Worksheet: Free Electron Model of Metals

In this worksheet, we will practice finding the density of a free electron model and electron quantum states of metals and defining metals' Fermi energy and temperature.

Q1:

In the free electron model of metals.

What is the percentage change in the energy between the 𝑛=𝑛=𝑛=4 state and the state with the next higher energy?

What is the percentage change in the energy between the 𝑛=𝑛=𝑛=400 state and the state with the next higher energy?

  • A 2 . 9 × 1 0 %
  • B 4 . 0 × 1 0 %
  • C 3 . 2 × 1 0 %
  • D 3 . 6 × 1 0 %
  • E 4 . 2 × 1 0 %

Q2:

A cube of copper has edges 1.50 mm long. Calculate the number of electron quantum states in this cube whose energies are in the range 3.75 eV to 3.77 eV.

  • A 4 . 2 7 × 1 0
  • B 4 . 5 5 × 1 0
  • C 4 . 1 5 × 1 0
  • D 4 . 4 0 × 1 0
  • E 4 . 7 0 × 1 0

Q3:

Copper at 𝑇=0K has a conduction band electron density of 8.47×10 m−3.

What is the Fermi energy of the copper?

What is the Fermi temperature of the copper?

  • A 8 . 2 0 × 1 0 K
  • B 8 . 3 5 × 1 0 K
  • C 8 . 0 2 × 1 0 K
  • D 8 . 6 5 × 1 0 K
  • E 8 . 4 9 × 1 0 K

Q4:

What is the ratio of the density of states for electrons at 2.7 eV and at 0.27 eV?

Q5:

Find the average energy of an electron in a Zn wire, consisting of one mole of zinc. Use a value of 266.49 pm for the lattice spacing of zinc.

  • A 2 . 5 6 × 1 0 J
  • B 1 . 7 0 × 1 0 J
  • C 1 . 6 1 × 1 0 J
  • D 4 . 2 4 × 1 0 J
  • E 8 . 4 8 × 1 0 J

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