**This section** contains five appendices concerning subjects in which the mathematical complexity can sometimes obscure the physics.

**Appendix 1** is a direct derivation of the Maxwellian speed distribution of a gas using the concepts of *density of states* and *occupation functions.*
**Appendix 2** concerns the derivation of speed of sound formulae and a discussion of the relationship between the different elastic moduli.
**Appendix 3** is a short demonstration that the Gibbs free energy is minimised when a substance is in equilibrium.
**Appendix 4** contains derivations of the formulas for the Einstein and Debye theories of the heat capacity of solids, emphasising their similarities and differences.
**Appendix 5** contains the derivation of a formula for the bulk modulus of a solid consisting of noble gas atoms interacting via a Lennar-Jones potential.

All can be downloaded here:

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