Problem 32 - Chemistry in 30 days

Question

In a cubic crystal, what is the total number of planes of symmetry ? (a) 1 (b) 9 (c) 13 (d) 23

Step-by-Step Solution

Answer

The total number of symmetry planes in a cubic crystal is 13.

Step 1: Understanding Symmetry Planes in Cubic Crystals

Cubic crystals are symmetrical in nature and have multiple planes of symmetry. It's important to start by recognizing the types of planes in a cubic crystal. These include axial planes, diagonal planes, and face-centered planes.

Step 2: Identify the Axial Planes

A cubic crystal has three axial symmetry planes. These are the planes that pass through the centers of opposite faces of the cube. Imagine cutting through the cube from front to back, left to right, and top to bottom.

Step 3: Identify the Diagonal Planes

There are six diagonal planes in a cubic crystal. These planes bisect the angles formed by the cube's edges. You can visualize these planes by considering cuts made through opposite edges, such as from one corner of the cube directly to the opposite corner.

Step 4: Identify the Face-Centered Planes

A cubic crystal has four face-centered planes, which cut through the centers of the opposite corners of the cube, essentially dividing the cube into two equal parts. These are more complex compared to the axial and diagonal planes.

Step 5: Total Count of Symmetry Planes

Add together the numbers of each type of plane: - Axial planes: 3 - Diagonal planes: 6 - Face-centered planes: 4 Thus, the total number of symmetry planes is the sum: 3 + 6 + 4.

Step 6: Conclusion

After adding all the planes: 3 (axial) + 6 (diagonal) + 4 (face-centered) = 13 planes of symmetry. Therefore, the correct answer to the options is (c) 13.

Key Concepts

Axial Symmetry Planes

In cubic crystals, axial symmetry planes are straightforward yet fundamental. These planes slice through the cube parallel to its principal axes. Visualizing helps. Imagine you have a dice (a classic cube shape), and you perform a simple magic trick: cut it in half through its faces.

• First, cut from the front face to the back face.
• Next, slice from the left face to the right face.
• Finally, divide from the top face to the bottom face.

These cuts give you three axial symmetry planes. They are essential because they split the cube into equal-sized mirror image halves, which is the essence of axial symmetry. Axial planes are easy to identify as they correspond directly to the primary axes of the cube.

Diagonal Symmetry Planes

The concept of diagonal symmetry planes in cubic crystals is slightly more complex but equally fascinating. A diagonal plane connects one corner of the cube to the opposite corner, slicing diagonally through the cube itself. In total, there are six diagonal symmetry planes within a cubic crystal.

• To envision these planes, think about drawing a line from the top-front-left corner of the cube to the bottom-back-right corner.
• Continue imagining other diagonal lines, connecting opposite corners passing through the cube’s center.
• Each diagonal line represents a plane that *bisects* the angle between any two axes of the cube.

These planes are crucial because they highlight the cube’s ability to maintain symmetrical properties, despite cuts that aren't aligned with the straightforward axes. Simply put, they demonstrate symmetry along diagonals, enhancing our understanding of crystal structures.

Face-Centered Symmetry Planes

Face-centered symmetry planes are more challenging to picture but are key features in the structure of cubic crystals. Unlike the simpler axial and diagonal planes, these planes cut across the cube's corners, literally passing through the centers of the faces.

• Visualize connecting the midpoints of edges instead of the entire edges themselves.
• These planes split the cube into two mirrored halves through opposing corners.
• In total, a cubic crystal has four face-centered symmetry planes.

The face-centered planes illustrate the cube’s deeper level of symmetry. They do not align merely with the cube’s faces or corners but rather divide the space within, highlighting the intricate balance of crystalline structures. Understanding these subtle, yet powerful planes provides deeper insights into material properties and their symmetry.