twice the radius of the atom:

a = 2r

Given that the radius r is 2.5 angstroms, we first convert this radius into centimeters. The conversion factor is:

1 \(\text{ angstrom}\) = 10^{-10} \(\text{ meters}\) = 10^{-8} \(\text{ centimeters}\)

Thus, the radius in centimeters is:

r = 2.5 \(\times\) 10^{-8} \(\text{ cm}\)

Now, we can calculate the edge length:

a = 2 \(\times\) r = 2 \(\times\) (2.5 \(\times\) 10^{-8}) = 5.0 \(\times\) 10^{-8} \(\text{ cm}\)

Next, we cube the edge length to find the volume:

V = (5.0 \(\times\) 10^{-8})^3 = 1.25 \(\times\) 10^{-23} \(\text{ cm}\)^3

Therefore, the volume of the simple cubic unit cell is:

V \(\approx\) 1.3 \(\times\) 10^{-22} \(\text{ cm}\)^3

Polonium crystallizes with a primitive cubic structure. It has a density of 9.4 g/cm³, a radius of 167 pm, and a molar mass of 209 g/mol. Calculate the number of atoms in one mole of Polonium.

• A. 6.0×10^-23
• B. 3.2×10¹¹
• C. 6.0×10²³
• D. 1.7×10¹⁵

Answer: (C)

Here's what students ask on this topic:

What is a simple cubic unit cell and how are atoms arranged in it?

A simple cubic unit cell, also known as a primitive cubic unit cell, is a basic crystal structure where atoms are positioned at each of the eight corners of a cube. Each corner atom is shared among eight adjacent unit cells, so only one-eighth of each corner atom belongs to a single unit cell. Therefore, the total number of atoms per simple cubic unit cell is 1 (8 corners × 1/8 atom per corner). There are no atoms located inside the cube's center in this structure. This arrangement results in a relatively low packing efficiency and coordination number compared to more complex cubic structures.

How is the edge length of a simple cubic unit cell related to the atomic radius?

In a simple cubic unit cell, the edge length $a$ is directly related to the atomic radius $r$. Since atoms touch each other along the edges of the cube, the edge length is equal to twice the atomic radius. This relationship can be expressed as $a=2r$. This means that the length of one side of the cube is the sum of the radii of two adjacent atoms touching along that edge.

What is the packing efficiency of a simple cubic unit cell and why is it considered low?

The packing efficiency of a simple cubic unit cell is approximately 52%. Packing efficiency refers to the fraction of volume in the unit cell that is actually occupied by atoms. In the simple cubic structure, atoms are arranged at the corners with significant empty space inside the cube, leading to a low packing efficiency. This means that nearly half of the volume in the unit cell is empty space, making it less efficient in terms of atomic packing compared to other cubic structures like body-centered cubic (68%) or face-centered cubic (74%).

What is the coordination number of a simple cubic unit cell and what does it signify?

The coordination number of a simple cubic unit cell is 6. The coordination number represents the number of nearest neighbor atoms surrounding a given atom. In the simple cubic structure, each atom touches six other atoms—one on each face of the cube along the x, y, and z directions. This is the lowest coordination number among cubic unit cells, reflecting the relatively simple and less densely packed arrangement of atoms in this structure.

How does the simple cubic unit cell compare to body-centered cubic (BCC) in terms of atoms per unit cell and packing efficiency?

The simple cubic unit cell contains 1 atom per unit cell, while the body-centered cubic (BCC) unit cell contains 2 atoms per unit cell—8 corner atoms each contributing 1/8 atom plus 1 atom fully inside the center. The packing efficiency of the simple cubic unit cell is about 52%, which is lower than the 68% packing efficiency of the BCC structure. This difference arises because the BCC has an additional atom at the center, allowing atoms to pack more closely and efficiently. Consequently, BCC has a higher coordination number of 8 compared to 6 in the simple cubic structure, indicating more nearest neighbors per atom.