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

## TL;DR

This paper uses a lattice model to explain the ideal gas law and thermodynamic concepts without relying on traditional assumptions about particle density or size.

## Contribution

A novel lattice-based derivation of the ideal gas law that avoids assumptions about lattice size or density.

## Key findings

- The ideal gas law can be derived from a lattice model without assuming low particle density.
- A statistical mechanical analysis with quantized volume aligns with the lattice model's results.
- The model provides insights into the assumptions behind thermodynamic laws and equilibrium processes.

## Abstract

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.