If copper, density = 9.0 g/cm3 and atomic mass 63.5, bears face-centered unit cells then what is the ratio of surface area to volume of each copper atom?

Options

A. 0.0028
B. 0.0235
C. 0.0011
D. 0.0323

Answer

To find the ratio of surface area to volume of each copper atom, we can follow these steps:

### Step 1: Understand the given data
- Density of copper (ρ) = 9.0 g/cm³
- Atomic mass of copper (M) = 63.5 g/mol
- For face-centered cubic (FCC) structure, the number of atoms per unit cell (Z) = 4
- Avogadro's number (Nₐ) = 6.02 x 10²³ atoms/mol

### Step 2: Calculate the volume of the unit cell (A³)
The formula for density is given by:
\[ \rho = \frac{Z \cdot M}{A^3 \cdot N_a} \]
Rearranging this formula to find \(A^3\):
\[ A^3 = \frac{Z \cdot M}{\rho \cdot N_a} \]

### Step 3: Substitute the values into the equation
Substituting the known values:
\[ A^3 = \frac{4 \cdot 63.5 \, \text{g/mol}}{9.0 \, \text{g/cm}^3 \cdot 6.02 \times 10^{23} \, \text{atoms/mol}} \]

### Step 4: Calculate \(A^3\)
Calculating the numerator:
\[ 4 \cdot 63.5 = 254 \, \text{g/mol} \]
Calculating the denominator:
\[ 9.0 \cdot 6.02 \times 10^{23} = 5.418 \times 10^{24} \, \text{g/cm}^3 \]
Now substituting these values:
\[ A^3 = \frac{254}{5.418 \times 10^{24}} \approx 4.68 \times 10^{-23} \, \text{cm}^3 \]

### Step 5: Calculate \(A\)
Taking the cube root to find \(A\):
\[ A = (4.68 \times 10^{-23})^{1/3} \approx 3.60 \times 10^{-8} \, \text{cm} \approx 360.5 \, \text{pm} \]

### Step 6: Calculate the radius \(R\) of the copper atom
For FCC, the relationship between edge length \(A\) and atomic radius \(R\) is:
\[ A = 2\sqrt{2}R \]
Rearranging for \(R\):
\[ R = \frac{A}{2\sqrt{2}} = \frac{360.5 \, \text{pm}}{2\sqrt{2}} \approx 127.46 \, \text{pm} \]

### Step 7: Calculate the surface area and volume of the atom
- Surface area \(S\) of a sphere:
\[ S = 4\pi R^2 \]
- Volume \(V\) of a sphere:
\[ V = \frac{4}{3}\pi R^3 \]

### Step 8: Calculate the ratio of surface area to volume
The ratio of surface area to volume is:
\[ \text{Ratio} = \frac{S}{V} = \frac{4\pi R^2}{\frac{4}{3}\pi R^3} = \frac{3}{R} \]

### Step 9: Substitute the value of \(R\)
Substituting \(R = 127.46 \, \text{pm}\):
\[ \text{Ratio} = \frac{3}{127.46 \times 10^{-10} \, \text{m}} \approx 0.0235 \, \text{cm}^{-1} \]

### Final Answer
The ratio of surface area to volume of each copper atom is approximately **0.0235**.

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