Question

Use the result of Problem 23 to calculate the total translational kinetic energy of the conduction electrons in $\mathbf{1}\mathbf{.}\mathbf{00}\mathit{c}{\mathit{m}}^{\mathbf{3}}$ of copper at T = 0K.

Short answer

The total translational kinetic energy of the conduction electrons is $57.1\mathrm{KJ}$.

Step-by-step solution

The given data

1. The number of conduction electrons per unit volume of copper,$n=8.43\times {10}^{28}{m}^{-3}$
2. Fermi energy of copper,$E_F=7.00eV$
3. Volume of copper,$V=1.00c{m}^3=1.00\times {10}^{-6}{m}^3$
4. Temperature,$T=0K$

Understanding the concept of translational kinetic energy

The energy content of a body due to its motion is known as kinetic energy. The total work done in moving an object from one point to another is equal to the change in kinetic energy of the substance.

Formulae:

The average energy of conduction electrons,

$E_{avg}=\frac{3}{5}{E}_F............................\left(1\right)$

The total energy of the conduction electrons in a volume,

$$\begin{gathered} E_{avg}=N{E}_{avg} \\ =n{V}_{avg}............................\left(2\right) \end{gathered}$$

Calculation of the translational kinetic energy

Using the given data and substituting the value of average energy of equation in equation, we can get the value of the translational kinetic energy as follows:

$\begin{array}{l}E=nV\left(\frac{3}{5}{E}_F\right)\\ =\left(8.43\times {10}^{28}{m}^{-3}\right)\left(1.00\times {10}^{-6}{m}^{-3}\right)\left(\frac{3}{5}\left(7eV\right)\left(1.6\times {10}^{-19}J/eV\right)\right)\\ =5.71\times {10}^{4}J\\ =5.71kJ\end{array}$

Hence, the value of the energy is 57.1 kJ .