Section 39–2 The pressure of a gas

Force per unit area / Energy per unit volume / Quasi-static adiabatic compression

 

In this section, Feynman relates the pressure of a gas to force per unit area, energy per unit volume, and quasi-static adiabatic compression. Most of these concepts trace back to Clausius’ (1857) paper, “On the Nature of the Motion which we call Heat.” In that seminal work, Clausius laid the foundation for the kinetic theory of gases, linking macroscopic properties like pressure and temperature to the microscopic motion of molecules. However, Feynman’s discussion goes beyond a basic explanation of gas pressure and includes a derivation of the adiabatic law.

 

1. Force per unit area

“We define the pressure, then, as equal to the force that we have to apply on a piston, divided by the area of the piston: P = F/A……. So we see that the force, which we already have said is the pressure times the area, is equal to the momentum per second delivered to the piston by the colliding molecules (Feynman et al., 1963, p. 39-3).”

 

The pressure of a gas is a macroscopic property that arises from the collective motion of microscopic particles. Its physical origin can be understood from three perspectives:

1. Macroscopic definition: Pressure (P) is defined as the force (F) exerted perpendicularly on a surface, divided by the area (A) of that surface: P = F / A.

2. Microscopic Origin: At the molecular level, gas particles move randomly at high speeds. When they collide with the walls of a container, they transfer momentum to the surface—producing a measurable force.

3. Statistical Average: The net pressure arises from averaging the momentum changes of many molecular collisions, as described by kinetic theory.

Feynman shows how macroscopic quantities like force and pressure arise from the statistical behavior of microscopic particles, bridging Newtonian mechanics with kinetic theory of gases and linking individual molecular motion to thermodynamic properties.

 

“… but eventually, when equilibrium has set in, the net result is that the collisions are effectively perfectly elastic. On the average, every particle that comes in leaves with the same energy. So we shall imagine that the gas is in a steady condition, and we lose no energy to the piston because the piston is standing still (Feynman et al., 1963, p. 39-3).”

 

In his 1857 paper, Clausius writes: “In order that Mariotte's and Gay-Lussac's laws, as well as others in connexion with the same, may be strictly fulfilled, the gas must satisfy the following conditions with respect to its molecular condition:

(1) The space actually filled by the molecules of the gas must be infinitesimal in comparison to the whole space occupied by the gas itself.

(2) The duration of an impact, that is to say, the time required to produce the actually occurring change in the motion of a molecule when it strikes another molecule or a fixed surface, must be infinitesimal in comparison to the interval of time between two successive collisions.

(3) The influence of the molecular forces must be infinitesimal (p. 116).”

In essence, Clausius stated three key simplifying assumptions to model ideal gases:

1. Infinitesimal Molecular Volume: The volume occupied by gas molecules is negligible compared to the container’s volume.
2. Infinitesimal Collision Duration: The time a collision takes is negligible compared to the time between collisions.
3. Infinitesimal Molecular Forces: Intermolecular forces are negligible except during collisions.

While Clausius restricted his model to these three assumptions, kinetic theory has refined them to account for real-gas effects (e.g., van der Waals forces). However, Clausius's framework remains a milestone in the development of statistical mechanics.

 

The kinetic theory of gases is built on a set of simplifying assumptions that ensure analytical solvability while offering reasonable agreement with experimental observations for gases:

1. Point Particles: Gas molecules are idealized as point masses since their individual volumes are negligible compared to the container size (L). Thus, the time between collisions with the same wall can be simplified as Δt = 2L/v_x, where vₓ is the average velocity component in the x-direction.

2. No Intermolecular Forces (Except During Collisions): Molecules are assumed not to exert forces on each other except during brief, elastic collisions. Between collisions, they move in straight lines at constant speeds.

3. Short Collision Duration: Collisions are assumed to occur instantaneously, allowing the momentum change to be treated as abrupt without the need of modeling the detailed interaction over time.

4. Perfectly Elastic Collisions: All collisions—whether between molecules or with the container walls—are assumed to be perfectly elastic. This implies:

(a) Kinetic energy is conserved, with no energy loss to heat or deformation.

(b) When a molecule collides with a wall, its momentum changes from +p​ to –p​, but its speed remains unchanged.

5. Large Number of Particles: The gas consists a large number of molecules (e.g., 10²³ or more), allowing statistical averaging. This enables definitions of macroscopic quantities such as pressure and temperature.

6. Random Motion: Molecules move randomly, following the Maxwell-Boltzmann distribution. At any moment, molecules are equally likely to move in any direction. The mean square velocity is distributed evenly among the three spatial dimensions: ⟨v²⟩=⟨v_x²⟩+⟨v_y²⟩+⟨v_z²⟩ where ⟨·⟩ denotes an ensemble averaging.

7. Negligible Gravitational Effects: Gravitational forces are considered too weak to significantly influence molecular motion. As a result, the velocity distribution remains isotropic: ⟨v_x²⟩ = ⟨v_y²⟩ = ⟨v_z²⟩ = ⟨v²⟩/3.

Some physicists introduce additional simplifying assumptions, such as identical particle masses (m), negligible relativistic effects (valid at low to moderate temperatures), and the absence of quantum effects (valid at high temperatures and low densities). These assumptions underpin the derivation of the ideal gas law and help connect microscopic particle dynamics to macroscopic thermodynamic observables. While real gases deviate from ideal behavior—especially at high densities or low temperatures—the kinetic theory remains a foundational framework for understanding gas behavior under most practical conditions.

 

2. Energy per unit volume

“For a monatomic gas we will suppose that the total energy U is equal to a number of atoms times the average kinetic energy of each, because we are disregarding any possibility of excitation or motion inside the atoms themselves. Then, in these circumstances, we would have PV= (2/3)U (Feynman et al., 1963, p. 39-5).”

 

Feynman showed that the product PV of a monatomic ideal gas corresponds is directly proportional to the internal energy U. This does not mean PV represents the work done by compressing a gas to zero volume at constant pressure—such a process would be physically unattainable. Rather, pressure and volume are interdependent, governed by the ideal gas law, and cannot be varied independently without altering other state variables. Notably, kinetic theory offers a more fundamental view of pressure—not merely as force per unit area, but as the rate of momentum transfer per unit area due to molecular collisions. This perspective also allows pressure to be interpreted as an energy density (energy per unit volume), revealing a deep connection between the mechanical origin of pressure and its thermodynamic role.

 

Note: A single particle cannot exert pressure in the thermodynamic sense—this pressure is a statistical property that emerges only from the collective behavior of many particles.

 

“It is only a matter of rather tricky mathematics to notice, therefore, that they are each equal to one-third of their sum, which is of course the square of the magnitude of the velocity: ⟨v_x²⟩= (1/3)⟨v_x²+ v_y²+ v_z²⟩=⟨v²⟩/3 (Feynman et al., 1963, p. 39-4).”

 

The "trick" Feynman highlights goes beyond the familiar Pythagorean theorem (or famous theorem of Greek*); it lies in bridging geometric symmetry with statistical reasoning. First, the equation v²=v_x²​+v_y²​+v_z²​, which relies on the three-dimensional Pythagorean theorem, applies to the velocity of a single atom or molecule. Second, the expression <v²> = <v_x²​ + v_y² + v_z²> may resemble the Pythagorean theorem, but its physical meaning is about statistical averaging over a large number of atoms. Third, a key insight comes from exploiting spherical symmetry. In an idealized isotropic system—where no direction is preferred and external forces such as gravity are absent—the average kinetic energy is evenly distributed across all spatial dimensions. This symmetry allows us to reduce the complexity of three-directional motion to a simple relation: <v_x²​> = <v²>/3. The one-third factor, a consequence of isotropy, underpins the derivation of the equipartition theorem and ultimately leads to the ideal gas law.

 

*In the audio recording [18:00], Feynman mentions the 'famous theorem of the Greeks,' more commonly known today as the Pythagorean theorem—though in higher dimensions, it is sometimes associated with de Gua’s theorem.

 

3. Quasi-static adiabatic compression

“A compression in which there is no heat energy added or removed is called an adiabatic compression, from the Greek a (not) + dia (through) + bainein (to go). (The word adiabatic is used in physics in several ways, and it is sometimes hard to see what is common about them.) (Feynman et al., 1963, p. 39-5).”

 

The term adiabatic derives from the Greek a- (not) and diabatos (passable), meaning “not passable”—in this context, referring to the absence of thermal energy transfer. In thermodynamics, a process is considered adiabatic if there is no heat transfer between a system and its surroundings, formally expressed as Q = 0 where Q is the heat transfer. This condition can be achieved—or more accurately, approximated—in two main ways:

1.      Perfect insulation – The system is thermally isolated, preventing any heat flow.

2.      Rapid process – The process occurs so quickly that there is insufficient time for significant heat transfer (e.g., a sudden gas expansion).

In general, adiabatic conditions are idealized approximations—used to simplify the analysis of systems in which heat transfer is minimal or intentionally ignored. In reality, perfectly adiabatic processes are physically unattainable; even under highly controlled conditions, some degree of thermal interaction inevitably occurs. As such, the term adiabatic can be somewhat misleading when applied to complex real-world systems—such as quantum systems—where complete isolation from thermal exchange is practically impossible.

 

“That is, for an adiabatic compression all the work done goes into changing the internal energy. That is the key—that there are no other losses of energy—for then we have PdV=−dU (Feynman et al., 1963, p. 39-5).”

 

Specifically, it is a quasi-static adiabatic compression, which has the following key features:

1. Quasi-static (Reversible): The process is carried out infinitely slowly, allowing the system to remain in thermal equilibrium at every stage. This ensures the process is reversible and that pressure P and volume V are well-defined throughout, allowing work to be calculated as ∫P dV.

2. Adiabatic condition (no heat transfer): The system is perfectly insulated, so no heat transfer with the surroundings. This implies ΔQ=0, and all energy transfer occurs only through mechanical work.

3. Compression (external work): Work is done on the gas by compressing it, which increases its internal energy. Since ΔQ = 0, the first law of thermodynamics reduces to dU = −PdV, where the change in internal energy dU results entirely from volume change under pressure.

This idealized model is fundamental in thermodynamics, providing insight into processes such as the temperature rise of a gas during compression and forming the theoretical basis for thermodynamic cycles, including those in heat engines.

 

Everyday Connection

“In banging against the eardrums they make an irregular tattoo—boom, boom, boom—which we do not hear because the atoms are so small, and the sensitivity of the ear is not quite enough to notice it. The result of this perpetual bombardment is to push the drum away, but of course there is an equal perpetual bombardment of atoms on the other side of the eardrum, so the net force on it is zero…... We sometimes feel this uncomfortable effect when we go up too fast in an elevator or an airplane…... (Feynman et al., 1963, p. 39-3).”

 

Another notable example is swimming-induced vertigo, often caused by air pressure imbalances in the ear—a condition known as alternobaric vertigo (swimming-induced vertigo). This type of vertigo can occur in several scenarios:

Shallow-water diving: Even small changes in depth (1–2 meters) can cause discomfort due to unequal pressure in the ears.

Uneven pressure equalization: Clearing pressure in one ear but not the other, sometimes due to nasal congestion or poor technique.

Tight swim goggles: Excessive external pressure on the outer ear may worsen air pressure imbalances.

While swimming may improve posture (e.g., reverse hunchback), relieve neck discomfort or reduce back pain, addressing one issue can sometimes introduce another.

 

Review Questions:

1. What is the minimum number of assumptions needed for the kinetic theory of (ideal) gases?

2. How does the mechanical definition of pressure as force per unit area (F/A) relate to its thermodynamic interpretation as energy per unit volume (U/V)?

3. Can a Zen master releasing intestinal gas be considered an example of an adiabatic process? What conditions (e.g., rapid expansion, thermal isolation) would be necessary for it to approximate an adiabatic process?

 

The moral of the lesson (in Feynman’s spirit): While one might theoretically liken flatulence to a quasi-static process, biological reality imposes strict constraints—no human can regulate the release slowly enough to maintain thermal equilibrium. In practice, achieving 'no heat transfer' requires either near-perfect insulation or a process so rapid that heat exchange is negligible—such as an adiabatic compression.

Note: The term 'adiabatic' was first introduced by William John Macquorn Rankine in his 1866 publication as shown below.

(Rankine, 1866)

References:

Clausius, R. (1857). Ueber die Art der Bewegung, welche wir Wärme nennen [On the nature of the motion which we call heat]. Annalen der Physik und Chemie, 176(3), 353–380.

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

Rankine, W. J. M. (1866). On the theory of explosive gas engines. Proceedings of the Institution of Civil Engineers, 25, 509–539.