Temperature, Pressure, Chemical Potential, and All That

The Maxwell–Boltzmann distribution of momentum is obtained from statistical mechanics. Expressions for the temperature, pressure, and chemical potential are formulated as partial derivatives of the entropy with respect to energy, volume, and particle-number. The temperature scale is derived from comparison with the ideal gas law. The concept of the fundamental relation is defined as an expression that contains all thermodynamic information about the system of interest. Its differential form is introduced. Equations of state contain partial information about the thermal properties of a system and can be expressed as partial derivatives of the fundamental relation. The function of thermometers, pressure gauges, and thermal reservoirs are derived from these principles.