Kinetic Theory

KINETIC THEORY (from Greek kinesis - movement), is based on the concept of matter as a collection of molecules bound by molecular forces and in continuous motion. The simplest case is that of a gas, which according to K. t. is approximately considered as consisting of completely free, point-like molecules not bound by internal forces, chaotically moving at high speed in all possible directions and interacting upon collision according to the laws of impact of elastic bodies. From these simple premises, all the basic properties of gases can be easily obtained. The most characteristic feature of a gas is its ability to expand indefinitely: gases practically instantly occupy 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 point of view of K. t., this pressure is the result of the "bombardment" to which the walls of the vessel are subjected by continuously incoming gas molecules. From this concept directly follows the basic law of gas state - the Boyle-Mariotte law. Indeed, by reducing the volume of the gas by half or a third, we force the flying molecules to hit the walls of the vessel twice or three times as often, and consequently the pressure should increase by two or three times. These qualitative considerations can easily be given quantitative form and the basic formula of K. t. of gases obtained. To do this, let us calculate the magnitude of the pressure p, caused by the impacts of molecules. 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 obviously in turn proportional to the number of molecules in 1 cm³ of gas (N) and the speed of these molecules (v). The intensity of the impulse imparted by each molecule, or momentum, as is known from mechanics, is proportional to the amount of motion of the molecule, i.e., the product of its mass by speed - mv. Thus we see that pressure p is proportional to 2Nmv = 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) With the help of this simple formula, one can first determine the speed of gas molecules. Indeed N (number of molecules in 1 cm³) × m (mass of molecule) = density of gas ρ; therefore ρ = ¹/₃ρv². Thus, to calculate the speeds of molecules, knowledge of only densities is necessary. Such calculations give, for example, the following values of v. Oxygen Nitrogen Hydrogen Carbon dioxide 425 1.844 m/sec. These values far exceed the speed of wind in the most severe hurricanes and approach the speed of projectiles from artillery. If this molecular bombardment does not have a destructive effect, it is only because the impacts are directed completely chaotically in all directions and on average balance each other out. Let the gas occupy 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 = mN/V. Substituting this value for density into the basic formula, we get pV = ¹/₃N¹mv², or pV = ²/₃N¹(¹/₂mv²). On the other hand, the basic law of gas state - the Boyle-Mariotte-Gay-Lussac law (see Gases) states: pV = BT, where T is absolute temperature. Comparing this expression with the formula just obtained, we find: pV = ²/₃N¹(¹/₂mv²) = BT.

(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 molecules is proportional to the absolute temperature of the gas. We see that the kinetic theory gives the concept of t^D a very concrete physical meaning: it is nothing other than the energy of motion of gas molecules; there are no 'hot' and 'cold' molecules, only molecules moving with greater or lesser speed. Up to now it has been assumed that all molecules move with the same speed, denoted by the letter v. Simple considerations show that this is incorrect. Indeed, if at some moment all molecules had the same speed, then already at the next moment, due to collisions occurring at the most varied angles, the molecules would acquire different speeds. Above we saw that the speed for nitrogen molecules-the main component of air-at 0° is approximately 500 m/sec. If such a molecule were to fly 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 northern lights still occur at an altitude of 500 km, meteors burn up at an altitude of 200 m. The reason for the contradiction is precisely that the average speed v is identified with the true speed. In reality, however, there are molecules with both smaller and significantly greater speeds. Maxwell was the first to introduce into the kinetic theory the 'law of distribution of molecular speeds,' i.e. a formula that allows calculating 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 often there are molecules with some speed that is the most probable, but a certain percentage of molecules possess much greater speeds and a certain percentage-much smaller speeds. Since the number of molecules is enormous and microscopic, the state of a gas is characterized as complete chaos, then with even greater right for constructing the kinetic theory of gases one can use the methods of statistics, the methods of probability theory. Maxwell's famous distribution formula, graphically illustrated by the figure, has exactly such a statistical character. A completely analogous law governs the distribution of all random phenomena in general, for example the distribution of random observational errors, etc. (see Variational Statistics). The most probable value of the speed of gas molecules, i.e. the speed possessed by the largest number of gas molecules, as mentioned earlier, is measured by many hundreds of meters per second. Despite this, molecules move translationally relatively slowly: this is at least confirmed by the relative slowness of the spread of odors. The reason for the apparent contradiction is that due to the enormous number of molecules, each of them continuously collides with surrounding molecules and therefore constantly changes the direction of its path. The distance traveled by a molecule between two collisions-the so-called mean free path-therefore has an insignificant value. This mean path length plays an outstanding role in a series of phenomena occurring in gases. These are: 1. Diffusion (see); if two gases are brought into contact, then as is known, immediately begins the mutual penetration of gases, caused by the motion of molecules-diffusion. It is clear that the greater the mean free path, the greater the speed of diffusion. 2. Thermal conductivity. In a stationary gas, in the absence of any flows, heat is transmitted, although extremely slowly, by thermal conductivity. From the point of view of kinetic theory, this must be understood as follows: molecules of the more heated part of the gas, through constant collisions, transmit their excess of momentum to the other molecules. But the same phenomenon can also be interpreted as the diffusion of molecules from the more heated part of the gas into the colder part and vice versa. 3. Internal friction. When one layer of gas slides along another, molecules diffuse through the interface of both layers; in this case, molecules diffusing from the stationary or more slowly moving layer slightly slow down the faster layer, and vice versa-molecules passing from the sliding layer into the stationary one 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 three phenomena are closely related to the mean free path; all of them allow this length to be calculated. However, due to mathematical difficulties, the calculation can be performed only on the condition that molecules interact like smooth billiard balls. Since this condition suffers from excessive simplification, the results do not possess particular accuracy. However, the order of magnitude is correct in all cases. For the best-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^-5 cm. If we compare the magnitude 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 are a whole series of methods that allow finding 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 due to 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 largest 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 one can calculate L. This and other methods according to 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 solid, liquid, and gaseous states. It is difficult to form a visual representation of the enormity of these numbers. It is sufficient to say that the best 'vacuum' (the emptiness of a Coolidge X-ray tube) achievable with modern technical means still contains 27 billion molecules in 1 cm³. Knowing the mean path length and the number of molecules in a unit volume, one can also calculate the size of molecules. Such calculations give for example for the diameter of the following numbers: Oxygen Nitrogen Hydrogen 2.9, 3.1, 2.3. Large organic molecules, for example protein molecules, have much larger dimensions; however, the smallest bacilli still contain 10⁵-10⁶ molecules, a red blood cell-10⁹, and a human spermatozoon-10⁸ molecules.--Everything presented 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 to each other by any interaction forces; in deriving the mean path length and number of collisions it was assumed that molecules are elastic spheres. If we take into account the finite volume of 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: pv=RT (see Gases), is replaced by the more complex equation of van der Waals (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-a correction for the true volume of molecules. For the very latest time, methods have been developed that allow determining molecular speeds and mean free path directly, not indirectly, by calculation as was presented above. In this case, all numbers obtained by calculation and presented above, as well as Maxwell's law of distribution of molecular speeds, have found excellent confirmation. The kinetic theory of solids has been developed so far very weakly, and the kinetic theory of liquids in essence does not yet exist: the mathematical difficulties encountered in transferring the kinetic theory of gases to other states are so great. (See also Brownian motion.)