At `100^(@)C`, copper (Cu) has FCC unit cell structure will cell edge length of `x Å`. What is the approximate density of Cu (in g `cm^(-3)`) at this temperature? [Atomic mass of Cu = 63.55 u]

Options

A. `(205)/(x^(3))`
B. `(105)/(x^(3))`
C. `(422)/(x^(3))`
D. `(211)/(x^(3))`

Answer

To find the approximate density of copper (Cu) at 100°C with an FCC unit cell structure and edge length of \( x \) Å, we can follow these steps:

### Step 1: Understand the FCC Structure
In a face-centered cubic (FCC) unit cell, there are 4 atoms per unit cell. This is because:
- There are 8 corner atoms, each contributing \( \frac{1}{8} \) of an atom to the unit cell (8 corners × \( \frac{1}{8} \) = 1 atom).
- There are 6 face-centered atoms, each contributing \( \frac{1}{2} \) of an atom to the unit cell (6 faces × \( \frac{1}{2} \) = 3 atoms).
- Total contribution = 1 + 3 = 4 atoms.

### Step 2: Calculate the Mass of the Unit Cell
The mass of the unit cell can be calculated using the formula:
\[ \text{Mass of unit cell} = z \times \text{Atomic mass} \]
Where \( z = 4 \) (number of atoms in FCC) and the atomic mass of Cu = 63.55 u.

\[ \text{Mass of unit cell} = 4 \times 63.55 \, \text{g/mol} = 254.2 \, \text{g/mol} \]

### Step 3: Convert the Mass to Grams per Unit Cell
Since the mass we calculated is in grams per mole, we need to convert it to grams per unit cell using Avogadro's number (\( N_A = 6.022 \times 10^{23} \, \text{mol}^{-1} \)):
\[ \text{Mass per unit cell} = \frac{254.2 \, \text{g/mol}}{6.022 \times 10^{23} \, \text{mol}^{-1}} \approx 4.22 \times 10^{-22} \, \text{g} \]

### Step 4: Convert the Edge Length from Ångstroms to Centimeters
The edge length \( x \) is given in Ångstroms. To convert it to centimeters:
\[ 1 \, \text{Å} = 10^{-8} \, \text{cm} \]
Thus, the edge length in centimeters is:
\[ \text{Edge length} = x \, \text{Å} = x \times 10^{-8} \, \text{cm} \]

### Step 5: Calculate the Volume of the Unit Cell
The volume \( V \) of the unit cell can be calculated using the formula:
\[ V = a^3 \]
Where \( a \) is the edge length. Therefore:
\[ V = (x \times 10^{-8} \, \text{cm})^3 = x^3 \times 10^{-24} \, \text{cm}^3 \]

### Step 6: Calculate the Density
Density \( d \) is given by the formula:
\[ d = \frac{\text{Mass of unit cell}}{\text{Volume of unit cell}} \]
Substituting the values we have:
\[ d = \frac{4.22 \times 10^{-22} \, \text{g}}{x^3 \times 10^{-24} \, \text{cm}^3} \]
This simplifies to:
\[ d = \frac{4.22}{x^3} \, \text{g/cm}^3 \]

### Final Result
Thus, the approximate density of copper at 100°C is:
\[ d \approx \frac{422}{x^3} \, \text{g/cm}^3 \]