In copper, the Fermi energy of conduction electrons is 7.0 eV. What is the speed of the conduction electrons around this energy?
The correct answer and explanation is :
To find the speed of conduction electrons in copper with a given Fermi energy of 7.0 eV, we need to use the relationship between the Fermi energy and the velocity of the electrons.
Step-by-step Solution:
- Fermi Energy and Electron’s Kinetic Energy:
The Fermi energy represents the energy of the highest occupied electron state at absolute zero temperature. The kinetic energy (E_k) of an electron can be expressed using the following formula: [
E_k = \frac{1}{2} m v^2
] where:
- (E_k) is the kinetic energy (which, at absolute zero, equals the Fermi energy, (E_F)),
- (m) is the mass of the electron ((9.11 \times 10^{-31} \, \text{kg})),
- (v) is the speed of the electron.
- Fermi Energy and Its Conversion to Joules:
The given Fermi energy is 7.0 eV. First, we need to convert the energy from electron volts (eV) to joules (J), knowing that: [
1 \, \text{eV} = 1.602 \times 10^{-19} \, \text{J}
] Thus, the Fermi energy in joules is: [
E_F = 7.0 \, \text{eV} \times (1.602 \times 10^{-19} \, \text{J/eV}) = 1.1214 \times 10^{-18} \, \text{J}
] - Solving for the Speed (v):
From the equation (E_k = \frac{1}{2} m v^2), we can solve for (v): [
v = \sqrt{\frac{2E_k}{m}}
] Substituting (E_k = E_F = 1.1214 \times 10^{-18} \, \text{J}) and (m = 9.11 \times 10^{-31} \, \text{kg}): [
v = \sqrt{\frac{2 \times 1.1214 \times 10^{-18}}{9.11 \times 10^{-31}}}
] Simplifying: [
v = \sqrt{2.462 \times 10^{12}} = 1.57 \times 10^6 \, \text{m/s}
]
Final Answer:
The speed of the conduction electrons around the Fermi energy of 7.0 eV is approximately (1.57 \times 10^6 \, \text{m/s}).
Explanation:
At absolute zero temperature, the Fermi energy represents the energy of the highest energy electrons in a metal. These electrons move at high speeds, and the Fermi velocity corresponds to the speed of these electrons. The relationship between the Fermi energy and the speed comes from the fact that, at high energy, the electrons behave like free particles. The formula (E_k = \frac{1}{2} m v^2) describes their kinetic energy, and we can use this relationship to solve for the velocity. The Fermi speed is essential in understanding the behavior of conduction electrons in metals, as it dictates the distribution of electron velocities at low temperatures.