Calculate the Fermi energy and Fermi temperature for copper.
The correct answer and explanation is :
To calculate the Fermi energy and Fermi temperature for copper, we need to use the following relationships:
1. Fermi Energy (E_F):
The Fermi energy is the energy of the highest occupied state at absolute zero temperature. It can be calculated using the formula:
[
E_F = \frac{{\hbar^2 (3\pi^2 n)^{2/3}}}{{2m}}
]
Where:
- (\hbar) is the reduced Planck’s constant (\hbar = 1.055 \times 10^{-34} \, \text{J·s}),
- (n) is the electron number density (electrons per unit volume),
- (m) is the mass of the electron (m = 9.11 \times 10^{-31} \, \text{kg}).
The number density (n) for copper can be calculated from the molar volume and Avogadro’s number.
2. Fermi Temperature (T_F):
The Fermi temperature is related to the Fermi energy by the formula:
[
T_F = \frac{E_F}{k_B}
]
Where:
- (k_B) is the Boltzmann constant (k_B = 1.38 \times 10^{-23} \, \text{J/K}).
Now, let’s go through the calculations.
Step 1: Calculate Electron Density (n)
Copper has one conduction electron per atom, and its atomic mass is 63.55 g/mol. The molar volume (V_m) of copper is 7.1 × 10^-6 m³/mol. The number of electrons per unit volume is given by:
[
n = \frac{N_A}{V_m} = \frac{6.022 \times 10^{23}}{7.1 \times 10^{-6}} = 8.48 \times 10^{28} \, \text{electrons/m}^3
]
Step 2: Calculate Fermi Energy (E_F)
Using the formula for (E_F):
[
E_F = \frac{{(1.055 \times 10^{-34})^2 (3\pi^2 \times 8.48 \times 10^{28})^{2/3}}}{{2 \times 9.11 \times 10^{-31}}}
]
After performing the calculation, the Fermi energy (E_F) is approximately:
[
E_F \approx 7.0 \, \text{eV}
]
Step 3: Calculate Fermi Temperature (T_F)
Using the relationship between Fermi energy and Fermi temperature:
[
T_F = \frac{7.0 \, \text{eV}}{8.617 \times 10^{-5} \, \text{eV/K}} \approx 81,100 \, \text{K}
]
Conclusion:
- Fermi Energy of Copper: (E_F \approx 7.0 \, \text{eV})
- Fermi Temperature of Copper: (T_F \approx 81,100 \, \text{K})
These results indicate that at room temperature, copper is well below its Fermi temperature, meaning most of its electrons are in the lowest energy states, and quantum mechanical effects dominate its behavior.