Solid-state kinetic models: basics and mathematical fundamentals

The kinetic model of gases explains the properties of a perfect gas using three main assumptions: molecules move in ceaseless random motion, their size is negligible compared to the distances between them, and they do not interact except during perfectly elastic collisions. The total energy of a gas is the sum of the kinetic energies of all its constituent molecules. The pressure exerted by the gas arises from the continual bombardment of the container walls by billions of colliding molecules.

The ideal-gas equation, which is empirical, describes the behavior of gases by establishing relationships between their macroscopic properties. For example, Charles’ law states that volume and temperature are directly related. Gases, therefore, expand when heated at constant pressure. Although gas laws explain how the macroscopic properties change relative to one another, it does not explain the rationale behind it.

Imagine a solid object involved in a general planar movement, with its center of mass pinpointed at a spot labeled G. The object's kinetic energy relative to an arbitrary point A can be quantified for each of its particles - the ith particle in this case. This measurement is achieved through the employment of the relative velocity definition. The position vector, known as rA, extends from point A to the mass element i.

The kinetic molecular theory qualitatively explains the behaviors described by the various gas laws. The postulates of this theory may be applied in a more quantitative fashion to derive these individual laws.

A mole is defined as the amount of any substance that contains as many molecules as there are atoms in exactly 12 grams of carbon-12. An Italian scientist Amedeo Avogadro (1776–1856) formed the  hypothesis that equal volumes of gas at equal pressure and temperature contain equal numbers of molecules, independent of the type of gas. Later, the hypothesis was developed to form the SI unit for measuring the amount of any substance.
The number of molecules in one mole is called Avogadro's number...

Mechanistic models, a category encompassing both physiological and compartmental modeling, differ from empirical models' approaches to incorporating known factors about the systems being modeled. Empirical models describe data with minimal assumptions, while mechanistic models aim to provide a robust description of available data by specifying assumptions and integrating known factors about the system. Compartmental analysis is a key example of a mechanistic model in pharmacokinetics and...