Section 39–1 Properties of matter

Idealizations / Approximations / Limitations

 

Though titled Properties of Matter, this section might be more fittingly named Introduction to Thermodynamics, as Feynman briefly discusses the theory’s foundational idealizations, mathematical approximations, and intrinsic limitations. To appreciate the power and elegance of thermodynamics, one might recall Einstein’s often-quoted words:

 

“A theory is the more impressive the greater the simplicity of its premises, the more different kinds of things it relates, and the more extended its area of applicability. Therefore the deep impression that classical thermodynamics made upon me. It is the only physical theory of universal content which I am convinced will never be overthrown, within the framework of applicability of its basic concepts.”

 

From a modern standpoint, Einstein’s admiration—though justified for classical systems— overlooks an inherent limitation. Here lies classical thermodynamics’ subtle paradox: its strength is its wide generality, yet that same generality narrow its power. While it offers a small number of universal principles applicable to diverse physical systems, it cannot capture their richer complexities—thermal fluctuations, quantized energy exchange, or nonequilibrium dynamics. However, Feynman’s discussion serves as a valuable entry point into thermodynamics and it prompts questions on how the classical theory must adapt to remain relevant in contemporary physics.

 

1. Idealizations: The Foundations of Thermodynamics

“It is the first part of the analysis of the properties of matter from the physical point of view, in which, recognizing that matter is made out of a great many atoms, or elementary parts, which interact electrically and obey the laws of mechanics, we try to understand why various aggregates of atoms behave the way they do (Feynman et al., 1963, p. 39-1).”

 

According to Feynman, analyzing the properties of matter from a physical point of view requires at least three key idealizations. They form the foundation of the kinetic theory of gases, which bridges microscopic particle dynamics with macroscopic observables. The key idealizations are:

1. Atomic Model of Matter – Gas molecules are modeled as a large number of randomly moving particles, enabling a statistical treatment of their behavior.

2. Newton’s Laws of motion – They govern particle collisions (momentum-conserving), e.g., linking microscopic motion to pressure, via momentum transfer to container walls.

3. Electrostatic Interactions – Intermolecular forces are assumed to be predominantly Coulombic, while gravitational and magnetic effects are negligible. In certain scenarios, Coulomb forces may be ignored, simplifying the distribution of energy (or velocity).

These idealizations allow for a simplified but insightful framework, especially when applied to macroscopically homogeneous, isotropic, and uncharged systems (Callen, 1985).

 

“For instance, when we compress something, it heats; if we heat it, it expands. There is a relationship between these two facts which can be deduced independently of the machinery underneath. This subject is called thermodynamics (Feynman et al., 1963, p. 39-2).”

 

Feynman’s description of thermodynamics emphasizes the interplay among heat and work, and the physical properties of matter. More broadly, thermodynamics is the branch of physics that explores how energy—particularly in the forms of heat and work—relates to state variables such as temperature, pressure, and entropy, thereby determining the behavior of physical systems. The term thermodynamics—derived from the Greek therme (heat) and dynamis (power)—means “heat in motion” or “heat power.” This name can be somewhat misleading: classical thermodynamics primarily addresses equilibrium states, which are static or time-independent, rather than dynamic processes (Atkins & de Paula, 2010). Despite its wide-ranging applicability, thermodynamics is a phenomenological theory—its laws are grounded in empirical observation, instead of derived from microscopic principles. Kinetic theory, by contrast, provides a statistical foundation for thermodynamic behavior and bridges the gap to statistical mechanics.

 

“We shall also find that the subject can be attacked from a nonatomic point of view, and that there are many interrelationships of the properties of substances… The deepest understanding of thermodynamics comes, of course, from understanding the actual machinery underneath, and that is what we shall do: we shall take the atomic viewpoint from the beginning and use it to understand the various properties of matter and the laws of thermodynamics (Feynman et al., 1963, p. 39-2).”

 

An automotive analogy may help clarify the distinctions among thermodynamics, kinetic theory, and statistical mechanics:

• Thermodynamics is like reading a car’s speedometer—it offers a macroscopic description (nonatomic viewpoint) based on observable quantities, without reference to microscopic mechanisms.
• Kinetic theory is akin to analyzing the engine’s revolutions per minute (RPM) to explain the car’s speed—it adopts a microscopic perspective (atomic viewpoint) to understand macroscopic behavior in terms of particle motion.
• Statistical mechanics is like the general theory of engines—it provides a unifying framework that applies probabilistic principles to a wide range of systems, but not limited to gases.

Note that Feynman’s terms ‘nonatomic point of view’ and ‘atomic viewpoint’ correspond to what is commonly known as macroscopic and microscopic perspectives respectively.

 

2. Approximations: From Microscopic Chaos to Macroscopic Order

“Anyone who wants to analyze the properties of matter in a real problem might want to start by writing down the fundamental equations and then try to solve them mathematically. Although there are people who try to use such an approach, these people are the failures in this field; the real successes come to those who start from a physical point of view, people who have a rough idea where they are going and then begin by making the right kind of approximations, knowing what is big and what is small in a given complicated situation (Feynman et al., 1963, p. 39-2).”

 

Thermodynamics describes macroscopic properties—such as temperature and entropy—without invoking the microscopic details of matter. Statistical mechanics, in contrast, provides a deeper foundation by using probability theory and statistical methods to connect the microscopic behavior of particles to these observable thermodynamic quantities. Since tracking the motion of every individual particle is practically impossible, statistical approaches serve as a bridge between microscopic disorder (e.g., particle velocities and collisions) and macroscopic regularity (e.g., temperature and pressure). For example, the ideal gas law models gases as collections of non-interacting point particles, while the Van der Waals equation incorporates molecular size and intermolecular forces, offering a more realistic description of real gases. In this context, Feynman’s remark about “knowing what is big and what is small” can be interpreted not only in terms of physical size, but as an invitation to identify which variables significantly affect a system’s macroscopic behavior and which can be safely ignored.

 

“As an interesting example, we all know that equal volumes of gases, at the same pressure and temperature, contain the same number of molecules. The law of multiple proportions, that when two gases combine in a chemical reaction the volumes needed always stand in simple integral proportions, was understood ultimately by Avogadro to mean that equal volumes have equal numbers of atoms. Now why do they have equal numbers of atoms? (Feynman et al., 1963, p. 39-2).”

 

In 1808, Joseph Gay-Lussac observed that gases react in simple whole-number volume ratios—e.g., 2 volumes of hydrogen combine with 1 volume of oxygen to form 2 volumes of water vapor (2H₂ + O₂ → 2H₂O). However, John Dalton (1808) rejected Gay-Lussac’s findings because Dalton incorrectly assumed water had the formula HO. The turning point came in 1811 when Amedeo Avogadro proposed two revolutionary ideas:

(1) equal gas volumes, under the same temperature and pressure, contain equal number of molecules;

(2) Many gases, including hydrogen and oxygen, exist as diatomic molecules.

Avogadro’s insight resolved the discrepancy in Dalton’s model by correctly identifying the composition of water and explaining why volume ratios align with molecular stoichiometry. Yet his ideas were largely ignored until 1860, when Stanislao Cannizzaro revived them at the Karlsruhe Congress, enabling chemists to determine atomic masses with consistency.

 

“Now why do they have equal numbers of atoms? Can we deduce from Newton’s laws that the number of atoms should be equal? (Feynman et al., 1963, p. 39-2).”

 

Newton’s laws of motion alone cannot explain Avogadro’s principle, because they deal with how every particle move—not with the big picture of how a large number of particles behave together. Avogadro’s insight depended on recognizing gases as composed of discrete molecules with definite stoichiometries—a chemical idea beyond the scope of Newton’s laws. While classical mechanics can explain how particle collisions generate pressure, but connecting equal volumes to equal numbers of molecules requires statistical averaging. (You need to average over a large number of particles to get meaningful results.) The ideal gas law (PV = nRT), which formalizes Avogadro’s principle*, emerges only when Newton’s laws are combined with molecular assumptions and statistical reasoning. Thus, Feynman’s question underscores this gap: classical physics alone cannot explain macroscopic gas behavior without invoking probability and the atomic nature of matter.

 

* Avogadro’s principle (n ∝ V at fixed P, T) is embedded in the ideal gas law PV = nRT and this requires assuming gases are composed of discrete molecules (a non-Newtonian idea).

 

3. Limitations: Where Classical Thermodynamics Fails

“… from a physical standpoint, the actual behavior of the atoms is not according to classical mechanics, but according to quantum mechanics, and a correct understanding of the subject cannot be attained until we understand quantum mechanics (Feynman et al., 1963, p. 39-1).”

 

Feynman rightly remarks that classical mechanics is inadequate for understanding atomic behavior, which fundamentally requires quantum mechanics. However, he could have been more precise about the limitations of classical thermodynamics, particularly its failure to accurately describe systems under certain conditions. While classical thermodynamics is a widely applicable theory, it breaks down when applied to quantum systems such as photons, ultracold atoms, or few-particle system. For example, quantum systems exhibit non-classical behavior due to their sensitivity to environmental interactions, which can induce decoherence and destroy their coherent quantum states. Thus, quantum thermodynamics has emerged as a suitable framework that redefines heat, work, and temperature in contexts where fluctuations, discreteness, and quantum correlations are unavoidable. It extends thermodynamic principles into the quantum domain, where basic assumptions underpinning the classical theory no longer apply.

 

“Here, unlike the case of billiard balls and automobiles, the difference between the classical mechanical laws and the quantum-mechanical laws is very important and very significant, so that many things that we will deduce by classical physics will be fundamentally incorrect. Therefore there will be certain things to be partially unlearned; however, we shall indicate in every case when a result is incorrect, so that we will know just where the ‘edges’ are (Feynman et al., 1963, p. 39-1).”

 

In his textbook Statistical Mechanics, Feynman writes: “If a system is very weakly coupled to a heat bath at a given 'temperature,' if the coupling is indefinite or not known precisely, if the coupling has been on for a long time, and if all the 'fast' things have happened and all the 'slow' things not, the system is said to be in thermal equilibrium.” This emphasizes that thermal equilibrium is not just about fast processes such as molecules settling into a Maxwell-Boltzmann distribution. It also requires that the system has interacted long enough with its environment to reach a stable state, while slower processes (e.g., container erosion) remain negligible. Thermodynamics holds under such conditions, but its accuracy depends on system size. In large systems (over 1,000 particles), fluctuations average out, making macroscopic quantities well-defined. In small systems (under 100 particles), fluctuations dominate, and corrections from statistical mechanics are needed. While thermodynamics may apply at the nanoscale, its assumptions must be used with care, as the boundary between large and small systems is inherently fuzzy.

Review Questions:

1. What idealizations underpin classical thermodynamics, and how do they simplify reality?

2. How does statistical mechanics employ approximations, and what justifies them?

3. When does classical thermodynamics fail, and how do modern frameworks address its limitations?

 

The moral of the Lesson (In Feynman’s spirit):

What makes a theory great? Simple principles, surprising connections, and wide applicability. Thermodynamics has all three—it’s the one theory I’d bet my life on.

But here’s the twist: a theory that powerful comes with its own limits. Thermodynamics is like a brilliant, no-nonsense old professor—it delivers rock-solid answers, but only to the big, timeless questions. Ask it about the messy details, the fluctuations, the microscopic chaos, and it just shrugs and says: "Not my department!"

 

References:

Avogadro, A. (1811). Essay on a Manner of Determining the Relative Masses of the Elementary Molecules of Bodies, and the Proportions in Which They Enter Into These Compounds. Journal de Physique.

Callen, H. B. (1985). Thermodynamics and an introduction to thermostatistics (2nd ed.). Wiley.

Cannizzaro, S. (1860). Sunto di un Corso di Filosofia Chimica. Paper presented at the Karlsruhe Congress.

Dalton, J. (1808). A New System of Chemical Philosophy. Manchester: Bickerstaff.

Einstein, A. (1970). Autobiographical notes. In P. A. Schilpp (Ed.), Albert Einstein: Philosopher-scientist (pp. 1–95). Open Court. (Original work published 1949).

Feynman, R. P. (2018). Statistical mechanics: a set of lectures. CRC press.

Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman lectures on physics, Vol. I: Mainly mechanics, radiation, and heat. Addison-Wesley.

Gay-Lussac, J. L. (1808). Mémoire sur la combinaison des substances gazeuses entre elles. Annales de Chimie.