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Chapter 37: Problem 50

You're trying to explain to your classmates how classical and quantum descriptions of electrical conduction in metals differ. Using copper's Fermi energy (7.0 eV), you calculate the associated electron speed, then compare your result with the classical thermal speed for an electron at room temperature \((300 \mathrm{K}) .\) What do you find, and how does this help with your explanation?

Short Answer

Expert verified
The result will indicate that the Fermi speed (quantum) is significantly larger than the classical thermal speed at room temperature. This helps explain that quantum effects are significant at this scale and classical concepts cannot fully account for the behavior of electrons. The Fermi speed represents the speed of the 'fastest' electrons in the metal, while the thermal speed represents an average speed derived from the kinetic energy at room temperature.

Step by step solution

01

Calculate Electron Speed Using Fermi Energy

Fermi energy (\(E_f\)) of an electron in copper is provided as 7.0 eV. We can convert this to Joules using the conversion factor 1 eV = \(1.6 \times 10^{-19}\) Joules. Therefore, Fermi energy \(E_f = 7.0 \times 1.6 \times 10^{-19}\) Joules. The speed of the electron can then be found using the formula: \(v = \sqrt{2E_f / m}\), where \(m\) is the mass of an electron, \(9.11 \times 10^{-31}\) kg.
02

Calculate Thermal Speed at Room Temperature

We can calculate the thermal speed using the formula \(v_thermal = \sqrt{2k_BT/m}\), where \(k_B\) is the Boltzmann constant \((1.38 \times 10^{-23}\) J/K), \(T\) is the temperature in Kelvin and \(m\) is the mass of an electron.
03

Comparing the speeds

Compare the result from the Fermi energy calculation with the result from the classical thermal speed calculation. These results will show the difference between the quantum and classical concepts.

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