Kinetic Theory

KINETIC THEORY (from the Greek kinesis—motion), is based on the concept of matter as an aggregate of molecules bound by molecular forces and in continuous motion. The simplest case is that of a gas, which according to the kinetic theory is represented in the first approximation as consisting of completely free point-like molecules, unbound by internal forces, rushing chaotically at high speed in all possible directions and interacting during collisions according to the laws of impact of elastic bodies. From these simple assumptions, all the basic properties of gases can be easily derived. The most characteristic feature of a gas is its ability to expand indefinitely: gases occupy almost instantaneously any volume provided to them. From this follows another property of gases—the pressure they exert on the walls of the vessels containing them. From the standpoint of the kinetic theory, this pressure is the result of the "bombardment" to which the walls of the vessel are subjected by the continuously colliding gas molecules. The basic law of the gaseous state—Boyle's Law—follows directly from this concept. Indeed, by reducing the volume of a gas by half or to one-third, we force the flying molecules to strike the walls of the vessel twice or thrice as often, and consequently, the pressure must increase twofold or threefold. These qualitative arguments can easily be given a quantitative form to obtain the basic formula of the kinetic theory of gases. For this purpose, let us calculate the magnitude of the pressure p caused by molecular impacts. It is clear that this pressure must be proportional to the number of impacts experienced by 1 cm² of the wall and the intensity of the impulse imparted by each impact. The number of impacts is, in turn, obviously proportional to the number of molecules in 1 cm³ of gas (N) and the velocity of these molecules (v). The intensity of the impulse imparted by each molecule, or momentum, as is known from mechanics, is proportional to the momentum of the molecule, i.e., the product of its mass and velocity—mv. Thus, we see that the pressure p is proportional to Nv · mv = Nmv². In this formula, only a constant numerical factor is missing, which, as a simple calculation shows, is equal to ¹/₃. Thus, p = ¹/₃Nmv².

(1) Using this simple formula, one can first of all determine the velocity of gas molecules. Indeed, N (the number of molecules in 1 cm³) × m (the mass of a molecule) = the density of the gas ρ; hence p = ¹/₃ρv². Thus, to calculate the velocities of molecules, only the densities need to be known. Such calculations yield, for example, the following values of v: Oxygen — 425 m/s, Nitrogen — 454 m/s, Hydrogen — 1,844 m/s, Carbon dioxide — 362 m/s.

These values greatly exceed the wind speed during the most severe hurricanes and approach the speed of artillery shells. If this molecular bombardment does not produce a destructive effect, it is only because the impacts are directed completely chaotically in all directions and, on average, balance each other out.

Let a gas occupy a volume V; according to the basic formula, the pressure exerted by this gas will be p = ¹/₃ρv²; but ρ = density = mass / volume; if the number of molecules in volume V is N¹, and the mass of each is m, then M = N¹m, and ρ = N¹m / V. Substituting this value for density into the basic formula, we get p = ¹/₃ · (N¹m / V) · v², or pV = ¹/₃N¹mv² = ²/₃N¹(mv² / 2). On the other hand, the basic law of the gaseous state—the Boyle-Charles-Gay-Lussac law (see Gases)—states: pV = RT, where T is the absolute temperature. Comparing this expression with the formula just obtained, we find: pV = ²/₃N¹(mv² / 2) = RT.

(2) But is the kinetic energy of an individual molecule, and B is a constant value. Thus, from formula (2) it follows that the kinetic energy of the molecules is proportional to the absolute temperature of the gas. We see that the Kinetic Theory gives the concept of temperature a completely concrete physical meaning: it is nothing other than the energy of motion of the gas molecules; there are no "hot" and "cold" molecules, there are only molecules moving with greater or lesser speed. Up to now, it has been assumed that all molecules move with one and the same speed, which is designated by the letter v. Simple considerations show that this is incorrect. Indeed, even if at some moment all molecules had the same speed in magnitude, then in the very next moment, due to collisions occurring at the most diverse angles, the molecules would acquire different speeds. We have seen above that the speed for nitrogen molecules—the main constituent of air—at 0° is approximately 500 m/sec. If such a molecule flew vertically upward, it could rise to a height of only 12.5 km. Thus, if all molecules moved with the same speed, the maximum height to which air molecules could rise, i.e., the theoretical limit of the atmosphere, would be equal to the same value of 12.5 km. In reality, it is known that the highest auroras still occur at a height of 500 km, meteorites ignite at a height of 200 km, etc. The reason for the contradiction is precisely that the average speed v is identified with the true speed. In reality, molecules are found with both lower and significantly higher speeds. Maxwell first introduced into the Kinetic Theory the "law of distribution of molecular velocities," i.e., a formula allowing one to calculate what fraction of the total number of molecules possesses a given speed. According to Maxwell, the state of a gas can be characterized as complete chaos; molecules possess all possible speeds; most frequently, molecules are found with a certain speed which is the most probable, but a certain percentage of molecules possesses much higher and a certain percentage much lower speeds. Since the number of molecules is huge and the microscopic state of the gas is characterized as complete chaos, one is all the more justified in using the methods of statistics, the methods of the theory of probability, to construct the Kinetic Theory of gases. The famous Maxwell distribution formula, graphically illustrated in the figure, has precisely such a statistical character. The distribution of all random phenomena in general, for example, the distribution of random errors of observation, etc., obeys a completely analogous law (see Variational Statistics). The most probable value of the speed of gas molecules, i.e., the speed possessed by the greatest number of gas molecules, as stated earlier, is measured in many hundreds of meters per second. Despite this, molecules move translationally relatively slowly: we are convinced of this, if only by the relative slowness of the propagation of odors. The reason for the apparent contradiction is that, due to the huge number of molecules, each of them continuously collides with surrounding ones and therefore constantly changes the direction of its path. The space traversed by a molecule between two collisions—the so-called mean free path—therefore has a negligible value. This mean free path plays an outstanding role in a whole series of phenomena occurring in gases. These are: 1. Diffusion (see); if two gases are brought into contact, then, as is known, the mutual penetration of the gases, caused by the motion of the molecules—diffusion—immediately begins. It is clear that the greater the mean free path, the greater the rate of diffusion. 2. Thermal conductivity. In a gas at rest, in the absence of any currents, heat is transferred, although extremely slowly, by thermal conductivity. From the point of view of the Kinetic Theory, this must be understood as meaning that the molecules of the hotter part of the gas, through constant collisions, transfer their excess momentum to the remaining molecules. But the same phenomenon can also be treated as the diffusion of molecules from the hotter part of the gas to the colder part and vice versa. 3. Internal friction (viscosity). When one layer of gas slides along another, molecules diffuse through the interface of both layers; in this process, molecules diffusing from the stationary or more slowly moving layer slightly retard the motion of the faster layer, and conversely, molecules entering the stationary layer from the sliding layer tend to set the latter in motion. As a result, the sliding layer experiences exactly the same friction as a solid body moving over a rough surface. Thus, all these three phenomena are closely related to the mean free path; they all allow this length to be calculated. However, in view of mathematical difficulties, the calculation can be performed only under the condition that the molecules interact like smooth billiard balls. Since this condition suffers from excessive simplification, the results do not differ in particular accuracy. However, the order of magnitude in all cases turns out to be correct. For the most well-known gases at normal pressure (760 mm Hg), the mean free path has the following values in millionths of a cm: Hydrogen, Oxygen, Nitrogen, Carbon dioxide, Chlorine: 18.3,

9.95,

9.44,

6.29,

4.57 × 10^-6 cm. If we compare the value of the mean path of an O₂ molecule with its speed, it is easy to calculate that this molecule must experience approximately 4,500 million collisions per second. At present, there exists a whole series of methods allowing one to find the number of molecules in 1 cm³ of gas. Without dwelling on their exposition, we will mention only one of them. The blue color of the sky, as is known, is caused by the scattering of sunlight by air molecules. According to Rayleigh, the intensity of scattered light is inversely proportional to the fourth power of the wavelength (which explains the fact that scattered light contains the greatest amount of rays of short wavelength, i.e., blue and light blue) and directly proportional to L—the number of molecules in one cm³. Thus, from observations of the intensity of the blue color of the sky, L can be calculated. This and other methods consistently give for L (the so-called Loschmidt number) a value of about 27 × 10¹⁸, while the number of molecules in a gram-molecule (Avogadro's constant) is equal to 60.64 × 10²². The latter number is more convenient to use in work, since it does not depend on pressure and is the same for a substance in the solid, liquid, and gaseous states. It is difficult to form a visual representation of the enormity of these numbers. Suffice it to say that the best "vacuum" achievable by modern technical means (the void of a Coolidge X-ray tube) still contains 27 billion molecules in 1 cm³. Knowing the mean path and the number of molecules in a unit volume, one can also calculate the sizes of the molecules. Such calculations give, for example, the following numbers for the diameter of molecules: Oxygen, Nitrogen, Hydrogen: 2.9,

3.1,

2.3,

Large organic molecules, e.g., protein molecules, have much larger sizes; however, the smallest bacilli still contain 10⁶–10⁷ molecules, a red blood cell—10⁹, and human spermatozoa—10⁸ molecules. All that has been set forth so far represents only a "first approximation" to reality. Indeed, in deriving the basic equation, we assumed that molecules are points having no volume and not connected with each other by any forces of interaction; in deriving the mean free path and the number of collisions, it was assumed that molecules are elastic spheres. If we take into account the finite volume of the molecules and the presence of cohesive forces, which can be detected, although to a weak degree, even in gases, then the simple equation of state of a gas, expressed by the Boyle-Mariotte-Gay-Lussac law: p v = RT (see Gases), is replaced by the more complex Van der Waals equation (see Gases, Van der Waals Law, Liquids): (p + a/v²)(v - b) = RT, where the term a/v² represents a correction for molecular interactions, and the term b is a correction for the true volume of the molecules. Very recently, methods have been developed that allow molecular velocities and the mean free path to be determined directly, rather than indirectly by calculation, as was described above. In this process, all the numbers obtained by calculation and cited above, as well as Maxwell's law of distribution of molecular velocities, found excellent confirmation. The Kinetic Theory of solids is still very weakly developed, and the Kinetic Theory of liquids essentially does not yet exist at all: so great are the mathematical difficulties encountered in transferring the Kinetic Theory of gases to other states. (See also Brownian Motion.)