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The experimental gas laws — Boyle's, Charles's and the pressure law — describe the macroscopic behaviour of gases very accurately. The ideal gas equation pV = nRT encapsulates them in one line. But none of these results explains why gases behave this way. Why is pressure proportional to temperature? Why does halving the volume double the pressure? Why does the equation involve temperature at all, rather than some other property?
The answer comes from the kinetic theory of gases, a powerful microscopic model that pictures a gas as a very large collection of tiny particles in rapid random motion. From a small number of simple assumptions about these particles, kinetic theory derives the ideal gas laws from first principles and — along the way — reveals a deep connection between temperature and molecular kinetic energy that unifies thermodynamics with Newtonian mechanics.
Module 5.1.4 of the OCR A-Level Physics A specification (H556) requires you to know and be able to state the assumptions of the kinetic theory of gases, and to use them in qualitative and quantitative arguments. This lesson focuses exclusively on the assumptions themselves. In Lesson 9 we will use them to derive the pressure formula pV = (1/3)Nm<c²>. In Lesson 10 we will connect the result to temperature.
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.
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