# 2: The Kinetic Theory of Gases


Gases are literally all around us—the air that we breathe is a mixture of gases. Other gases include those that make breads and cakes soft, those that make drinks fizzy, and those that burn to heat many homes. Engines and refrigerators depend on the behaviors of gases, as we will see in later chapters. As we discussed in the preceding chapter, the study of heat and temperature is part of an area of physics known as thermodynamics, in which we require a system to be macroscopic, that is, to consist of a huge number (such as $$10^{23}$$) of molecules. We begin by considering some macroscopic properties of gases: volume, pressure, and temperature. The simple model of a hypothetical “ideal gas” describes these properties of a gas very accurately under many conditions. We move from the ideal gas model to a more widely applicable approximation, called the Van der Waals model. To understand gases even better, we must also look at them on the microscopic scale of molecules. In gases, the molecules interact weakly, so the microscopic behavior of gases is relatively simple, and they serve as a good introduction to systems of many molecules. The molecular model of gases is called the kinetic theory of gases and is one of the classic examples of a molecular model that explains everyday behavior.

• 2.1: Prelude to The Kinetic Theory of Gases
Gases are literally all around us—the air that we breathe is a mixture of gases. Other gases include those that make breads and cakes soft, those that make drinks fizzy, and those that burn to heat many homes. Engines and refrigerators depend on the behaviors of gases, as we will see in later chapters.
• 2.2: Molecular Model of an Ideal Gas
The ideal gas law relates the pressure and volume of a gas to the number of gas molecules and the temperature of the gas. A mole of any substance has a number of molecules equal to the number of atoms in a 12-g sample of carbon-12. The number of molecules in a mole is called Avogadro’s number. The ideal gas law can also be written and solved in terms of the number of moles of gas: pV=nRT and is generally valid at temperatures well above the boiling temperature.
• 2.3: Pressure, Temperature, and RMS Speed
Kinetic theory is the atomic description of gases as well as liquids and solids. It models the properties of matter in terms of continuous random motion of molecules. The temperature of gases is proportional to the average translational kinetic energy of molecules. Hence, the typical speed of gas molecules vrm is proportional to the square root of the temperature and inversely proportional to the square root of the molecular mass.
• 2.4: Heat Capacity and Equipartition of Energy
Summary Every degree of freedom of an ideal gas contributes $$\frac{1}{2}k_BT$$  per atom or molecule to its changes in internal energy. Every degree of freedom contributes $$\frac{1}{2}R$$ to its molar heat capacity at constant volume $$C_V$$ and do not contribute if the temperature is too low to excite the minimum energy dictated by quantum mechanics. Therefore, at ordinary temperatures $$d = 3$$ for monatomic gases, $$d = 5$$ for diatomic gases, and $$d \approx 6$$ for polyatomic gases.
• 2.5: Distribution of Molecular Speeds
The motion of individual molecules in a gas is random in magnitude and direction. However, a gas of many molecules has a predictable distribution of molecular speeds, known as the Maxwell-Boltzmann distribution. The average and most probable velocities of molecules having the Maxwell-Boltzmann speed distribution, as well as the rms velocity, can be calculated from the temperature and molecular mass.
• 2.A: The Kinetic Theory of Gases (Answer)
• 2.E: The Kinetic Theory of Gases Introduction (Exercises)
• 2.S: The Kinetic Theory of Gases (Summary)

Thumbnail: In an ordinary gas, so many molecules move so fast that they collide billions of times every second. (Public Domain; Greg L via Wikipedia)

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