We shall find that we can derive all kinds of things — marvelous things — from the kinetic theory, and it is most interesting that we can apparently get so much from so little – Richard Feynman [1]
Setting the stage
We sense the air around us every day. We feel wind on our faces. We feel the weight of humid summer air. We watch a sealed empty water bottle collapse in an airplane seat during descent and can’t help but wonder — why? What is happening inside that bottle? What is happening inside the air itself?
We haven’t changed as a species over the past many thousands of years. We are drawn to understand. Darwinian evolution rewarded us for this impulse. So it was regarding air. We were fascinated by the “sea of air” we live in — and we still are. Otto von Guericke’s famous experiment captured this fascination perhaps better than any other. To witness a team of horses fail to pull apart two copper hemispheres held together by nothing but air pushing against a vacuum — this was not merely a spectacle. It was a call to action. One could not witness such a thing without asking why.
The answer, it turned out, could be expressed with startling simplicity.
PV = nRT
Four variables. One equation. The pressure of a gas (P), its volume (V), the number of molecules it contains (n), and its temperature (T) are all connected by a single clean relationship. No exceptions for different gases. No correction for the shape of the container. No dependence on what the gas is made of — whether it is oxygen, nitrogen, hydrogen, or carbon dioxide — only on how many molecules are present and how hot they are.
This simplicity is remarkable. And remarkable simplicity, in science, is rarely an accident.
The equation itself was a long time coming. Two centuries of careful experimental work — Boyle’s pressure and volume studies, Gay-Lussac’s temperature relationships, Avogadro’s counting hypothesis, Clapeyron’s synthesis — were required to assemble these four variables into one law. We will briefly trace that history in the next section. But the law, once assembled, immediately raised a deeper question. One that the experimentalists had not needed to ask, but that the theorists could not avoid.
Why does this work?
What is it about a gas that produces this exact relationship? What is actually happening inside that bottle, inside that room, inside the atmosphere pressing down on Otto von Guericke’s horses? If PV = nRT is a fact about nature, then nature must have a reason for it. There must be a physical story — a mechanism — that, if you understood it, would make the law feel not just true but obvious.
This is the question that drives this chapter. And it is the question that drives this book.
The answer, when it came, was beautiful in its simplicity. A gas is a collection of atoms moving at high speed, colliding with each other and with the walls of their container, with nothing but Newton’s laws governing each collision. That is all. Pressure is the accumulated force of countless molecular impacts on the wall. Temperature is the average kinetic energy of those moving molecules. The ideal gas law is not a mystery — it is Newton’s laws applied to an unseen world.
This physical picture — first proposed by Daniel Bernoulli in 1738, largely ignored, and then rebuilt by Rudolf Clausius in 1857 — is the foundation of the kinetic theory of gases. It is, as Richard Feynman observed, a case in which we can apparently get so much from so little.
The reach of this simple model is extraordinary. Once you accept that a gas is made of moving, non-interacting atoms, the ideal gas law follows. But so does much more. The viscosity of a gas — how it resists flow — follows from the same picture. So does thermal conductivity, diffusion, and the mean free path of motion. So does the speed of sound. So does the temperature rise when you compress a gas in a cylinder. So does the basis for separating molecules by mass. All of these phenomena, drawn from a single physical hypothesis about what a gas is, will be the subject of the chapters that follow this one.
We begin, however, at the beginning. With the experiments that established the law. With the theorists who asked why. And with the physical model that, once accepted, transformed our understanding of the invisible world.
* * * * *
The ideal gas law is a rare thing in science: a quantitative relationship between measurable properties of nature arrived at purely through experiment, with no physical model required and no assumptions about what a gas actually is. The scientists involved did not need to know that gases were made of atoms to establish PV = nRT — the equation emerged directly from what they measured. In this sense the ideal gas law was, unknowingly, one of the first quantitative consequences of the atomic theory of matter, discovered before the theory itself existed to explain it. So before sharing the model behind the equation, let’s first pay our respects to those behind the history.
The experimental evolution of the ideal gas law
The great thing about gases is that their properties lend themselves readily to direct measurement. Pressure, volume, temperature, and quantity — each of these can be observed, varied, and recorded in a laboratory. This directness made gases the natural starting point for the experimental investigation of heat and matter. And it made the ideal gas law not merely a scientific result but a gift to science — a clean, empirically grounded relationship that theorists could test their ideas against.
But the law did not arrive all at once. It was assembled piece by piece over nearly two centuries, each piece contributed by a different experimenter working in a different country, often unaware of the others, each holding one corner of a picture that none of them could yet see whole.
Boyle and the spring of air [2]
The first piece came from Robert Boyle (1627–1691). During his travels through Europe and upon reading of Otto von Guericke’s work on the power of atmospheric pressure, Boyle obtained early threads of knowledge regarding the behavior of air. Upon return to his Oxford laboratories, he and his assistant Robert Hooke (1635–1703) studied the relationship between pressure and volume of a gas trapped in an upside-down tube immersed in a bath of mercury at constant ambient temperature. As the height of mercury was changed, so changed the volume of the gas. From this, Boyle determined in 1662 that gas pressure is inversely proportional to volume—or, alternatively, that PV = constant—at constant temperature.
P 1/V Boyle’s Law (fixed temperature and mass)
Gay-Lussac and the role of temperature [3]
Pressure and volume were tractable. Temperature was harder. One could envision zero volume and zero pressure, but zero temperature — what could this mean, and where would it be found? This conceptual difficulty meant that the temperature relationship lagged behind.
While Guillaume Amontons (1663–1705) seems to have been the first to recognize the importance of measuring how gas volume and pressure vary with temperature, it took another century before the necessary experimental accuracy was achieved. By 1802, Joseph-Louis Gay-Lussac (1778–1850) determined that air and other gases — oxygen, nitrogen, hydrogen, carbon dioxide — all expand by the same fraction when heated through the same temperature interval. He also determined the coefficient of thermal expansion to be 1/267, establishing the following relationship.
V (t + 267) Gay-Lussac’s Law (volume–temperature in oC)
In 1807, Gay-Lussac continued his experimental studies by examining the relationship between the specific heats of gases and their densities, in the course of which he found that the change of gas temperature is directly proportional to the change of pressure at constant volume.
P (t + 267) Gay-Lussac’s Law (pressure–temperature in oC)
Avogadro and the counting of molecules [4]
Mass remained the final gas property requiring study. In 1811, Amedeo Avogadro (1776–1856), building on Gay-Lussac’s work, proposed his famed hypothesis: that equal volumes of any gas at the same temperature and pressure contain equal numbers of molecules. This was a daring claim. It could not be directly verified at the time. But it gave experimentalists a powerful new tool for determining molecular weights — for a given number of molecules, the ratio of gas weights equals the ratio of molecular weights— and it told us the final proportionality.
V n Avogadro’s Law
Clapeyron and the synthesis [5]
It was Émile Clapeyron who, in 1834, tied these threads together into a single equation. Working with the proportionalities established by Boyle, Gay-Lussac, and Avogadro, he arrived at what we now call the ideal gas law.
PV = nRT The Ideal Gas Law
where R is the universal gas constant, i.e., a proportionality constant, the same for ideal gases. Clapeyron’s original equation was, PV = R (t+267). In 1857 Rudolf Clausius added the “n” and also replaced (t+267) with absolute temperature, T, following William Thomson’s 1854 introduction of the absolute thermodynamic temperature scale, which defined absolute temperature as the centigrade temperature plus 273.7, placing absolute zero at −273.7 °C. [6]
What the law gave us
By the mid-nineteenth century, the ideal gas law stood as one of the most reliable and widely used results in all of science. Thomson, Clausius and others all relied on equation PV = nRT and its inherent simplicity to guide their analyses of the steam engine and the foundations of thermodynamics.
The beauty of the law lies not only in its simplicity but in what that simplicity implies. Pressure, volume, temperature, and quantity — four variables, one equation, no exceptions for the type of gas, so long as it is ideal. This universality was a signal. It was saying something deep about the nature of gases — something that transcended the chemical identity of any particular substance and pointed toward a common physical mechanism.
That mechanism was waiting to be found.
The Physical Model: from assumption to derivation, and what follows
The ideal gas law stood on solid experimental ground by the 1830s. But it said nothing about why. It connected four measurable properties of nature — pressure, volume, temperature, and quantity — with an exactness that demanded explanation. The equation was waiting for a physical story.
That story required a leap. Not a mathematical leap, but a physical one. Someone had to be willing to say or at least hypothesize: a gas is made of atoms. Those atoms move. And the macroscopic properties we measure — pressure, temperature — are the consequences of those motions. This was not an obvious thing to propose in a scientific community still debating whether atoms existed at all.
The prehistory: Bernoulli, Herapath, Waterston
Daniel Bernoulli was arguably the first to make this leap in print. In 1738 he envisioned “minute corpuscles moving hither and thither with a very rapid motion” [7] and brought calculations to bear on such movements, arriving at the first published proposal of what we now call the kinetic theory of gases. His work went largely unnoticed for over a century.
Others independently arrived at the same physical picture. John Herapath (1790–1868) submitted a comprehensive paper on the kinetic theory to the Royal Society in 1820, where it was rejected by Humphry Davy as being “too speculative.” [8] It eventually appeared in the Annals of Philosophy the following year, but without the recognition it deserved. John James Waterston (1811–1883) suffered a similar fate: his submission to the Royal Society was rejected, described by one referee as “nothing but nonsense, unfit even for reading before the Society.” [9] His paper remained in the Royal Society’s archives until Lord Rayleigh discovered it in 1892.
The pattern is worth noting. The physical model was sound. The mathematics was workable. What the kinetic theory lacked was not correctness — it was credibility. In a scientific community built around the caloric theory and Newton’s static-atom model of gases [10], a billiard-ball world of freely moving molecules colliding with each other was simply too radical to take seriously.
As Stephen G. Brush well summarized [11], what was needed was not a connection between heat and molecular motion, because even with a static-atom model one could make that connection, but a connection saying that heat is nothing but molecular motion and that regarding heat of a gas, this motion is the free movement of molecules through space. That was the leap that kept being rejected.
August Krönig’s 1856 paper Elements of a Theory of Gases [12] helped revive the discussion, offering a clear physical vision of what the kinetic theory should look like. It was not a complete treatment — the mathematics did not go far enough, and no serious comparison with experimental data was attempted. But it lit a competitive fire. Rudolf Clausius read it and responded.
Clausius and the first complete kinetic theory
Clausius had completed his nine-memoir Mechanical Theory of Heat, which presented the First and Second Laws of Thermodynamics, without invoking any assumptions about the microscopic nature of matter — not because he wasn’t thinking about such things, but because his scientific rigor kept the two efforts deliberately separate. His need to understand why, however, was running in parallel all along. As he later acknowledged: “Before writing my first memoir on heat, which was published in 1850, and in which heat is assumed to be a motion, I had already formed for myself a distinct conception of the nature of this motion… In my former memoirs I intentionally avoided mentioning this conception.” [13] Krönig’s 1856 paper provided the competitive motivation to finally publish, and in 1857 Clausius published The Nature of the Motion which we call Heat with characteristic care. He laid out his assumptions explicitly before employing any mathematics: that only a small fraction of the total volume is occupied by molecules, that intermolecular forces are insignificant, and that all molecules share the same velocity — the last of these acknowledged openly as a simplification: “There is no doubt that actually the greatest possible variety exists amongst the velocities.” It was a deliberate strategy to simplify the mathematics while still allowing meaningful comparison with experiment.
The derivation: pressure from atomic motion
Clausius’s first goal was to derive the relationship between the translational motion of molecules and the macroscopic pressure of the gas.
Pressure, physically, is force per unit area. Force, by Newton’s second law, is the rate of change of momentum; hence, the element of time is required. Every time a molecule strikes the wall of its container and bounces back elastically, it transfers momentum to that wall. The accumulation of countless such impacts is what we measure as pressure.
In plain english,
Force = (change in momentum of single molecule) × (rate of collision of molecules against wall)
For a single molecule striking a wall directly, with speed vx in the x-direction:
Before collision: momentum = mvx
After elastic collision: momentum = −mvx
Change in momentum of molecule = 2mvx
The calculation of the rate of collision of molecules against the wall is interesting. Clausius considered a molecule traveling an infinitesimal distance dx in an infinitesimal time dt. The molecules that will strike the wall in the next instant dt are precisely those currently within distance dx of it — meaning that molecules with different velocities will arrive at the wall at the same moment only if their distance from the wall is proportional to their velocity. The total number of molecules striking the wall at any moment is therefore the density of the gas (N/V) multiplied by the volume of gas that will reach the wall — which is the wall area A multiplied by the velocity component directed toward the wall, vx. Since molecules move equally in all three spatial directions, the mean-square velocity in any one direction equals one-third of the total mean-square velocity. And since at any moment half of the molecules moving in the x-direction are heading toward the wall while the other half are heading away, only one-sixth of the total molecules are contributing to the impact rate at any given wall. These factors combine to give the collision rate used in Clausius’s pressure derivation.
Force = 2mvx × (rate of collision of molecules against wall)
Force = 2mvx x [(N/V)(Avx)(1/2)]
Force = 2mvx2 x [(N/V)(A)(1/2)]
Force = (1/3) mv2 x [(N/V)(A)]
Pressure = (Force/A) = (1/3) mv2 (N/V)
Which rearranges to:
PV = 2/3 N (1/2 mv²)
where N is the total number of molecules, m is the mass of each, and v2 is the mean-square speed.
In sum, the force generated against the wall by a single molecular impact is the change in momentum of that molecule, and pressure is what you get when you multiply that single-impact force by the rate at which molecules are delivering it. What’s interesting here is that the kinetic energy itself is not initially integral to the math; it evolves out of the math. The pressure is not directly caused by a change in kinetic energy.
Temperature as kinetic energy
Clausius then compared the prior equation against the experimentally established ideal gas law PV = nRT and concluded that the average translational kinetic energy of a gas molecule is directly proportional to the absolute temperature.
This derivation transformed temperature from a mysterious property of matter to a direct measure of the average kinetic energy of molecular motion. Higher temperature means faster-moving molecules. Lower temperature means slower-moving molecules. At absolute zero, in the classical picture, all translational motion ceases.
Clausius used this result, plus the densities of known gases at a given pressure, to calculate the mean translational speeds of gas molecules at known conditions — a number no experimentalist had ever directly measured. His results, reflecting that velocity varies with the square root of mass:
Oxygen: 461 m/s
Nitrogen: 492 m/s
Hydrogen: 1,844 m/s
The mean free path
One immediate challenge to the kinetic theory came from an unexpected direction. If molecules in air are moving at nearly 500 m/s, why does it take minutes to smell a gas released across a room? The Dutch meteorologist C. H. D. Buys-Ballot (1817–1890) posed this question directly.
Clausius welcomed the challenge — “I rejoice at the discussion of this point by M. Buijs-Ballot” [14]— and responded by adding a new layer to his physical model. Molecules are not traveling in straight unimpeded lines. They are constantly colliding with each other, and each collision redirects their path. The relevant question becomes: how far, on average, can a molecule travel before colliding with another? Clausius called this the mean free path.
Working from the geometry of a moving sphere passing through a field of stationary obstacles, and using an educated estimate that the ratio of total volume to molecular volume is approximately 1000:1 — consistent with the volume change observed when a liquid vaporizes — Clausius concluded that a molecule travels approximately 1000 times its own radius between collisions. For nitrogen or oxygen, with a molecular radius of approximately 8 × 10⁻⁹ cm, this gives a mean free path of approximately 8 × 10⁻⁶ cm. Given the molecular speeds calculated above, this translates into billions of collisions per second.
Slow diffusion, in other words, is not a problem for the kinetic theory. It is a consequence of it. Molecules move fast but travel only tiny distances before being redirected. The macroscopic phenomenon of slow diffusion emerges naturally from the microscopic picture of constant molecular collisions.
What Clausius established — and what remained
Clausius’s 1857 paper did something that Bernoulli, Herapath, and Waterston had not been able to do: it gave the kinetic theory credibility. His standing in the scientific community, his rigorous treatment of assumptions, his willingness to compare predictions with experimental data, and the connection he drew between molecular motion and the already-established laws of thermodynamics combined to make the kinetic theory impossible to dismiss. As E.E. Daub later described it, the paper marked “the watershed between the primitive and sophisticated versions of the kinetic theory of gases.” [15]
But Clausius’s work fell short in one critical way. It failed to link its predictions to a directly measurable macroscopic property in a manner that could serve as a decisive experimental test. Molecular speed, radius, and mean free path were theoretical quantities. How could one measure the invisible?
The bridge from micro to macro needed something more. It needed a prediction — derived from the kinetic theory — that could be tested in the laboratory against a property that experimentalists already knew how to measure. That bridge was built by James Clerk Maxwell, whose work is the subject of the chapters that follow.
A final comment on the ideal gas law
The beauty of the ideal gas law lies not only in its simplicity but also in the power of what it shows, specifically the relationship between PV and T, for in this relationship one catches a glimpse of the relationship between work and heat which became the basis for the mechanical theory of heat and the subsequent higher-level theory of energy and its conservation.
References
[1] Feynman, Richard Phillips, Robert B. Leighton, Matthew L. Sands, and Richard Phillips Feynman. 1989a. The Feynman Lectures on Physics. Volume I. Mainly Mechanics, Radiation, and Heat. Vol. 1. The Feynman Lectures on Physics 1. Redwood City, Calif.: Addison-Wesley, p. 41-1.
[2] Brush, Stephen G. 1986. The Kind of Motion We Call Heat:: A History of the Kinetic Theory of Gases in the 19th Century. Book 1: Physics and the Atomists. 3rd print. North-Holland Personal Library. pp. 10-12.
[3] Crosland, M. P. 2008. “Gay-Lussac, Joseph Louis.” In Complete Dictionary of Scientific Biography, 5:317–27. Charles Scribner’s Sons.
[4] Crosland, M.P. 2008. “Avogadro, Amedeo.” In Complete Dictionary of Scientific Biography, 1:343–50. Charles Scribner’s Sons.
[5] Carnot, S., Clausius, R., and Clapeyron, É., Reflections on the Motive Power of Fire, edited by E. Mendoza, Dover, 1988, p. 82.
[6] Thomson, William. “On the Dynamical Theory of Heat, with Numerical Results Deduced from Mr. Joule’s Equivalent of a Thermal Unit, and M. Regnault’s Observations on Steam.” Transactions of the Royal Society of Edinburgh 20 (1853): 261–288, §100.
[7] Coopersmith, Jennifer. 2010. Energy, the Subtle Concept: The Discovery of Feynman’s Blocks from Leibniz to Einstein. New York: Oxford University Press. p. 72.
[8] Brush, Stephen G. 2003. “Introduction.” In The Kinetic Theory of Gases: An Anthology of Classic Papers with Historical Commentary, edited by Stephen G. Brush and Nancy S. Hall, 1–42; 179–96. History of Modern Physical Sciences 1. London : River Edge, NJ: Imperial College Press ; Distributed by World Scientific Pub. p. 14.
[9] Ibid. p. 17
[10] The static-atom concept unintentionally originated with Sir Isaac Newton when he used it as the basis for a mathematical model of the relationship between pressure and volume in a gas. He was not proposing that this model reflect reality but this message was lost when later on others latched onto his static-atom approach.
[11] Brush, 2003. p. 14.
[12] Krönig, August. 1856. “Grundzüge Einer Theorie Der Gase.” Annalen Der Physik 175 (10): 315–22.
[13] Brush, 2003. pp. 111-134.
[14] Brush, 2003. pp. 135-147.
[15] Daub, E.E. 1970. “Waterston, Rankine, and Clausius on the Kinetic Theory of Gases.” Isis (The History of Science Society) 61 (1): 105–6. p. 106.
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