The Ideal Gas Equation

Summary:
This class introduces the Ideal Gas Equation, derived from Boyle-Mariotte’s, Charles’, and Gay-Lussac’s laws, and grounded in the Kinetic Theory of Gases under idealized assumptions, such as non-interacting particles and elastic collisions. It also discusses its limitations under extreme conditions, such as relativistic or quantum gases, highlighting its importance in thermodynamics and its application to various physical systems.

Learning Objectives:
At the end of this class, students will be able to:

1. Understand the Ideal Gas Equation and its formulation based on Boyle-Mariotte’s, Charles’, and Gay-Lussac’s laws.
2. Identify the fundamental relationships between pressure, volume, and temperature in ideal gases.

TABLE OF CONTENTS:
Empirical Formulation of the Ideal Gas
Comments on the Ideal Gas Law
Limitations of the Ideal Gas Model

Empirical Formulation of the Ideal Gas

Experiments conducted on gases show a relationship between the pressure P of a gas contained in a volume V and its temperature T. For example, if the temperature remains constant, it is observed that:

P \propto \dfrac{1}{V}

This result is known as Boyle-Mariotte’s Law. On the other hand, if we work at constant pressure, the following holds:

V \propto T

where T is the temperature measured in Kelvin. This is known as Charles’ Law. Additionally, if we keep the volume constant, the relationship:

P \propto T

is verified. This is called Gay-Lussac’s Law. These three laws can be combined into a single expression, yielding:

PV \propto T

If we consider a gas composed of N particles, the resulting equation is:

\boxed{PV = Nk_B T}

where k_B = 1.3807 \cdot 10^{-23} \, [J \cdot K^{-1}] is the Boltzmann Constant. This formula is related to the famous equation PV = nRT, mnemonically remembered as “Pancho Villa Never Failed Thermodynamics,” where R = 8.314472 \, [J/(mol \cdot K)] is the universal ideal gas constant, and n represents the number of moles.

Comments on the Ideal Gas Law

Although this law was initially presented from an empirical perspective, it can also be derived from first principles through the Kinetic Theory of Gases. In this theory, the gas is modeled as a collection of tiny particles colliding with each other and the container walls. The term “ideal” arises because it relies on certain assumptions:

1. No attractive or repulsive forces at a distance exist between the particles (such as electromagnetic forces).
2. The particles are assumed to be point-like, with negligible size and spherical shape.
3. Collisions between particles and between particles and the container walls are perfectly elastic.

These idealizations are not strictly met in reality, but they simplify calculations and yield useful results that describe gas behavior over a broad range of conditions.

Moreover, the Ideal Gas Equation serves as the foundation of Classical Thermodynamics. Its relevance spans from astrophysics to atmospheric physics and engine analysis, whose study drove the development of thermodynamics. Therefore, the Ideal Gas Equation is fundamental and worth remembering.

It is important to mention that thermodynamics also applies to non-gaseous systems, such as strings, bubbles, or magnets.

Limitations of the Ideal Gas Model

It is necessary to recognize that the Ideal Gas Law has limitations, as it does not adequately describe all gases under all conditions. For example, when gas particles are relativistic (moving at speeds close to the speed of light) or when quantum effects are relevant, the model is no longer valid. It also fails at very low temperatures and high densities, where particles begin to interact significantly, as in liquids and solids.

In these situations, more advanced models, such as quantum gas models or more complex equations of state, must be used.