The density of copper is `9xx10^3 kg m^(-3)` and its atomic mass is 63.5 u. Each copper atom provides one free electron. Estimate the number of free electrons per cubic metre in copper.

Options

A. `10^(19)`
B. `10^(23)`
C. `10^(25)`
D. `10^(28)`

Answer

To estimate the number of free electrons per cubic meter in copper, we can follow these steps:

### Step 1: Understand the relationship between density, mass, and volume
The density (\(\rho\)) of a substance is defined as its mass (\(m\)) per unit volume (\(V\)):
\[ \rho = \frac{m}{V} \]

### Step 2: Calculate the number of moles per unit volume
We can express the mass in terms of the number of moles (\(n\)) and the molar mass (\(M\)):
\[ m = n \cdot M \]
Substituting this into the density equation gives:
\[ \rho = \frac{n \cdot M}{V} \]
Rearranging this, we find the number of moles per unit volume (\(n/V\)):
\[ \frac{n}{V} = \frac{\rho}{M} \]

### Step 3: Substitute the values
Given:
- Density of copper, \(\rho = 9 \times 10^3 \, \text{kg/m}^3\)
- Atomic mass of copper, \(M = 63.5 \, \text{u} = 63.5 \, \text{g/mol} = 0.0635 \, \text{kg/mol}\)

Now, substituting these values into the equation for \(n/V\):
\[ \frac{n}{V} = \frac{9 \times 10^3 \, \text{kg/m}^3}{0.0635 \, \text{kg/mol}} \]

### Step 4: Perform the calculation
Calculating \(n/V\):
\[ \frac{n}{V} = \frac{9 \times 10^3}{0.0635} \approx 141.73 \times 10^3 \, \text{mol/m}^3 \]

### Step 5: Convert moles to number of atoms
To find the number of atoms per cubic meter, we multiply the number of moles per unit volume by Avogadro's number (\(N_A = 6.022 \times 10^{23} \, \text{atoms/mol}\)):
\[ N = \left(\frac{n}{V}\right) \cdot N_A = 141.73 \times 10^3 \, \text{mol/m}^3 \times 6.022 \times 10^{23} \, \text{atoms/mol} \]

### Step 6: Calculate the total number of atoms
Calculating this gives:
\[ N \approx 141.73 \times 10^3 \times 6.022 \times 10^{23} \approx 8.53 \times 10^{28} \, \text{atoms/m}^3 \]

### Step 7: Relate atoms to free electrons
Since each copper atom provides one free electron, the number of free electrons per cubic meter is equal to the number of atoms:
\[ \text{Number of free electrons} = N \approx 8.53 \times 10^{28} \, \text{electrons/m}^3 \]

### Conclusion
The estimated number of free electrons per cubic meter in copper is approximately:
\[ \boxed{8.53 \times 10^{28} \, \text{electrons/m}^3} \]