Copper has one conduction electron per atom and a density of$\mathbf{8}\mathbf{.9}\mathbf{\times }{\mathbf{10}}^{\mathbf{3}}\mathbf{}\mathbf{\text{kg}}\mathbf{/}{\mathbf{\text{m}}}^{\mathbf{3}}$. By the criteria of equation$\mathbf{(}\mathbf{9}\mathbf{-}\mathbf{43}\mathbf{)}$, show that at room temperature$\mathbf{(}\mathbf{300}\mathbf{}\mathbf{\text{K}}\mathbf{)}$, the conduction electron gas must be treated as a quantum gas of indistinguishable particles.

We will use the condition for classical behaviour to see thatin copper the conduction electron constitutes a quantum gas-

$\frac{N}{V}\frac{{\hslash }^3}{{\left(m{k}_{B}T\right)}^{\frac{3}{2}}}<<1$

Here,

$N\to$Number of particles

$V\to$Volume

$\hslash \to$Planck's reduced constant

$m\to$Mass of the object

$k_B\to$Boltzmann' constant

$T\to$Temperature.

Converting mass of copper from u to unit

$\begin{array}{c}{m}_A=63.546u\\ =\left(63.546u\right)\left(\frac{1.66\times {10}^{-27}\text{kg}}{1u}\right)\\ =1.055\times {10}^{-25}\text{kg}\end{array}$

$\begin{array}{c}\frac{N}{V}\frac{{\hslash }^3}{{\left(m{k}_{B}T\right)}^{\frac{3}{2}}}<<1\\ \frac{D}{m_A}\frac{{\hslash }^3}{{\left(m{k}_{B}T\right)}^{\frac{3}{2}}}<<1\end{array}$

Substitute$1.38\times {10}^{-23}\text{J}/\text{K}$ for $k_B,8.9\times {10}^3\text{kg}/{\text{m}}^3$for density of copper$\left(D\right)\cdot 1.055\times {10}^{-31}\text{J}.\text{s}$ for $\hslash ,1.055\times {10}^{-25}\text{kg}$for $m_A$and $300\text{K}$for $\text{T}$,

$\begin{array}{c}\frac{D}{m_A}\frac{{\text{h}}^3}{{\left(m{k}_{B}T\right)}^{\frac{3}{2}}}=\frac{(8.9\times {10}^3\text{kg}/{\text{m}}^3)}{(1.055\times {10}^{-25}\text{kg})}\frac{{(1.055\times {10}^{-31}\text{J}.\text{s})}^3}{{\left[(9.1\times {10}^{-25}\text{kg})(1.38\times {10}^{-23}\text{J}/K)\left(300K\right)\right]}^{\frac{3}{2}}}\\ =428.4\end{array}$

Since 428.4 is not less than less than 1, so the statement isn't true. It means that the conduction electrons don't satisfy the conditions to be conditions a classical gas. Therefore, since they aren't considered to be a classical gas, they're considered a quantum gas.