The laws of thermodynamics describe the behaviour of gases at the macroscopic level through quantities like pressure, volume, and temperature. But why does a gas exert pressure, and why is the temperature of a gas related to the motion of its molecules? The kinetic theory of gases answers these questions by explaining macroscopic behaviour in terms of the microscopic motion of molecules.
The kinetic theory models a gas as a collection of a very large number of molecules in constant random motion, colliding elastically with each other and with the walls of the container. From this model, the theory derives the pressure of a gas, the ideal gas equation, and the kinetic interpretation of temperature.
This chapter also introduces the distribution of molecular speeds, the concept of degrees of freedom, the law of equipartition of energy, and the mean free path of a molecule. The kinetic theory beautifully connects the observable world of pressure and temperature with the invisible world of moving molecules, and it provides the molecular basis for the specific heats of gases studied in thermodynamics.
An ideal gas is a model in which the gas molecules are treated as point particles that have negligible volume and exert no intermolecular forces on each other except during brief elastic collisions. The kinetic theory assumes: the gas consists of a very large number of molecules; the molecules move randomly in all directions; the collisions between molecules and with the walls are perfectly elastic; the molecular size is negligible compared with the average distance between molecules; and the time spent in collisions is negligible compared with the time of free motion.
Real gases approach ideal behaviour at low pressures and high temperatures, where the molecular volume and intermolecular forces become relatively unimportant. At high pressures and low temperatures, real gases deviate from ideal behaviour, and equations of state like van der Waals' equation are needed to describe them.
The number of molecules in one mole of any gas is Avogadro's number, NA = 6.022 x 10^23. The number of molecules in n moles is n x NA.
The pressure of a gas arises from the collisions of the molecules with the walls of the container. Each collision imparts momentum to the wall, and the rate of change of momentum per unit area is the pressure. If m is the mass of one molecule and v_rms^2 is the mean square speed of the molecules, the pressure of an ideal gas containing N molecules in a volume V is:
P = (1/3) * (N m / V) * v_rms^2
Since N m / V = rho, the density of the gas, we can write P = (1/3) rho v_rms^2. This result shows that the pressure is proportional to the mean square speed of the molecules; the faster the molecules move, the greater the pressure they exert.
The mean square speed is the average of the squares of the molecular speeds, and the root mean square speed is:
v_rms = sqrt(v_rms^2) = sqrt(3 P / rho) = sqrt(3 P V / (N m))
Using the ideal gas equation PV = nRT and the expression for pressure, we can relate the average kinetic energy of a molecule to the absolute temperature. For a gas containing N molecules in n moles, PV = nRT = N k_B T, where k_B is Boltzmann's constant:
k_B = R / N_A = 1.38 x 10^-23 J/K
The average translational kinetic energy of a single molecule is:
(1/2) m v_rms^2 = (3/2) k_B T
This result is the kinetic interpretation of temperature: the absolute temperature of a gas is a measure of the average translational kinetic energy of its molecules. At absolute zero (0 K), the average kinetic energy of the molecules would be zero.
The root mean square speed of molecules of molar mass M at temperature T is:
v_rms = sqrt(3 R T / M)
This formula shows that at a given temperature, lighter molecules move faster than heavier ones, which is why hydrogen molecules move faster than oxygen molecules at the same temperature.
The molecules in a gas do not all move with the same speed; they have a distribution of speeds. Maxwell and Boltzmann derived the distribution of molecular speeds in a gas, which depends on temperature and the molecular mass. Most molecules have speeds near the most probable speed, and only a few have speeds very low or very high.
Three characteristic speeds are defined. The root mean square speed is:
v_rms = sqrt(3 R T / M)
The average speed is:
v_avg = sqrt(8 R T / (pi M))
and the most probable speed, at which the distribution has its maximum, is:
v_mp = sqrt(2 R T / M)
The three speeds are related as v_rms > v_avg > v_mp. As the temperature rises, the distribution broadens and all three speeds increase. This explains why a gas diffuses faster at higher temperature.
The degrees of freedom of a gas molecule are the number of independent ways in which it can store energy. A monatomic molecule such as helium has only three translational degrees of freedom, corresponding to motion along the three axes. A diatomic molecule such as oxygen has, in addition to three translational degrees, two rotational degrees of freedom (rotation about two perpendicular axes). A polyatomic molecule has three translational and three rotational degrees of freedom.
The law of equipartition of energy states that in thermal equilibrium, the total energy of a system is equally divided among all its degrees of freedom, with each degree contributing an average energy of (1/2)k_B T per molecule. Thus the average energy of a monatomic molecule is (3/2)k_B T, and of a diatomic molecule is (5/2)k_B T.
The equipartition law explains the specific heats of gases. For one mole of a monatomic gas, Cv = (3/2)R; for a diatomic gas, Cv = (5/2)R. These values match the results obtained from thermodynamics, providing strong confirmation of the kinetic theory.
Molecules in a gas are in constant motion and collide frequently with each other. The mean free path is the average distance travelled by a molecule between two successive collisions. For a gas of molecules with diameter d and number density n (number per unit volume), the mean free path is:
lambda = 1 / (sqrt(2) * pi * d^2 * n)
Since the number density n = P/(k_B T), the mean free path is inversely proportional to the pressure and directly proportional to the temperature. At atmospheric pressure, the mean free path of air molecules is about 10^-7 m. The mean free path increases at low pressures, which is why at very low pressures molecules can travel across a container without colliding.
The molar specific heat at constant volume and constant pressure for gases can be derived from the kinetic theory. For one mole of a monatomic gas, Cv = (3/2)R and Cp = (5/2)R, giving gamma = Cp/Cv = 5/3. For a diatomic gas, Cv = (5/2)R and Cp = (7/2)R, giving gamma = 7/5. For a polyatomic gas, Cv = 3R and Cp = 4R, giving gamma = 4/3.
These predictions agree very well with experiment for monatomic gases. For diatomic gases, the rotational degrees of freedom may not be fully active at low temperatures, which leads to deviations at low temperatures. The kinetic theory thus provides a molecular explanation for the thermodynamic relations studied earlier.
| Quantity | Formula | Value |
|---|---|---|
| Boltzmann's constant | k_B = R / N_A | 1.38 x 10^-23 J/K |
| Avogadro's number | N_A | 6.022 x 10^23 /mol |
| Pressure | P = (1/3) rho v_rms^2 | Pa |
| Average KE of a molecule | (3/2) k_B T | J |
| RMS speed | sqrt(3RT/M) | m/s |
| Mean free path | lambda = 1/(sqrt(2) pi d^2 n) | m |
| Type of Gas | Degrees of Freedom | Cv | gamma |
|---|---|---|---|
| Monatomic | 3 | (3/2)R | 5/3 |
| Diatomic | 5 | (5/2)R | 7/5 |
| Polyatomic | 6 | 3R | 4/3 |
In this chapter we connected the macroscopic properties of gases to molecular motion through the kinetic theory. We saw that gas pressure arises from molecular collisions, P = (1/3)rho v_rms^2, and that temperature is a measure of the average translational kinetic energy, (1/2)mv_rms^2 = (3/2)k_B T. The Maxwell speed distribution showed that molecules have a range of speeds, with v_rms > v_avg > v_mp, and the equipartition theorem distributed energy equally among degrees of freedom, explaining the specific heats of monatomic, diatomic, and polyatomic gases. The mean free path described how far molecules travel between collisions. The kinetic theory thus provides the molecular foundation for thermodynamics, completing the bridge between the invisible world of molecules and the observable behaviour of gases.