Copper, a monovalent, has molar mass 63.54 g/mol and density `8.96 g//cm^3`. What is the number density n of conduction electron in copper?

Options

A. `1.1 xx 10^26 m^(-1)`
B. `2.8 xx 10^25m^(-1)`
C. `8.49 xx 10^28 m^(-1)`
D. None of these

Explanation

To find the number density \( n \) of conduction electrons in copper, we will follow these steps:

### Step 1: Understand the Given Data
- Molar mass of copper \( M = 63.54 \, \text{g/mol} \)
- Density of copper \( \rho = 8.96 \, \text{g/cm}^3 \)
- Avogadro's number \( N_A = 6.02 \times 10^{23} \, \text{atoms/mol} \)

### Step 2: Calculate the Volume of 1 Mole of Copper
Using the formula for density:
\[ \rho = \frac{m}{V} \]
where \( m \) is the mass and \( V \) is the volume. Rearranging gives:
\[ V = \frac{m}{\rho} \]
For 1 mole of copper, the mass \( m = M = 63.54 \, \text{g} \). First, convert the mass into kg:
\[ m = 63.54 \, \text{g} = 0.06354 \, \text{kg} \]
Now calculate the volume:
\[ V = \frac{0.06354 \, \text{kg}}{8.96 \, \text{g/cm}^3} = \frac{0.06354 \, \text{kg}}{8.96 \times 10^{-3} \, \text{kg/cm}^3} = 7.09 \, \text{cm}^3 \]

### Step 3: Calculate the Number of Atoms per Unit Volume
The number of atoms per unit volume \( n' \) can be calculated using:
\[ n' = \frac{N_A}{V} \]
Substituting the values:
\[ n' = \frac{6.02 \times 10^{23} \, \text{atoms/mol}}{7.09 \times 10^{-6} \, \text{m}^3} = 8.49 \times 10^{28} \, \text{atoms/m}^3 \]

### Step 4: Calculate the Number Density of Conduction Electrons
Since copper is a monovalent metal, each atom contributes one conduction electron. Thus, the number density \( n \) of conduction electrons is equal to the number density of atoms:
\[ n = n' = 8.49 \times 10^{28} \, \text{electrons/m}^3 \]

### Final Answer
The number density \( n \) of conduction electrons in copper is:
\[ n = 8.49 \times 10^{28} \, \text{electrons/m}^3 \]

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