Question

Many substances crystallize in a cubic structure. The unit cell for such crystals is a cube having an edge with a length cqual to \(d_{\circ}\) 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 corner, through the center of the cubc, to the other comer? 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.

Short answer

The length of the face diagonal is \(d_f = d_0\sqrt{2}\) and the length of the cube diagonal is \(d_c = d_0\sqrt{3}\).

Step-by-step solution

Find the length of the face diagonal

To find the length of the face diagonal, we can use the Pythagorean theorem since the diagonal is the hypotenuse of a right triangle formed by two adjacent edges of the cubic unit cell. In this case, each of the edges of the face has length equal to \(d_0\). Let \(d_f\) be the length of the face diagonal. We can write the Pythagorean theorem for the right triangle as follows: \(d_f^2 = d_0^2 + d_0^2\)

Solve for the face diagonal length \(d_f\)

Solving for \(d_f\) from the equation, we get: \(d_f^2 = 2d_0^2\) \(d_f = \sqrt{2d_0^2}\) So, the length of the face diagonal is \(d_f = d_0\sqrt{2}\).

Find the length of the cube diagonal

To find the length of the cube diagonal, we need to use the Pythagorean theorem again, this time using the face diagonal length and one of the edges of the cube as sides of a new right triangle. We can use the hint provided in the exercise: Let \(d_c\) be the cube diagonal, then we can write the Pythagorean theorem for this new right triangle: \(d_c^2 = (d_0\sqrt{2})^2 + d_0^2\)

Solve for the cube diagonal length \(d_c\)

Now, solving for \(d_c\) from the equation, we get: \(d_c^2=d_0^2(2+1)\) \(d_c=\sqrt{3d_0^2}\) So, the length of the cube diagonal is \(d_c = d_0\sqrt{3}\). In conclusion, the length of the face diagonal is \(d_f = d_0\sqrt{2}\) and the length of the cube diagonal is \(d_c = d_0\sqrt{3}\).

Key concepts

Pythagorean Theorem

Understanding the Pythagorean theorem is key to solving many geometrical problems, including those related to crystal structures. The theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. This can be expressed mathematically as:
\[ c^2 = a^2 + b^2 \] where \(c\) is the length of the hypotenuse, and \(a\) and \(b\) are the lengths of the triangle's other two sides.
In the context of cubic crystal systems, the Pythagorean theorem allows us to calculate the length of the face and cube diagonals by considering them as hypotenuses of right-angled triangles formed by the cube's edges and face diagonals, respectively. As demonstrated in the step-by-step solution, using this fundamental theorem effortlessly unlocks the understanding of cube geometry required for crystallography.

Cubic Crystal Systems

Cubic crystal systems are one of the simplest and most symmetrical crystal systems in crystallography. The structure is defined by three axes of equal lengths that are perpendicular to each other. Because of this unique symmetry, the cubic system can be broken down into smaller cubes, known as unit cells. Each side of a unit cell, or the edge length, is often denoted as \(d_0\) in chemistry.

Importance of Edge Length and Diagonals

The edge length is crucial as it is used to calculate other important parameters like the face diagonal and the cube diagonal, which are essential for understanding the full geometry of the crystal. The face diagonal spans the distance between two opposite corners of a square face on the cube, while the cube diagonal connects two opposing corners of the cube itself.
Calculating these diagonals not only supports the study of crystal morphology but also aids in determining the arrangement of atoms within the crystal and the calculation of atomic packing factors, which are vital for material characterization.

Crystallography in Chemistry

Crystallography, a branch of chemistry that studies the arrangement of atoms in crystalline solids, relies heavily on geometric principles like those of the Pythagorean theorem. By examining the unit cell crystal structure, chemists can infer the properties of the material, such as its density and how it will interact with light or electricity.

Understanding Through Geometry

In the case of the cubic crystal system, the geometric understanding provided by the theorem reveals the spatial distribution of atoms and is essential for the calculation of inter-atomic distances. This knowledge is not just academic; it has practical applications in the development of new materials and the exploration of substance properties. For example, knowing the length of the cube diagonal, which might represent the distance between atoms at far ends of the unit cell, can help in the prediction of electronic behavior and bonding characteristics.
In summary, the combination of crystallography and geometry creates a powerful toolkit for chemists and material scientists, enabling them to visualize and understand the intricate lattices that make up the materials of our world.