Formula mass of `NaCl` is `58.45g mol^(-1)` and density of its pure form is `2.167g cm^(-3)`. The average distance between adjacent sodium and chloride ions in the crystal is `2.814xx10^(-8)cm`. Calculate Avogadro constant.

Explanation

To calculate Avogadro's constant using the given data for sodium chloride (NaCl), we will follow these steps:

### Step 1: Understand the given data
- Formula mass of NaCl (m) = 58.45 g/mol
- Density of NaCl (D) = 2.167 g/cm³
- Average distance between adjacent sodium and chloride ions (d) = 2.814 x 10^(-8) cm

### Step 2: Calculate the edge length (a) of the unit cell
The average distance between adjacent sodium (Na⁺) and chloride (Cl⁻) ions is given as:
\[ d = R_{Cl^-} + R_{Na^+} \]
Since the distance between adjacent ions is half the edge length of the unit cell:
\[ d = \frac{a}{2} \]
Thus, we can find the edge length (a):
\[ a = 2 \times d = 2 \times 2.814 \times 10^{-8} \, \text{cm} \]
\[ a = 5.628 \times 10^{-8} \, \text{cm} \]

### Step 3: Calculate the volume of the unit cell (V)
The volume of the cubic unit cell is given by:
\[ V = a^3 \]
Substituting the value of a:
\[ V = (5.628 \times 10^{-8} \, \text{cm})^3 \]
\[ V = 1.786 \times 10^{-22} \, \text{cm}^3 \]

### Step 4: Calculate the number of formula units (Z) in the unit cell
In NaCl, each unit cell contains 4 Na⁺ ions and 4 Cl⁻ ions, so:
\[ Z = 4 \]

### Step 5: Use the density formula to find Avogadro's constant (N_A)
The density (D) of the unit cell can be expressed as:
\[ D = \frac{Z \cdot m}{V \cdot N_A} \]
Rearranging this to solve for Avogadro's constant (N_A):
\[ N_A = \frac{Z \cdot m}{D \cdot V} \]
Substituting the known values:
- Z = 4
- m = 58.45 g/mol
- D = 2.167 g/cm³
- V = 1.786 x 10^(-22) cm³

Now substituting these values into the equation:
\[ N_A = \frac{4 \cdot 58.45 \, \text{g/mol}}{2.167 \, \text{g/cm}^3 \cdot 1.786 \times 10^{-22} \, \text{cm}^3} \]

### Step 6: Calculate N_A
Calculating the denominator:
\[ D \cdot V = 2.167 \cdot 1.786 \times 10^{-22} = 3.867 \times 10^{-22} \, \text{g} \]

Now substituting back:
\[ N_A = \frac{233.8}{3.867 \times 10^{-22}} \]
\[ N_A \approx 6.0522 \times 10^{23} \, \text{mol}^{-1} \]

### Final Answer
Thus, Avogadro's constant is approximately:
\[ N_A \approx 6.0522 \times 10^{23} \, \text{mol}^{-1} \]

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