Electron standing waves and the confined electron

Overview

Recall questions on why confinement gives discrete energies, the allowed wavelengths and resulting momentum and energy, nodes in higher states, why an atom has a definite size, and the model's limits.

What you need to know

Answer each one before you reveal it.

Questions from this subtopic

Each one with the model answer underneath.

1. 1

   Describe the simple model of an atom as an electron confined in a box.

   Model answer

   The electron is treated as a wave trapped between the walls of a small region, so only those wavelengths that fit a whole number of half-wavelengths into the box are allowed.

2. 2

   Explain why confining an electron leads to a set of discrete energies.

   Model answer

   Only certain standing wave patterns fit inside the confining region, and each allowed wavelength corresponds through p = h/λ to one particular momentum and therefore one particular energy.

3. 3

   Write the expression for the allowed wavelengths of an electron standing wave in a box of length L.

   Model answer

   The allowed wavelengths are λ = 2L/n, where n is a whole number counting the standing wave patterns.

4. 4

   Calculate the wavelength of the lowest-energy standing wave of an electron confined in a box of length 1.0 × 10⁻¹⁰ m.

   Model answer

   With n = 1, λ = 2L = 2 × 1.0 × 10⁻¹⁰ = 2.0 × 10⁻¹⁰ m.

5. 5

   Calculate the momentum of an electron whose de Broglie wavelength is 2.0 × 10⁻¹⁰ m.

   Model answer

   p = h/λ = 6.63 × 10⁻³⁴ ÷ 2.0 × 10⁻¹⁰ = 3.3 × 10⁻²⁴ N s.

6. 6

   Calculate the kinetic energy of an electron of momentum 3.3 × 10⁻²⁴ N s, and express it in electronvolts.

   Model answer

   Ek = p²/2m = (3.3 × 10⁻²⁴)² ÷ (2 × 9.11 × 10⁻³¹) = 6.0 × 10⁻¹⁸ J, which is about 38 eV.

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