The serious scientific application of the atomic theory began in the eighteenth century, with calculations of the properties of gases, which had been studied experimentally since the century before. This is the topic with which we begin this chapter. Applications to chemistry and electrolysis followed in the nineteenth century, and are considered in subsequent sections. The final section of this chapter describes how the nature of atoms began to be clarified with the discovery of the electron.

The canonical ensemble describes systems which can exchange energy with their surroundings, which may be modelled as a heat bath.The statistical mechanical quantity that characterizes systems in the canonical ensemble is the partition function, which is shown to be related to the Helmholtz free energy.The connections between statistical mechanics and the laws of thermodynamics are discussed.The application of the canonical ensemble is illustrated through a variety of examples: two-level systems, the quantum and classical simple harmonic oscillator, rigid rotors and a particle in a box.The differences in the statistical properties of distinguishable and indistinguishable particles are considered and used to derive the thermodynamic properties of ideal and non-ideal gases, including the ideal gas equation, the Sackur--Tetrode equation and the Van der Waals equation.The chapter concludes with a discussion of the equipartition theorem and its application to the Dulong--Petit Law.

Kinetic theory is a framework for calculating macroscopic physical properties of systems from their microscopic degrees of freedom.This idea is applied to an ideal gas to derive the Maxwell--Boltzmann velocity distribution, which is demonstrated to be compatible with the ideal gas law and is used to calculate the rate of effusion of an ideal gas.When molecular collisions are important, the mean free path and collision time are quantities that can characterize these collisions.Situations in which collisions are important, such as Brownian motion and diffusion, are presented, along with relevant equations: the Langevin equation and Fick's Law.

The grand canonical ensemble applies to open systems that can exchange both energy and particles with their environment.The grand canonical partition function and its relation to the grand potential are derived, with an emphasis on the chemical potential.Examples in which the grand canonical ensemble applies are presented, including two-level systems, Langmuir adsorption isotherms, chemical equilibrium and the law of mass action.