A Pedagogical Reinforcement of the Ideal (Hard Sphere) Gas Using a Lattice Model: From Quantized Volume to Mechanical Equilibrium

Due to their simplicity and ease of visualization, lattice models can be useful to illustrate basic concepts in thermodynamics. The recipe to obtain classical thermodynamic expressions from lattice models is usually based on invoking the thermodynamic limit, and the ideal gas law can easily be obtained as the density of non-interacting particles vanishes. We present a lattice-based analysis that shows that, when a gas consisting of non-interacting particles evolves towards mechanical equilibrium with the environment, the ideal gas law can be obtained with no recourse to unnecessary assumptions regarding the size or particle density of the lattice. We also present a statistical mechanical analysis that considers a quantized volume and reproduces the process obtained for the discrete lattice model. We show how the alternative use of a well-known and accessible model (the non-interacting lattice gas) can give microscopic insights into thermal systems and the assumptions that underlie the laws used to describe them, including local vs. global equilibrium, irreversible processes, and the sometimes subtle difference between physical assumptions and mathematically convenient approximations.