# Appendices

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.