# Atomic models

Quantitative models can be used to calculate approximate electron energies. Electron energies and angular momenta in an atom are quantised.

Key Concepts

Bohr model

The Bohr model predicts the hydrogen spectrum by applying the equations for circular motion for electron orbits that have quantised angular momentum.

The energy of an electron in an energy level is given by:

$$E=-{13.6\over n^2}$$

• $$E$$ is the energy associated with the $$n^{th}$$ energy level (eV)
• $$n$$ is the number of the energy level

The consequence is that, moving out from the nucleus, orbits get further apart but the energy difference gets less.

Hence, the energy of an emitted photon when an electron moves down between energy levels is:

$$E = 13.6({1\over {n_f}^2} - {1\over {n_i}^2})$$

Electron in a box model

A helpful model for understanding the probability of an electron's position is a standing wave. Just as an electron cannot move outside an atom, a wave cannot move from a string clamped at both ends. A standing wave may only oscillate at particular harmonic frequencies; an electron may only have particular discrete energies.

Substituting into the de Broglie wavelength equation, $$\lambda = {h\over mv}$$, we can find the kinetic energy, $$E_k={1\over 2}mv^2$$:

$$\Rightarrow v={h\over m\lambda}=\sqrt{{2E_k\over m}}$$

$$E_k={h^2\over 2\lambda^2m}$$

The possible wavelengths for a clamped string of length $$L$$ are $$2L, {2L\over 2}, {2L\over 3}... {2L\over n}$$ etc. For the $$n^{th}$$ harmonic:

$$E_k={n^2h^2\over 8L^2m}$$

This 'electron in a box' model predicts discrete energy levels.

Essentials

Quantised properties

#### Angular momentum

In the Bohr model, angular momentum is quantised. We commence by:

1. Recalling the de Broglie wavelength of an electron, $$\lambda={h\over mv}$$
2. Using the standing wave condition that a circumference of an orbit must be equal to a whole number of wavelengths, $$2\pi r=n\lambda_n$$

Combining these conditions:

$$\Rightarrow 2\pi r=n{h\over mv}$$

$$mvr={nh\over 2\pi}$$

• $$mvr$$ is angular momentum (kgm2s-2)
• $$n$$ is an integer equal to 1 minus the quantum number
• $$h$$ is Planck's constant (Js)

This calculation reveals the flaw in the Bohr model. We cannot know both the position and momentum of a particle.

#### Spin

Observations from the Stern-Gerlach experiment demonstrate that electron spin is another quantised property. Electron spin can take one of two values.

Electron tunnelling

Tunnelling is the term given to a particle passing through a potential energy barrier without having sufficient energy to surmount the barrier. It is a conceptually similar phenomenon to a ball rolling through a physical tunnel in a hill, rather than being given an energetic kick to climb over. But in quantum tunnelling barriers, no such physical tunnel exists.

Instead, the continuous nature wave function of the particle means that there is a finite probability of the particle's position being on the far side of the potential barrier.

Many factors affect the likelihood of tunnelling. Those which would reduce the likelihood of tunnelling include:

• Increasing the width of the barrier
• Increasing the mass of the particle
• Increasing the energy deficit between the particle and the barrier energy potential

Test Yourself

Use quizzes to practise application of theory.