Calculate the Fermi energy in zinc at 0 K. The density and atomic mass of zinc is 7.14 g/cm3 and 65.38 am3.
The Fermi energy of copper at 0 K is 7.04 eV. What is the average energy of free electrons in copper at 0 K?
The Correct Answer and Explanation is :
Part 1: Calculation of Fermi Energy in Zinc
The Fermi energy ( E_F ) can be estimated using the following formula:
[
E_F = \left( \frac{3}{5} \right) \cdot \left( \frac{3 \pi^2 n}{V} \right)^{2/3}
]
Where:
- ( n ) is the number of free electrons per unit volume (in electrons per cubic meter),
- ( V ) is the volume of a unit cell.
To calculate the Fermi energy of zinc at 0 K, we need to determine the electron concentration ( n ).
- Density of Zinc:
The density ( \rho ) of zinc is given as 7.14 g/cm³, which is equivalent to 7140 kg/m³. - Molar mass and Avogadro’s number:
The atomic mass of zinc is 65.38 amu, or 65.38 g/mol. Using Avogadro’s number ( N_A = 6.022 \times 10^{23} ) atoms/mol, the number of atoms per unit volume ( n_{\text{atoms}} ) is given by:
[
n_{\text{atoms}} = \frac{\rho}{\text{atomic mass}} \times N_A = \frac{7140 \, \text{kg/m³}}{65.38 \, \text{g/mol}} \times \frac{1000}{1} \times 6.022 \times 10^{23} \, \text{atoms/mol}
]
- Electron concentration:
Zinc has 2 free electrons per atom, so the electron concentration ( n_{\text{el}} ) is:
[
n_{\text{el}} = 2 \times n_{\text{atoms}}
]
- Fermi energy formula:
With ( n ) determined, we can use the above formula to find the Fermi energy of zinc.
Part 2: Average Energy of Free Electrons in Copper at 0 K
The average energy ( \langle E \rangle ) of free electrons in a metal at 0 K can be approximated using the following relation:
[
\langle E \rangle = \frac{3}{5} E_F
]
Where ( E_F ) is the Fermi energy. Given that the Fermi energy of copper is 7.04 eV, the average energy is:
[
\langle E \rangle = \frac{3}{5} \times 7.04 \, \text{eV} = 4.22 \, \text{eV}
]
This result shows that, at 0 K, the average energy of free electrons in copper is approximately 4.22 eV.
Explanation
At absolute zero (0 K), the free electrons in a metal fill energy states up to the Fermi level ( E_F ), which is the highest occupied state. The Fermi energy corresponds to the energy of the highest energy electron at 0 K. The average energy of these electrons is related to the Fermi energy because most electrons occupy states near the Fermi energy, and the distribution is symmetric around it. Therefore, the average energy is approximately 60% of the Fermi energy.
This model assumes that the electrons behave as a free electron gas and that interactions between them are neglected. The average energy gives an estimate of how the electrons are distributed within the available energy states at absolute zero.
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