Problem 72 - Chemistry Central Science

Question

Cadmium telluride, CdTe, takes the zinc blende structure (Section 11.8) with a unit cell edge length of \(6.49 \AA\). There are four cadmium atoms and four tellurium atoms per unit cell. How many of each type of atom are there in a cubic crystal with an edge length of \(5 \mathrm{~nm}\) ?

Step-by-Step Solution

Answer

In a cubic crystal of Cadmium telluride (CdTe) with an edge length of 5 nm, there are 2048 Cadmium atoms and 2048 Tellurium atoms.

Step 1: Convert edge length of unit cell to nm

Convert the given edge length of the unit cell from angstroms (Å) to nanometers (nm) in order to match the given edge length of the cubic crystal. The edge length of the unit cell is given as \(6.49 \AA\). Since \(1 \AA = 0.1 nm\), we can convert the edge length of the unit cell: Edge length = \(6.49 \AA \times 0.1\) = 0.649 nm

Step 2: Calculate the number of unit cells in the cubic crystal

Now, we need to find the total number of unit cells in the cubic crystal. Separate the cubic crystal into unit cells and count them. Since the cubic crystal has an edge length of 5 nm: Number of unit cells along one edge = $$\frac{\text{edge length of cubic crystal}}{\text{edge length of unit cell}}$$ = $$\frac{5 \,\text{nm}}{0.649 \,\text{nm}}$$ = 7.71 ≈ 8 Since it's a cubic crystal, the number of unit cells is the same along all edges. So, the total number of unit cells in the cubic crystal would be: Total unit cells = $$8 \times 8 \times 8$$ = 512

Step 3: Calculate the number of cadmium and tellurium atoms in the cubic crystal

We are given that there are 4 Cadmium and 4 Tellurium atoms per unit cell. So, we can calculate the total number of Cadmium and Tellurium atoms by multiplying the total number of unit cells in the cubic crystal with the number of atoms per unit cell. Total Cadmium atoms = $$\text{total unit cells} \times \text{Cadmium atoms per unit cell}$$ = $$512 \times 4$$ = 2048 Total Tellurium atoms = $$\text{total unit cells} \times \text{Tellurium atoms per unit cell}$$ = $$512 \times 4$$ = 2048 Thus, in a cubic crystal of Cadmium telluride (CdTe) with an edge length of 5 nm, there are 2048 Cadmium atoms and 2048 Tellurium atoms.