. How many unit cells are present in a cubeshaped ideal crystal of NaCl of mass 1 00 g? [Atomic masses: Na = 23, Cl = 35.5]

To determine how many unit cells are present in a cube-shaped ideal crystal of NaCl with a mass of 100 grams, we need to follow a series of logical steps involving the molar mass of NaCl, Avogadro's number, and the structure of the crystal itself. Let's break this down step by step.

Step 1: Calculate the Molar Mass of NaCl

The molar mass of sodium chloride (NaCl) can be calculated by adding the atomic masses of sodium (Na) and chlorine (Cl).

• Atomic mass of Na = 23 g/mol
• Atomic mass of Cl = 35.5 g/mol

Thus, the molar mass of NaCl is:

Molar mass of NaCl = 23 g/mol + 35.5 g/mol = 58.5 g/mol

Step 2: Calculate the Number of Moles in 100 g of NaCl

Next, we need to find out how many moles of NaCl are present in 100 grams. This can be done using the formula:

Number of moles = Mass (g) / Molar mass (g/mol)

Substituting the values:

Number of moles = 100 g / 58.5 g/mol ≈ 1.71 moles

Step 3: Determine the Number of Formula Units

Each mole of NaCl contains Avogadro's number of formula units, which is approximately \(6.022 \times 10^{23}\) units/mol. Therefore, the total number of formula units in 1.71 moles is:

Number of formula units = Number of moles × Avogadro's number

Number of formula units = 1.71 moles × \(6.022 \times 10^{23}\) units/mol ≈ 1.03 \times 10^{24} units

Step 4: Relate Formula Units to Unit Cells

In a face-centered cubic (FCC) structure like that of NaCl, each unit cell contains 4 formula units of NaCl. This is because:

• Each corner atom is shared by 8 unit cells (1/8 of an atom per unit cell).
• Each face-centered atom is shared by 2 unit cells (1/2 of an atom per unit cell).

Thus, the total contribution per unit cell is:

4 NaCl formula units per unit cell

Step 5: Calculate the Number of Unit Cells

To find the total number of unit cells, we divide the total number of formula units by the number of formula units per unit cell:

Number of unit cells = Total formula units / Formula units per unit cell

Number of unit cells = \(1.03 \times 10^{24} \text{ units} / 4 \approx 2.57 \times 10^{23} \text{ unit cells}\)

In summary, a cube-shaped ideal crystal of NaCl with a mass of 100 grams contains approximately \(2.57 \times 10^{23}\) unit cells. This calculation illustrates the relationship between mass, moles, and the microscopic structure of a crystal, providing insight into the fascinating world of solid-state chemistry.