To calculate the density of a diamond crystal, we'll use the following formula:

$\text{Density} (\rho) = \frac{Z \times M}{a^3 \times N_A}$

where:

• $Z$ is the number of atoms per unit cell (which is 8 for diamond),
• $M$ is the atomic mass of carbon (which is 12 amu),
• $a$ is the lattice parameter (which is 3.57 Å),
• $N_A$ is Avogadro's number ($6.022 \times 10^{23}$ atoms/mol).

First, let's convert the lattice parameter from angstroms to centimeters:

$a = 3.57 \text{ Å} = 3.57 \times 10^{-8} \text{ cm}$

Now we can calculate the density:

$\rho = \frac{8 \times 12 \text{ g/mol}}{(3.57 \times 10^{-8} \text{ cm})^3 \times 6.022 \times 10^{23} \text{ atoms/mol}}$

Let's compute this step by step.

Step 1: Calculate the volume of the unit cell

$a^3 = (3.57 \times 10^{-8} \text{ cm})^3 = 4.546 \times 10^{-23} \text{ cm}^3$

Step 2: Calculate the density

$\rho = \frac{96 \text{ g/mol}}{4.546 \times 10^{-23} \text{ cm}^3 \times 6.022 \times 10^{23} \text{ atoms/mol}}$

$\rho = \frac{96 \text{ g/mol}}{2.737 \times 10^{1} \text{ g/mol cm}^3}$

$\rho = 3.51 \text{ g/cm}^3$

So, the density of diamond is approximately $3.51 \text{ g/cm}^3$.

Would you like more details or have any questions?

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Here are 5 related questions:

1. How does the density of diamond compare to other allotropes of carbon?
2. What factors affect the lattice parameter of a crystal structure?
3. Can you calculate the density of another crystal using the same method?
4. How does temperature affect the density of diamond?
5. How is Avogadro's number determined experimentally?

Tip: Always ensure units are consistent when calculating physical quantities like density.