1. The Ideal Gas Model is the entire set of ideas related to gases that scientists regularly use as whenever they reason about phenomena involving gases. Whether gases involved in the particular phenomenon “obey” the ideal gas law or not, the starting point for thinking about gases is the Ideal Gas Model. It embodies the ideas that gases are composed of extremely small particles with a lot of space between them that interact only like perfect billiard balls when they happen to collide with each other.
2. The Ideal Gas Model is so widely use by practically all scientists because on the one hand the mathematical relationships between the variables are straightforward and easy to reason with, and on the other hand, gas properties appearing in the model are empirically meaningful and can be rigorously determined.
where n is the number of moles and R is the gas constant (both macroscopic constructs) and N is the number of molecules and kB is the Boltzmann’s constant (both microscopic constructs).
4. The temperature term forms a bridge between the particle model of thermal energy and macroscopic thermodynamics. The temperature connects directly to the meaning of thermal energy through the fundamental relationship of temperature as a measure of the random thermal motion in any energy mode in thermal equilibrium at temperature T: Ethermal/mode = (1/2)kBT, while at the same time, temperature is seen to be directly proportional to the pressure of a gas.
5. When combined with the particle model of thermal energy, Ethermal (total) = (total number of active modes) ´ (1/2)kBT, the Ideal Gas Model provides substantial insight into what the fundamental properties of an ideal gas are from a PV perspective and how they differ from an internal energy perspective, as well as providing a bridge between the two.
when applied to an ideal gas gives the fundamental result that the molar heat capacity at constant pressure of any ideal gas is greater than the heat capacity at constant volume simply by the additive factor R.
For an ideal gas, the molar heat capacity at constant pressure is larger than at constant volume by exactly the value R. This is true for any ideal gas, whether monatomic, diatomic, or polyatomic, because the ideal gas law doesn’t depend on intra-atomic motions. Looking back at the table of experimentally determined heat capacities at the beginning of this chapter, we see indeed that the molar heat capacity measured at constant pressure is larger than the constant volume heat capacity by one R. Our model of matter does indeed work pretty well for gases.