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Spec mapping: OCR H556 Module 5.2 — Kinetic theory of gases (the assumptions of an ideal-gas kinetic model: large N, random motion, point molecules, elastic collisions, negligible collision time, no intermolecular forces between collisions; mean free path concept; conditions under which real gases approach ideal behaviour). Refer to the official OCR H556 specification document for exact wording.
Lessons 6 and 7 gave us the macroscopic equation of state of an ideal gas in two equivalent forms, pV=nRT and pV=NkT. Both forms are accurate to within fractions of a per cent for everyday gases at everyday conditions, but neither explains why a gas behaves this way. Why is pressure proportional to absolute temperature? Why does halving the volume double the pressure? Why does the equation involve temperature at all, rather than some other property of the gas?
The answer comes from the kinetic theory of gases — a microscopic model that pictures a gas as an enormous number of identical particles in perpetual random motion, colliding elastically with each other and with the container walls. From a small set of simple assumptions, kinetic theory derives the macroscopic gas laws from first principles, and — along the way — reveals a deep connection between absolute temperature and molecular kinetic energy that unifies classical mechanics with classical thermodynamics.
This lesson does the philosophy of the kinetic-theory model: it states the six standard assumptions in OCR H556 wording, explains what each one enables in the derivation, illustrates them with mean-free-path and rms-speed estimates, and prepares the ground for Lessons 9 and 10. The H556 specification expects you to be able to state the assumptions concisely (mark schemes typically award one mark for each of three to five assumptions, depending on the question), to justify them physically, and to apply them in qualitative arguments about real-gas departures from ideality.
The OCR specification lists six assumptions that characterise a kinetic-theory ideal gas. Together, they define what we mean by the word "ideal":
Let us examine each one carefully.
Assumption 1: A gas consists of a very large number of identical molecules.
This is vital for the statistical nature of kinetic theory. Individual molecules move in wildly different directions with wildly different speeds. It is meaningless to speak about the "pressure" or "temperature" of a single molecule. Pressure arises only as an average effect of billions upon billions of collisions with the container walls, happening so frequently that fluctuations are too small to detect. Temperature is a statistical measure of the average molecular kinetic energy.
At STP, one cubic centimetre of air contains about 2.7 × 10¹⁹ molecules (Loschmidt's number divided by 10⁶). Even the smallest cubic millimetre of air has 2.7 × 10¹⁶ molecules. Statistical averages over numbers this large are, for all practical purposes, exact.
Why is this assumption needed? Without it, the central-limit theorem of statistics would not apply, and the average quantities (pressure, temperature, density) would fluctuate wildly. The gas laws would not be simple equations but stochastic inequalities.
Assumption 2: The molecules are in constant, rapid, random motion.
"Constant" means the molecules never come to rest — they are perpetually moving. "Rapid" means their speeds are high (hundreds of metres per second at room temperature, as we shall calculate in Lesson 10). "Random" means three things:
Random motion is essential because it is only through the random distribution of directions that we can argue, in Lesson 9, that "one third of the molecules are moving in the x-direction, one third in the y-direction, and one third in the z-direction, on average". This factor of 1/3 ends up in the pressure formula.
Why is this assumption needed? Without randomness, the gas would behave directionally: it would exert more pressure on one wall of a container than on the opposite wall, and its temperature would depend on which direction you measured. In reality, gas pressure is the same in all directions, which is a macroscopic consequence of microscopic randomness.
Assumption 3: The total volume of the molecules themselves is negligible compared to the volume of the container.
In a gas at atmospheric pressure, molecules are separated by many molecular diameters. For air at STP, the typical separation is about 3.3 nm (about 10 molecular diameters). The molecules themselves occupy roughly 0.1% of the total volume.
Under this assumption, the entire volume of the container is "available" to each molecule. We do not need to subtract the space occupied by other molecules. This greatly simplifies the analysis.
Why is this assumption needed? It allows us to equate "volume of container" with "volume available for molecular motion" without correction. Real gases at high pressure, where the molecules are squeezed closer together, deviate from pV = nRT precisely because this assumption breaks down. Corrections are made in the more sophisticated van der Waals equation (p + a/V²)(V - b) = nRT, where the b term accounts for the non-zero volume of the molecules themselves.
Assumption 4: All collisions — between molecules, and between molecules and the walls of the container — are perfectly elastic.
An elastic collision is one that conserves both momentum and kinetic energy. The molecules' speeds after a collision may differ from their speeds before, but the total kinetic energy of the two molecules is unchanged (and momentum is conserved, as it must always be).
This matters because in Lesson 9 we will compute the momentum change when a molecule bounces off a wall. If the collisions were inelastic — if some kinetic energy were lost to heat, sound, or deformation of the wall — the molecules would gradually slow down, and the gas temperature would fall. We know this does not happen: a gas in a sealed container maintains its temperature indefinitely.
Why is this assumption needed? It guarantees that the molecules keep moving forever, never losing energy, so the gas has a stable temperature.
Real molecules do of course experience non-elastic interactions, especially in certain quantum regimes (e.g. excitation of rotational modes by collision), but at the level of translational kinetic energy involved in bouncing off a wall, the assumption of elastic collision is extremely accurate.
Assumption 5: The time duration of each collision is negligible compared to the time between collisions.
When two molecules collide, the interaction takes a very short time — much shorter than the typical time they spend moving freely between collisions. For air at STP, molecules travel about 68 nm between collisions (the mean free path) at a typical speed of 500 m s⁻¹, so the time between collisions is about 10⁻¹⁰ s. The time spent in a collision is a thousand times smaller.
Why does this matter? Because we want to say that at any given instant, "nearly all" the molecules are in free flight and "almost none" of them are in collision. This lets us treat collisions as instantaneous events that happen at specific points in time, separated by long stretches of free flight.
Why is this assumption needed? It allows us to neglect the time-averaged force a molecule feels from another molecule during a collision. We can treat the molecules as independent free particles most of the time, exchanging momentum in instantaneous collisions.
Assumption 6: Intermolecular forces (other than during collisions) are negligible.
The molecules exert no forces on each other except when they are in direct contact. Between collisions, each molecule flies in a straight line at constant speed, just like a projectile in free fall (except there is no gravity either in this idealisation).
This assumption explains why, for an ideal gas, the internal energy depends only on temperature — not on volume or pressure. If there are no intermolecular forces, there is no potential energy associated with the relative positions of the molecules; the only energy is kinetic. And kinetic energy depends only on speed, which in turn depends only on temperature.
Why is this assumption needed? It makes U depend on T alone, greatly simplifying the thermodynamics of an ideal gas, and it removes the possibility of condensation. Real gases near their liquefaction point have significant attractive forces that cause them to condense into liquids; ideal gases do not.
graph TD
KT[Kinetic theory of ideal gas] --> A1[1. Many molecules]
KT --> A2[2. Random rapid motion]
KT --> A3[3. Negligible molecular volume]
KT --> A4[4. Elastic collisions]
KT --> A5[5. Negligible collision time]
KT --> A6[6. No intermolecular forces]
A1 --> P1[Enables statistical averaging]
A2 --> P2[Isotropic pressure]
A3 --> P3[Full volume available to molecules]
A4 --> P4[Stable temperature]
A5 --> P5[Collisions as instant events]
A6 --> P6[Internal energy depends only on T]
These six assumptions are not arbitrary. Each one is there to enable a specific step in the derivation of pV = (1/3)Nm<c²> (Lesson 9) and its connection to temperature (Lesson 10). Skip any one of them and the derivation fails. Relax any one of them and you get a more complicated, less elegant result — often, the real behaviour of a non-ideal gas.
| Assumption | Purpose in the derivation |
|---|---|
| 1. Large N | Statistical averages are meaningful |
| 2. Random motion | Isotropic pressure; factor of 1/3 for one dimension |
| 3. Negligible volume | Container volume = volume available to molecules |
| 4. Elastic collisions | Total kinetic energy of gas conserved |
| 5. Short collision time | Treat molecules as free most of the time |
| 6. No intermolecular forces | Internal energy depends on T alone |
No real gas is perfectly ideal. But real gases approach ideal behaviour whenever the six assumptions above are approximately satisfied, which happens when the gas is:
As a rule of thumb, air at room temperature and atmospheric pressure is ideal to within about 0.1%. At 10 times atmospheric pressure, the error rises to about 1%. At 100 atmospheres or at temperatures near -100 °C, non-ideal effects become important.
Gases that deviate noticeably from ideal behaviour even at moderate pressure include:
A simple application of the assumptions: how far does a molecule travel, on average, between collisions, in air at STP?
Consider a molecule of diameter d ≈ 0.3 nm = 3 × 10⁻¹⁰ m travelling through a gas at number density n = 2.7 × 10²⁵ molecules m⁻³. It sweeps out a cylindrical volume per unit length equal to π d² /4 (roughly, although a more careful treatment with Maxwell-Boltzmann gives a factor of √2). For collisions, use the effective cross-section σ = π d² for a pair of molecules (this is the classical form that you will see in physical chemistry).
The mean distance between collisions — the mean free path — is approximately
λ≈1/(nσ)≈1/((2.7×1025)(π×(3×10−10)2))≈1/((2.7×1025)(2.83×10−19))≈1/(7.64×106)≈1.3×10−7m≈130nm(A more refined calculation gives about 68 nm; the answer depends slightly on exactly how you model the molecular cross-section.)
Either way, this is much larger than the molecular diameter (0.3 nm) — consistent with assumption 3 that the molecular volume is negligible. And it is much shorter than the container dimensions, consistent with the idea that wall collisions are rare compared with molecule-molecule collisions. But at a typical speed of 500 m s⁻¹, the average molecule undergoes about 10¹⁰ collisions per second — mind-bogglingly many.
A more sophisticated question, answered fully in Lesson 10: how fast do air molecules move on average at room temperature?
Using the result (1/2) m <c²> = (3/2) k T from Lesson 10, and the mass of a nitrogen molecule m ≈ 4.65 × 10⁻²⁶ kg at T = 293 K,
So nitrogen molecules zip around at around 500 m s⁻¹ at room temperature — roughly the speed of a rifle bullet. The extraordinary speed is a direct consequence of the small mass of the molecule.
A mercury atom has a mass of m = 3.3 × 10⁻²⁵ kg. Estimate the root-mean-square speed of mercury atoms in vapour at T = 600 K.
Mercury atoms are about seven times heavier than nitrogen molecules, so at a given temperature they move about √7 ≈ 2.6 times more slowly. This is a general feature of the Maxwell-Boltzmann distribution: heavier molecules move more slowly at the same temperature.
Kinetic theory was developed in the middle of the nineteenth century, primarily by James Clerk Maxwell, Ludwig Boltzmann and Rudolf Clausius. It was a bold hypothesis in its day: the reality of atoms was not universally accepted until the work of Jean Perrin on Brownian motion around 1905. Before Perrin's results, many scientists considered atoms mere mathematical fictions. Kinetic theory was therefore one of the first theories to predict macroscopic quantities from an explicit atomic picture, and its spectacular success — the derivation of pV = nRT from the assumption of random molecular motion — helped to convince sceptics that atoms must be real after all.
Boltzmann himself was sometimes rather depressed by the failure of his contemporaries to appreciate his work; he committed suicide in 1906, shortly before his ideas became universally accepted. His tombstone bears the equation S = k log W, which you will meet in university statistical mechanics: a testament to the central idea of statistical physics, that macroscopic order emerges from microscopic chaos.
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