Problem 1 - Chemical Principles in the Laboratory

Question

Many substances crystallize in a cubic structure. The unit cell for such crystals is a cube having an cdge with a length equal to \(d_{\mathrm{O}}\) . a. What is the length, in terms of \(d_{0}\), of the face diagonal, which runs diagonally across one face of the cube? _____________ . b. What is the length, again in terms of \(d_{0}\), of the cube diagonal, which runs from one comer, through the center of the cube, to the other corner? Hint: Make a right triangle having a face diagonal and an edge of the cube as its sides, with the hypotenuse equal to the cube diagonal. _________.

Step-by-Step Solution

Answer

Face diagonal: \(\sqrt{2}d_{0}\), Cube diagonal: \(\sqrt{3}d_{0}\).

Step 1: Understand the Face Diagonal

A face diagonal of a cube is found on one face of the cube. If each side of the cube is denoted by \(d_{0}\), the face diagonal can be calculated using the Pythagorean theorem because it creates a right triangle with two sides that are the edges of the cube.

Step 2: Calculate the Face Diagonal

The face diagonal \(d_{f}\) of a cube can be calculated using the Pythagorean theorem in a 2D plane:\[ d_{f} = \sqrt{d_{0}^2 + d_{0}^2} = \sqrt{2d_{0}^2} = \sqrt{2}d_{0} \]

Step 3: Understand the Cube Diagonal

The cube diagonal runs from one corner of the cube through the center to the opposite corner. To calculate this, we need to consider a right triangle where the sides are \(d_{0}\) (the edge), \(d_{f}\) (the face diagonal), and the hypotenuse is the cube diagonal.

Step 4: Calculate the Cube Diagonal

The cube diagonal \(d_{c}\) can be found using the Pythagorean theorem in 3D:\[ d_{c} = \sqrt{d_{0}^2 + d_{f}^2} = \sqrt{d_{0}^2 + (\sqrt{2}d_{0})^2} \= \sqrt{d_{0}^2 + 2d_{0}^2} = \sqrt{3d_{0}^2} = \sqrt{3}d_{0} \]

Key Concepts

Understanding the Unit Cell

The concept of a unit cell is key in the study of crystal structures. A unit cell is the smallest portion of a crystal lattice that, when repeated in all directions, results in the entire crystal structure. In the case of many crystals, including those crystallizing in a cubic structure, the unit cell takes the shape of a cube.

This cube has equal-length edges which are conventionally denoted by the symbol \(d_0\). By arranging these cubes side by side, you can visualize how they stack up to form the entire crystal lattice. Each unit cell can be thought of as a building block that combines to form the larger crystal.

• The cubic unit cell is characterized by all three of its edges being equal, while the angles between the edges are 90 degrees.
• This symmetry makes cubic structures particularly easy to study and calculate geometrically.

Understanding the dimensions and directions within a unit cell helps in deciphering more complex geometrical and physical properties of the material.

Applying the Pythagorean Theorem

The Pythagorean theorem is a fundamental principle used to derive relations in right triangles. This theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

When dealing with a cubic unit cell, the Pythagorean theorem becomes a powerful tool. Consider the face of a cube, which can be viewed as a square. The diagonal across this face forms a right triangle with the two edges of the cube as the other sides.

To find the length of the face diagonal, use the Pythagorean theorem as follows:

\[\text{Face Diagonal } (d_f) = \sqrt{d_0^2 + d_0^2} = \sqrt{2}d_0\]

Extending this idea further, you can calculate the cube diagonal, which includes the hypotenuse of a right triangle formed by the edge, face diagonal, and cube diagonal:

\[\text{Cube Diagonal } (d_c) = \sqrt{d_0^2 + d_f^2} = \sqrt{3}d_0\]

This showcases how the Pythagorean theorem helps relate different parts of the cube geometrically, which is essential for understanding and solving problems involving crystal structures.

Making Geometric Calculations

Geometric calculations in cubic crystal structures allow us to understand relationships between different parts of the cube. Cubes are inherently symmetrical, but deriving quantitative measurements requires an understanding of geometric principles and formulas.

In our context, geometric calculations are applied to find:

• Face Diagonal: Using the Pythagorean theorem in two dimensions, we can calculate the length of the face diagonal as \(\sqrt{2}d_0\).
• Cube Diagonal: By utilizing the same theorem in three dimensions, the diagonal that spans from one corner of the cube to the diagonally opposite corner can be evaluated as \(\sqrt{3}d_0\).

These calculations are crucial as they reveal how dimensions within a unit cell can be determined systematically. Understanding these distances also has practical implications in material science, such as determining the packing density of atoms within the crystal.

When conducting geometric calculations, visualization of the cube and its elements greatly aids in comprehending the spatial relationships, thereby making problem-solving more intuitive.