Chapter 12 ยท Introduction

Chapter 12 โ€” Kinetic Theory

A gas is an enormous crowd of tiny molecules racing about and colliding. Treat that crowd with Newton's laws and averages, and pressure, temperature and the heat capacities of gases all fall out.

Open a bottle of perfume in one corner of a room and, a little later, someone on the other side smells it. Pump up a bicycle tyre and it grows hard; leave it in the hot sun and it grows harder still. A balloon taken out into the winter cold shrinks. None of these surprise you, but each one asks a question. What is actually pushing on the inside of the tyre? Why should warming a gas raise its pressure? And why does the perfume take minutes, not milliseconds, to cross the room, when โ€” as you will find โ€” its molecules move faster than a rifle bullet?

This chapter answers all three by taking seriously the idea that matter is made of molecules in constant motion. You will start with the evidence for atoms and the behaviour of gases โ€” Boyle's law, Charles's law, Avogadro's hypothesis and Dalton's law of partial pressures โ€” and write the ideal-gas equation in its molecular form, $pV=Nk_BT$. Then you will build the kinetic theory of an ideal gas: applying Newton's second law to molecules bouncing off a wall gives $p=\tfrac13\rho\overline{v^2}$, and comparing this with the gas equation reveals what temperature really is โ€” a measure of the average kinetic energy of the molecules. Finally, the law of equipartition of energy shares that energy among the ways a molecule can move, which predicts the specific heats of gases and solids, and the mean free path explains why molecules travelling at hundreds of metres per second spread so slowly.

Moleculesโ†’Gas lawsโ†’$pV=Nk_BT$โ†’$p=\tfrac13\rho\overline{v^2}$โ†’$\tfrac12m\overline{v^2}=\tfrac32k_BT$โ†’$v_\text{rms}$โ†’Equipartitionโ†’$C_v$, $C_p$, $\gamma$โ†’Mean free path
๐ŸŽฏ By the end of this chapter you canโ€ฆ
  • describe the evidence for the molecular nature of matter and estimate the size and spacing of molecules in solids, liquids and gases;
  • use the mole, the Avogadro constant and $pV=nRT=Nk_BT$, with Boyle's law, Charles's law, Avogadro's hypothesis and Dalton's law of partial pressures, and say when a real gas behaves ideally;
  • state the assumptions of the kinetic theory and derive $p=\tfrac13nm\overline{v^2}=\tfrac13\rho\overline{v^2}$;
  • interpret temperature as average molecular kinetic energy, $\tfrac12m\overline{v^2}=\tfrac32k_BT$, and calculate $v_\text{rms}=\sqrt{3k_BT/m}=\sqrt{3RT/M}$;
  • count degrees of freedom and use the law of equipartition of energy to find $C_v$, $C_p$ and $\gamma$ for monatomic, diatomic and polyatomic gases, and $3R$ for solids;
  • calculate the mean free path $\lambda=\dfrac{1}{\sqrt2\,\pi d^2n}$ and the collision frequency;
  • tackle Board, JEE Main, JEE Advanced and NEET-style problems on kinetic theory (with the Maxwell speed distribution as a JEE Advanced extension).
Teaser โ€” How Fast Is the Air?

The box holds a gas at atmospheric pressure; each dot is a molecule, drawn with a trail showing its speed (the motion is slowed down enormously). Before you look at the readouts, guess: at room temperature, do the nitrogen molecules of the air move slower or faster than a jet airliner ($250\ \text{m/s}$)? Then change the gas and the temperature.

25 ยฐC
rms speed $v_\text{rms}$โ€”
mean KE per moleculeโ€”
molecules per cmยณโ€”
typical spacingโ€”

Sections in this chapter

12.1

The Molecular Nature of Matter and the Behaviour of Gases

Atoms and molecules, the mole, $pV=Nk_BT$, Boyle's and Charles's laws, Avogadro's hypothesis, Dalton's law and real gases.

$pV=nRT=Nk_BT$
Open section โ†’12.2

Kinetic Theory of an Ideal Gas: Pressure and Temperature

The assumptions of kinetic theory, the pressure of a gas from molecular collisions, temperature as kinetic energy, and the rms speed.

$v_\text{rms}=\sqrt{\tfrac{3RT}{M}}$
Open section โ†’12.3

Equipartition of Energy, Specific Heats and Mean Free Path

Degrees of freedom, $\tfrac12k_BT$ per mode, $C_v$, $C_p$ and $\gamma$ of gases, the $3R$ law for solids, and the mean free path.

$\lambda=\dfrac{1}{\sqrt2\,\pi d^2n}$
Open section โ†’
โœ… Before you start
You will need Newton's second law in the form $F=\dfrac{\Delta p}{\Delta t}$, impulse and momentum (Section 4.1), the third law and conservation of momentum (Section 4.2), kinetic energy and elastic collisions (Section 5.1, Section 5.3), and pressure as force per unit area, including atmospheric and gauge pressure (Section 9.1). From Class 11 Physics, Chapter 10 (Thermal Properties of Matter) you need the kelvin scale and the ideal-gas equation $pV=nRT$; Chapter 11 (Thermodynamics) gives internal energy, the first law and the molar heat capacities $C_v$ and $C_p$, which Section 12.3 explains from the molecular picture. From Class 9โ€“10 Chemistry and Science you should remember atoms, molecules and the mole. From mathematics you need powers of ten, square roots and averages.
โ† 11.3 The Second Law, Reversible Processes and the Carnot Engine12.1 The Molecular Nature of Matter and the Behaviour of Gases โ†’