Problem 23 - Chemistry: The Central Science

Question

Which of the three-dimensional primitive lattices has a unit cell where none of the internal angles is \(90^{\circ}\) ? (a) Orthorhombic, (b) hexagonal, (c) rhombohedral, (d) triclinic, (e) both rhombohedral and triclinic.

Step-by-Step Solution

Answer

Based on the analysis of the given lattice types, the short answer to the question is (e) both rhombohedral and triclinic, as these are the two three-dimensional primitive lattices where none of the internal angles is 90 degrees.

Step 1: Recall Characteristics of Given Lattice Types

Here's a basic rundown of each lattice type and their internal angles: (a) Orthorhombic: Internal angles are α = β = γ = 90°. (b) Hexagonal: Internal angles are α = β = 90°, and γ = 120°. (c) Rhombohedral: Internal angles are α = β = γ ≠ 90°. (d) Triclinic: Internal angles are α ≠ β ≠ γ ≠ 90°. Now that we have the characteristics of each lattice type, we can identify the lattice(s) that fit the criteria of having no internal angles equal to 90 degrees.

Step 2: Identify Lattices without 90° Internal Angles

We will go through each options and evaluate whether any of the internal angles is 90 degree or not. (a) Orthorhombic: All internal angles are 90°, so this does not meet the criteria. (b) Hexagonal: Two internal angles are 90°, so this does not meet the criteria. (c) Rhombohedral: No internal angles are 90°, so this meets the criteria. (d) Triclinic: No internal angles are 90°, so this meets the criteria.

Step 3: Conclusion

Considering the conditions provided in the problem, the answer is (e) both rhombohedral and triclinic. These are the two three-dimensional primitive lattices with a unit cell where none of the internal angles is 90 degrees.