Question

Copper, a monovalent metal, has molar mass 63.54 g/mol and density $\mathbf{8}\mathbf{.}\mathbf{96}\mathbf{g}\mathbf{/}{\mathbf{cm}}^{\mathbf{3}}$. What is the number density nof conduction electrons in copper?

Short answer

Conduction electrons, in copper, have a number density $n=8.49\times {10}^{28}{\mathrm{m}}^{-3}$.

Step-by-step solution

The given data

1. Molar mass of copper,$A=63.54\mathrm{g}/\mathrm{mol}$
2. Density of copper,$d=8.96\mathrm{g}/{\mathrm{cm}}^3$

Understanding the concept of density and molar mass

The mass of one mole of a substance is called molar mass. To get the mass of the copper atom, we divide the molar mass of copper by the Avogadro number. Now, the number of atoms per unit volume can be calculated by the formula for the density of the substance.

Formulae:

The mass of an atom, $M=A/N_A...............................\left(1\right)$

where $N_A=6.022\times {10}^{23}{\mathrm{mol}}^{-1}$ and A is the molar mass.

The number density of conduction electrons,$n=\frac{d}{M}..................................\left(2\right)$

Where d= density of the atom,M = mass of a single atom

Calculation of the number density of conductions electrons of copper

Since each atom contributes one conduction electron, the number of atoms per unit volume is equal to the number of conduction electrons per unit volume.

Now, the mass of the copper atom can be given using the data of molar mass in equation (1) as follows:

$$\begin{gathered} M=\frac{63.54\mathrm{g}/\mathrm{mol}}{6.022\times {10}^{23}{\mathrm{mol}}^{-1}} \\ =1.055\times {10}^{-22}\mathrm{g} \end{gathered}$$

Thus, the value of the number density of the conductions electrons of the copper atom can be calculated using the given data in equation (2) as follows:

$$\begin{gathered} n=\frac{8.96\mathrm{g}/{\mathrm{cm}}^3}{1.055\times {10}^{-22}\mathrm{g}} \\ =8.49\times {10}^{28}{\mathrm{m}}^{-3} \end{gathered}$$

Hence, the value of the number density of free electrons is $8.49\times {10}^{28}{\mathrm{m}}^{-3}$.