The historical importance of the physics of gases was not only that it provided a direct insight into the nature of heat but also that when heat was applied to gases and vapours its effects were more obvious and more easily quantifiable than when it was applied to solid and liquids. In other words, gases and vapours provided the means whereby heat could readily produce mechanical effects.
— D. S. L. Cardwell [1]
Thank goodness for the ideal gas. It lends itself so beautifully to study and education. The exactness of the relationship between its properties, the first equation of state to connect the fundamental thermodynamic properties, can’t get much simpler.
PV = nRT
But the simplicity is somewhat misleading. Many, many years were required to put this law together. Ingenuity, patience, accuracy, reliable equipment, new analytical technologies, meticulous experiments, and a thinking mind were all required to realize this tremendous accomplishment.
With this law, one can tease out aspects of nature. Daniel Bernoulli was the first to propose a gas to be a billiard-ball world of colliding atoms that adhere to Newton’s laws of motion. The success of this derivation supported the connected theories behind the assumptions — specifically the theory of energy, the kinetic theory of gas, and the atomic theory of matter. This is indeed what happened in the late 1800s.
The study of gases naturally branched out to explore other properties, such as heat capacity, and other behaviors, such as adiabatic compression and expansion. The gathering results found many applications in both theory and practice. The advent of the heat engine demanded such supportive studies, since the heart of the process involved the expansion of a gas against a piston to do work while simultaneously losing thermal energy. The understanding that this process actually represented the conversion of heat into work contributed greatly to the final completion of the theory of energy. To truly understand the history of energy, we must first understand the history of the study of gases, starting with the ideal gas.
The Experimental Foundation: Building the Ideal Gas Law
The great thing about ideal gases is that their properties — and especially the change in their properties, since energy is all about change — readily lend themselves to direct measurement. The first to demonstrate this was Robert Boyle (1627–91). During his travels through Europe and upon reading of Otto von Guericke’s (1602–1668) work on the power of atmospheric pressure, Boyle obtained threads of knowledge regarding the behavior of air. Upon return to his Oxford laboratories, he and his assistant Robert Hooke (1635–1703) [2] sought to probe the subject in greater detail.
As a dedicated and persistent experimentalist, Boyle together with Hooke 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.
P ∝ 1/V Boyle’s Law (fixed temperature and mass)
Boyle’s work, inspired in part by Guericke and motivated to its conclusion by Francis Line who challenged Boyle’s earlier results, [3] contributed to the increased understanding of the behavior of air and really of any gas, all of which would be used in subsequent analyses of the steam engine.
Others followed Boyle’s lead and pursued studies of the relationship between any two variables while keeping the third constant. Since Boyle developed the pressure-volume relationship, temperature remained to be addressed — which presented something of a conceptual challenge, since one could envision zero volume and zero pressure, but zero temperature? What could this mean, and where would it be found?
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 until Joseph-Louis Gay-Lussac (1778–1850) obtained the necessary accuracy in measurements to draw conclusions. By 1802 Gay-Lussac determined that air and other gases — oxygen, nitrogen, hydrogen, carbon dioxide — all expand by the same fraction when heated through the same temperature. [4] He also determined the coefficient of expansion to be 1/267, which meant that upon adding the constant of 267 to the temperature reading in degrees Celsius, the following relationship would hold.
V ∝ T Gay-Lussac’s Law (volume–temperature)
This law is sometimes referred to as Charles’ Law in the English-speaking world, based on the work of Jacques Charles (1746–1823) some fifteen years earlier. But as Cardwell noted, the historical record does not support this attribution, and the law properly belongs to Gay-Lussac. [5]
In 1807, Gay-Lussac continued his experimental studies by determining the relationship between the specific heats of gases and their densities. [6] In the course of these studies, he found that the change of gas temperature was directly proportional to the change of pressure.
P ∝ T Gay-Lussac’s Law (pressure–temperature)
Mass became the final gas property requiring study. In 1811 Amedeo Avogadro (1776–1856), building on Gay-Lussac’s work, proposed his famed hypothesis: [7] that the number of integral molecules in any gas is always the same for equal volumes at the same temperature and pressure. Fixed volume at the same temperature and pressure meant fixed number of molecules. This gave experimentalists a new approach to determining molecular weight and told us the following relationship.
V ∝ n Avogadro’s Law
It was finally Émile Clapeyron in 1834 who tied together the P-V-T properties into the ideal gas law, [8] which, when including Avogadro’s work allowing the law to be scaled by the number of molecules, became:
PV = nRT Ideal Gas Law
Ironically, John Dalton (1776–1844) used the gas laws in the development of his atomic theory even though, as a disciple of Lavoisier, he was also a staunch believer in the caloric theory. [9] In Dalton’s mind, gas was comprised of atoms — but his atoms were not the projectiles we know them to be today. Instead, they were static, fixed in space, stacked together as closely as possible, and surrounded by an atmosphere of caloric that was the source of the repulsive forces between them. Such ad hoc explanations provided the means to arrive at the right answer for the wrong reason and so (wrongly) served to keep caloric alive for quite some time.
The Kinetic Theory Derivation: What Atoms Tell Us About PV = nRT
Source note: The kinetic theory derivation below follows the textbook treatment in Atkins & de Paula, Physical Chemistry, 10th ed. (Oxford University Press, 2014), Chapter 1, and K. A. Dill & S. Bromberg, Molecular Driving Forces, 2nd ed. (Garland Science, 2011), Chapter 11. The original primary sources are Rudolf Clausius, ‘Über die Art der Bewegung, welche wir Wärme nennen,’ Annalen der Physik, 100, 353–380 (1857), and James Clerk Maxwell, ‘Illustrations of the Dynamical Theory of Gases,’ Philosophical Magazine, 19, 19–32 (1860). [Textbook-tier sourcing; primary sources identified.]
The experimental ideal gas law PV = nRT was built up piece by piece over two centuries of careful measurement. But the deeper question is: why does this law hold? What is happening at the atomic level that produces this clean relationship between pressure, volume, temperature, and the number of molecules?
The answer comes from the kinetic theory of gases, and it begins with a picture of startling simplicity. Imagine a container filled with atoms — each one moving in a straight line until it collides with a wall or another atom, bouncing away according to Newton’s laws. No attraction between atoms. No repulsion at a distance. Just motion, collision, and rebound. This is the ideal gas.
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What is pressure, physically?
Pressure is force per unit area. Force, by Newton’s second law, is the rate of change of momentum. Every time an atom strikes the wall of its container and bounces back, it transfers momentum to that wall. The accumulation of countless such impacts — billions of billions per second — is what we measure as pressure.
Working through the mathematics of this picture for a gas of N atoms each of mass m moving with average speed v, and considering only the component of motion perpendicular to each wall, one arrives at:
PV = (1/3) N m <v²>
where <v²> is the mean-square speed of the atoms. This result, derived from Newton’s laws alone and the assumption of elastic collisions, is the kinetic theory expression for the pressure of an ideal gas. [Atkins & de Paula, Ch. 1; Clausius, 1857]
What is temperature, physically?
To connect equation 18.2 with the empirical ideal gas law PV = nRT, we need to understand what temperature means at the atomic level. The connection is direct: the average translational kinetic energy of a gas atom is proportional to the absolute temperature.
(1/2) m <v²> = (3/2) kBT
where kB is Boltzmann’s constant (kB = 1.381 × 10⁻²³ J/K). Temperature, in other words, is a measure of the average kinetic energy of atomic motion. A higher temperature means faster-moving atoms. A lower temperature means slower-moving atoms. At absolute zero, in the classical picture, atomic motion ceases entirely. [Atkins & de Paula, Ch. 1; Dill & Bromberg, Ch. 11]
Substituting
PV = N kBT = nRT
where n is the number of moles and R = NAkB is the universal gas constant. The empirical ideal gas law emerges directly from Newton’s laws applied to a gas of non-interacting atoms. This is both a beautiful and a powerful result.
What the derivation tells us — and what it hides
The kinetic theory derivation illuminates something important: the ideal gas law rests on two physical assumptions that are never literally true but are often very good approximations.
First, atoms are treated as point masses — they occupy no volume themselves. In reality, every atom has a finite size. At very high pressures, when atoms are packed closely together, their own volume becomes a non-negligible fraction of the total container volume. The ideal gas law begins to fail.
Second, intermolecular forces are assumed to be negligible. In reality, all atoms attract each other at a distance through electromagnetic interactions — London dispersion forces, dipole-dipole interactions, and others. At very low temperatures and high densities, when atoms move slowly and spend more time near each other, these attractive forces matter. The ideal gas law again begins to fail.
The compressibility factor Z provides a clean diagnostic for where the ideal gas assumption holds and where it breaks down:
Z = PV / nRT
For an ideal gas, Z = 1 exactly. Deviations from unity are the fingerprints of real behavior. Z < 1 signals that attractive forces are pulling atoms together, effectively reducing the pressure below what an ideal gas would predict. Z > 1 signals that the finite volume of atoms is making the effective volume available for motion smaller than the container volume, pushing pressure above the ideal prediction. We will return to this diagnostic in Chapter 20 when we take up the physics of real gases.
Heat Capacity, Cp and Cv, and the Caloric Theory Episode
The success of the ideal gas law motivated more experiments to quantify other properties and behaviors of gases, liquids, and solids. One property of study was heat capacity. Recognizing its importance, experimentalists such as Joseph Black designed calorimetry experiments to quantify specific heat capacity: C = Q/ΔT. Others including Adair Crawford (1748–1795), Gay-Lussac, and Dulong and Petit subsequently built on Black’s work.
The challenge in measuring heat capacity for a gas is that as heat is added, temperature is not the only property that changes. If the container is rigid — as in a calorimeter bomb — volume is constant, pressure increases, and the resulting heat capacity is called CV. If the container is flexible and the boundary movable — as in a balloon or inflatable bladder — pressure is constant, volume increases, and the resulting heat capacity is called CP.
In the constant-pressure case, the gas pushes against a moving boundary as it expands. It does work on that boundary, and this costs energy. The total temperature rise is therefore less for constant-pressure heating than for constant-volume heating, which means CP must be greater than CV. The understanding of the significance of this difference would become one of the critical components of Julius Mayer’s calculation of the mechanical equivalent of heat.
From the kinetic theory perspective, the relationship between CP and CV for an ideal gas is exact:
Cp – Cv = R
The difference equals R, the universal gas constant, because the extra energy going into constant-pressure heating goes entirely into the work of expansion — PΔV — and for an ideal gas this work equals nRΔT. [Atkins & de Paula, Ch. 2; textbook-tier sourcing]
Adiabatic compression: atoms and the moving piston
Another approach to effecting temperature change in a gas is through adiabatic compression or expansion — compression or expansion fast enough that no heat is exchanged with the surroundings. This is where the atomic picture is most physically transparent.
Feynman put it best in his Lectures on Physics: [10]
Suppose a piston moves inward, so that the atoms are slowly compressed into a smaller space. What happens when an atom hits the moving piston? Evidently it picks up speed from the collision. You can try it by bouncing a ping-pong ball from a forward-moving paddle, for example, and you will find that it comes off with more speed than that with which it struck. So the atoms are ‘hotter’ when they come away from the piston than they were before they struck it. Therefore all the atoms which are in the vessel will have picked up speed. This means that when we compress a gas slowly, the temperature of the gas increases.
When gas atoms strike a receding boundary during expansion, they rebound at a lower speed and so cool. When they strike an approaching boundary during compression, they rebound at a higher speed and so heat. This is the entirety of the physical story behind adiabatic compression and expansion. No caloric. No mystery. Just Newton’s laws applied at the atomic scale.
The quantitative relationship for an ideal gas undergoing a reversible adiabatic process is:
TV(γ-1) = constant
where γ = CP/CV is the heat capacity ratio. For a monatomic ideal gas — helium, argon, neon — γ = 5/3. For a diatomic gas such as air (approximately), γ = 7/5 = 1.4. The physical meaning of γ is this: it quantifies how many ways a molecule can store energy. A monatomic gas can only translate — three translational degrees of freedom. A diatomic gas can also rotate — two additional rotational degrees of freedom — and so distributes any added energy across more modes, which means less of it appears as a temperature rise. [Atkins & de Paula, Ch. 2; textbook-tier sourcing]
The Caloric Theory Episode: How Bad Data Can Derail Good Science
In what should have been a clear-cut victory for the kinetic theory of gases — explaining at a molecular level exactly why heat and work are related — the caloric theory once again miraculously survived. The calorists argued that for a given amount of matter, the amount of caloric surrounding the atoms is governed by the total volume available. As gas volume decreased during compression, caloric shifted from ‘latent’ to ‘sensible’ and so increased temperature. This was the theory.
The importance of heat capacity measurements was very real in France at this time, so much so that in 1811 the French Institut proposed a competition to determine the most accurate experimental approach. One of the more significant mistakes in the history of thermodynamics ensued when F. Delaroche (1781–1813) and J.-E. Bérard (1789–1869) rose to the challenge by carrying out a series of investigations that seemed to work out well — with the exception of one single data point mistakenly suggesting that heat capacity varied with volume. [11]
They shared their results with the utmost confidence: “Everyone knows that when air is compressed heat is disengaged… The experiments which we have carried out seem to us to remove all doubts upon the subject.” [12]
These incorrect data and conclusions — suggesting that the specific heats of gases increase with their volumes — led both Carnot and Clapeyron astray, increasing the influence of the caloric theory on their thought processes. As Mendoza wrote, “probably no other bad data have upset the development of thermodynamics more than these.” [13]
The episode is a sobering reminder that even careful experimentalists, when working within a powerful paradigm, can see what they expect to see rather than what is actually there. The caloric theory had an answer for everything — which is precisely what made it so difficult to displace.
The Physical Picture: What the Ideal Gas Really Is
With both the historical and kinetic theory foundations now in place, it is worth stepping back to state clearly what the ideal gas is and is not — physically.
An ideal gas is a collection of atoms or molecules in which:
- Intermolecular forces are negligible — atoms interact only at the moment of collision, not at a distance.
- Atoms occupy negligible volume themselves — they are point masses for the purposes of calculating pressure and volume.
- Collisions are perfectly elastic — kinetic energy is conserved in every collision; no energy is lost to vibration, rotation of the collision partners, or photon emission.
- The time spent in collision is negligible compared with the time spent in free flight between collisions.
Under these conditions, all of the internal energy of the gas is kinetic — translational, rotational, and vibrational — with no potential energy from intermolecular interactions. Temperature directly measures this kinetic energy. Pressure directly measures the rate of momentum transfer to the walls. The ideal gas law PV = nRT is the exact consequence of these assumptions applied through Newton’s laws.
These assumptions are remarkably good for real gases at low pressures and high temperatures — conditions where atoms are far apart, moving fast, and spending almost no time near each other. Under such conditions, the London dispersion forces that attract all atoms to each other are irrelevant, and the finite volume of the atoms themselves is a tiny fraction of the container volume.
But as pressure increases or temperature decreases, these assumptions begin to fail, and with them the ideal gas law. The compressibility factor Z moves away from unity and carries with it a physical story about what intermolecular forces are doing. That story is the subject of Chapter 20.
Closing Thoughts
The ideal gas law provided an excellent quantified relationship between readily measurable properties of gas and served to help move discussions from the abstract to the concrete. Carnot, Clapeyron, Thomson, and Clausius all relied on equation 18.1 and its inherent simplicity to guide their analyses of the steam engine.
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. 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.
And now we understand why. The kinetic theory derivation shows that PV = nRT is not a mysterious empirical fact. It is the direct consequence of Newton’s laws applied to a gas of non-interacting atoms. Temperature is the kinetic energy of atomic motion. Pressure is the accumulated momentum of atomic impacts. The ideal gas law is atoms in action.
References
[1] Cardwell, D. S. L. From Watt to Clausius: The Rise of Thermodynamics in the Early Industrial Age. Cornell University Press, 1971, pp. 128–129.
[2] During the course of this work, Boyle and Hooke also invented their famed air pump, which they used to create a vacuum. See Shapin, S. and Schaffer, S. Leviathan and the Air-Pump. Princeton University Press, 2011.
[3] Brush, S. G. The Kinetic Theory of Gases: An Anthology of Classic Papers with Historical Commentary. Imperial College Press, 2003, pp. 422–424.
[4] Holbrow, C. H. and Amato, J. C. ‘Multimode rotational spectra and the Avogadro-Ampère hypothesis.’ American Journal of Physics, 79, 2011.
[5] Cardwell, 1971, pp. 130–131.
[6] Crosland, M. ‘Gay-Lussac.’ In Complete Dictionary of Scientific Biography. Charles Scribner’s Sons, 2008.
[7] Crosland, M. ‘Avogadro.’ In Complete Dictionary of Scientific Biography. Charles Scribner’s Sons, 2008.
[8] Carnot, S., Clausius, R., and Clapeyron, É. Reflections on the Motive Power of Fire and other Papers on the Second Law of Thermodynamics. Edited with an introduction by E. Mendoza. Dover, 1988, p. 82.
[9] Cardwell, 1971, Chapter 5.
[10] Feynman, R. P., Leighton, R. B., and Sands, M. The Feynman Lectures on Physics. Addison-Wesley, 1989, Vol. I, p. 39-5.
[11] Carnot et al., 1988. See E. Mendoza’s Introduction. Also Cardwell, 1971, pp. 135–137.
[12] Cardwell, 1971, pp. 136–137.
[13] Carnot et al., 1988. Mendoza, Introduction, p. xvi.
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