The Kinetic Theory of Gases

Why do gases behave the way they do In some of the previous chapters, we have used gases as examples with an implicit understanding that we are modeling their behavior, but we never really got into a discussion of why they have that behavior. 

In this chapter, we will review the kinetic theory of gases, focusing on ideal gases. This review allows us to revisit some of the topics from Chapter 1 (in which we discussed the nature of gases in a more phenomenological perspective) so that we can better apply these ideas when we consider chemical reactions in the gas phase in the Chapter 20. 

What we will find in this chapter is that the physical behavior of gases can be understood if some simple assumptions are made. Suppose we treat an individual gas particle as a hard piece of matter What are the properties of a collection of hard pieces of matter It turns out that we can predict some properties by applying classical, rather than quantum, mechanics. The physical behavior of gases can be considered as a statistical average of all of the individual gas particles, so some of the ideas in this chapter are reminiscent of statistical thermodynamics. In addition, we will be focusing on the physical behavior of gases, not their chemical behavior. Because chemistry depends on electrons, it is vital to understand how electrons behave in order to understand how chemicals behave that is, we need quantum mechanics. But in order to understand the physical behavior of matter, we are able to use more simple physical theories of nature. Some of these physical behaviors are relevant to chemical behavior, as we will see in chemical kinetics. 

In this chapter on kinetic theory, we will consider the origin of the pressure of gases. We will find that the speeds of gas particles can have many values but the distribution of their speeds can be calculated. So can an average speed—in several different ways. We will also consider how many times gas particles collide with each other, how far they travel between collisions, and how far they travel from an arbitrary starting point. One of the more curious things from kinetic theory is the prediction that gas particles are moving very fast indeed, but because of all their collisions their net displacements change rather slowly. 

Unless otheiwise noted, all art on this page is Cengage Learning 2014. 

FIGURE 9.11 Velocity Is shown by an arrow of length v. It can be separated Into three components, and along the three Cartesian coordinate axes and projected into the x-y plane. 

The underlying assumptions of the kinetic theory of gases are simple  

A pure gas consists of a large number of identical molecules separated by distances that are great compared with their size. 

The gas molecules are constantly moving in random directions with a distribution of speeds. 

The molecules exert no forces on one another between collisions, so between collisions they move in straight lines with constant velocities. 

E is the energy expressed in calories per mole. It is calculated from the velocity w expressed in meters per second by the relation E — 0.0129 w2. The quantity e EIRT is defined cn page 21 as the fraction of the molecules which have an energy of E calories per mole or greater (i.e. from E to infinity). The fraction of the molecules dn/na having velocities between w and w+0.01 as calculated by equation (23) is recorded in Table IV for various velocities and temperatures. They are shown graphically in Fig. 7 where dn/no is given in per cent. In Table IV, w is expressed in meters per 

The fraction of activated molecules, 5.49XI0-19 for example, seems at first sight to be utterly negligible but it must be remem- 

An example of consecutive first order reactions is given as follows  

At the beginning of the reaction there is 1 mole of A per unit volume and no B nor C. Since 6 = 0.1, A decomposes at the rate of 10 per cent per hour. After time /, x moles of A have decomposed into B leaving —x moles of A and producing x moles of B. But as fast as B is produced it, too, starts to decompose into the stable substance C at the rate of 5 per cent per hour. After time t, y moles of C have been produced. At time /, then, the amount of B is represented by jc—y. It is the result of the balance between formation and decomposition. 

Numerical values of the amounts of A, B and C at various times as calculated from these formulas are given in Table V. 

Nelkin and Yip (1966) suggested that light-scattering experiments on dilute gases could be used as a test of the validity of the Boltzmann equation for the description of time-dependent phenomena. Prior to this the equation had been checked by only measuring transport coefficients and sound propagation. 

Soon after Nelkin and Yip (1966) suggested the experiment, Greytak and Benedek (1966) performed experiments on xenon (and CO2) and observed the deviations from the hydrodynamic line shape predicted by Nelkin and Yip. More recently Clark (1970) studied Xe and found the theoretical spectra to be in very good agreement with experiment. The theoretical calculations have now been extended to mixtures of gases by Boley and Yip (1972) and to polyatomic molecules by Desai etal. (1972). Experiments on helium—xenon mixtures by Clark (1970) and Gornall and Wang (1972) confirm the theoretical calculations. These latter calculations reduce to the results of Sections 5.6 and 10.4 respectively in the y — 00 limit. 

Uhlenbeck, G. E. and Ford, G. W., Lectures in Statistical Mechanics, American Mathematical Society (1963). 

We will start by imagining that we have a cylinder with a piston at one end that can move without any frictional losses. (We told you would get sick of cylinders and pistons ) Then each collision of the perfectly elastic particles of the gas would move that piston a little bit. Of course, a real gas has an enormous number of atoms or particles, so the net effect of all the collisions is felt like a continuous force, rather than individual impulses. We want to calculate the force necessary to keep the piston stationary. The pressure is then this force divided by the cross-sectional area of the piston. To calculate this, we will need to sum up all the impulse forces delivered to the piston. Recalling our classical mechanics, we can write Equation 10-8. 

the force (F) delivered by the impact of a single particle is equal to the rate of change of momentum of that particle as a result of the collision. What we need to calculate is how much momentum is delivered as a result of all the collisions that occur in a given time, say, a second. 

The problem is that the molecules are moving chaotically, in different directions with different speeds. This problem can be overcome by dealing with average values. Let s first assume that a particle strikes the piston with a component of velocity in the x-direction (normal to the piston surface area) of v. Equation 10-9 then gives the change in momentum. 

But this is not quite right and this is where the averages come in. The molecules will not all have the same component of velocity vx in the -direction. Not only that, the molecules are equally likely to be moving away from the piston as towards it, so we replace vf with a value averaged over all the mol-ecnles, v and divide by 2, to get Equation 10-11. 

This is still not right, because if the particles are moving randomly, they are equally likely to be moving with a given velocity in any direction, which leads to Equations 10-12. 

The wide utility of the ideal gas law suggests that the gaseous state is a relatively simple state of matter to understand and that the physical properties of all ideal gases tend to be pretty much the same. Chemists and physicists summarize the physical properties of gases in a model that was developed in the middle of the nineteenth century, the kinetic theory of gases. 

The kinetic theory of gases gives us another way of defining an ideal gas A gas is said to be ideal if its properties satisfy the postulates of the kinetic theory of gases. 

During the nineteenth century the concepts that atoms and molecules are in continual motion and that the temperature of a body is a measure of the intensity of this motion were developed. The idea that the behavior of gases could be accounted for by considering the motion of the gas molecules had occurred to several people (Daniel Bernoulli in 1738, J. P. Joule in 1851, A. Kronig in 1856), and in the years following 1858 this idea was developed into a detailed kinetic theory of gases by Clausius, Maxwell, Boltzmann, and many later investigators. The subject is discussed in courses in physics and physical chemistry, and it forms an important part of the branch of theoretical science called statistical mechanics. 

In a gas at temperature T the molecules are moving about, different molecules having at a given time different speeds v and different kinetic energies of translational motion (m being the mass of a molecule). It 

The Effusion and Diffusion of Gases the Mean Free Paths of Molecules 

In the foregoing discussions we have ignored the appreciable sizes of gas molecules, which cause the molecules to collide often with one another. In an ordinary gas, such as air at standard conditions, a molecule moves only about 50 nm, on average, between collisions that is, its mean free path under these conditions is only about two hundred times its own diameter. 

The value of the mean free path is significant for phenomena that depend on molecular collisions, such as the viscosity and the thermal conductivity of gases. Another such phenomenon is the diffusion of one gas through another or through itself (such as of radioactive molecules of a 

Nothing exists except atoms and empty space everything else is opinion. 

Every chemistry student is familiar with the ideal gas equation PV = nRT. It turns out that this equation is a logical consequence of some basic assumptions about the nature of gases. These simple assumptions are the basis of the kinetic theory of gases, which shows that the collisions of individual molecules against the walls of a container creates pressure. This theory has been spectacularly successful in predicting the macroscopic properties of gases, yet it really uses little more than Newton s laws and the statistical properties discussed in the preceding chapters. 

The ideal gas law has been used in many examples in earlier chapters, and some of the important physical properties of gases (the one-dimensional velocity distribution, average speed, and diffusion) were presented in Chapter 4. This chapter puts all of these results into a more comprehensive framework. For example, in Section 7.3 we work out how the diffusion constant scales with pressure and temperature, and we explore corrections to the ideal gas law. 

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Langmuir adsorption isotherm A theoretical equation, derived from the kinetic theory of gases, which relates the amount of gas adsorbed at a plane solid surface to the pressure of gas in equilibrium with the surface. In the derivation it is assumed that the adsorption is restricted to a monolayer at the surface, which is considered to be energetically uniform. It is also assumed that there is no interaction between the adsorbed species. The equation shows that at a gas pressure, p, the fraction, 0, of the surface covered by the adsorbate is given by ... 

Substances at high dilution, e.g. a gas at low pressure or a solute in dilute solution, show simple behaviour. The ideal-gas law and Henry s law for dilute solutions antedate the development of the fonualism of classical themiodynamics. Earlier sections in this article have shown how these experimental laws lead to simple dieniiodynamic equations, but these results are added to therniodynaniics they are not part of the fonualism. Simple molecular theories, even if they are not always recognized as statistical mechanics, e.g. the kinetic theory of gases , make the experimental results seem trivially obvious. 

We will almost always treat the case of a dilute gas, and almost always consider the approximation that the gas particles obey classical, Flarniltonian mechanics. The effects of quantirm properties and/or of higher densities will be briefly commented upon. A number of books have been devoted to the kinetic theory of gases. Flere we note that some... 

Dorfman J R and van Bei]eren H 1977 The kinetic theory of gases Statisticai Mechanics, Part B Time-Dependent Processes ed B J Berne (New York Plenum)... 

It is worth remarking that the development of both types of model, like so many other aspects of the kinetic theory of gases, relies heavily on ideas of Clerk Maxwell. Some of these were rediscovered by later workers, but there is remarkably little that was not anticipated, at least in outline, by Maxwell. 

In the second part of hla memoir Reynolds gave a theoretical account of thermal transpiration, based on the kinetic theory of gases, and was able CO account for Che above "laws", Chough he was not able to calculate Che actual value of the pressure difference required Co prevent flow over Che whole range of densities. ... 

As a consequence of these simple deductions, Graham s experiments c effusion through an orifice came to be regarded as one of the earliest direct experimental checks on the kinetic theory of gases. However, a closer examination of his experimental conditions reveals that this view is mistaken. As mentioned earlier, his orifice diameters ranged upwards from 1/500 in., while the upstream pressure was never very much less thai atmospheric. Under these circumstances the molecular mean free path len ... 

If the fraction of sites occupied is 0, and the fraction of bare sites is 0q (so that 00 + 1 = 0 then the rate of condensation on unit area of surface is OikOo where p is the pressure and k is a constant given by the kinetic theory of gases (k = jL/(MRT) ) a, is the condensation coefficient, i.e. the fraction of incident molecules which actually condense on a surface. The evaporation of an adsorbed molecule from the surface is essentially an activated process in which the energy of activation may be equated to the isosteric heat of adsorption 4,. The rate of evaporation from unit area of surface is therefore equal to... 

For example, the measurements of solution osmotic pressure made with membranes by Traube and Pfeffer were used by van t Hoff in 1887 to develop his limit law, which explains the behavior of ideal dilute solutions. This work led direcdy to the van t Hoff equation. At about the same time, the concept of a perfectly selective semipermeable membrane was used by MaxweU and others in developing the kinetic theory of gases. 

In the late 1800s, the development of the kinetic theory of gases led to a method for calculating mmticomponent gas diffusion (e.g., the flux of each species in a mixture). The methods were developed simnlta-neonsly by Stefan and Maxwell. The problem is to determine the diffusion coefficient D, . The Stefan-Maxwell equations are simpler in principle since they employ binary diffnsivities ... 

All ciyogenic hquids except hydrogen and helium have thermal conductivities that increase as the temperature is decreased. For these two exceptions, the thermal conductivity decreases with a decrease in temperature. The kinetic theory of gases correc tly predicts the decrease in thermal conductivity or all gases when the temperature is lowered. 

The kinetic theory of gases has been used so far, the assumption being that gas molecules are non-interacting particles in a state of random motion. This... 

This description of the dynamics of solute equilibrium is oversimplified, but is sufficiently accurate for the reader to understand the basic principles of solute distribution between two phases. For a more detailed explanation of dynamic equilibrium between immiscible phases the reader is referred to the kinetic theory of gases and liquids. 

Here, a molecule of C is formed only when a collision between molecules of A and B occurs. The rate of reaction r. (that is, rate of appearance of species C) depends on this collision frequency. Using the kinetic theory of gases, the reaction rate is proportional to the product of the concentration of the reactants and to the square root of the absolute temperature ... 

The traditional unipolar diffusion charging model is based on the kinetic theory of gases i.e., ions are assumed to behave as an ideal gas, the properties of which can described by the kinetic gas theory. According to this theory, the particle-charging rate is a function of the square of the particle size dp, particle charge numbers and mean thermal velocity of tons c,. The relationship between particle charge and time according White s... 

White s equation is widely used mainly because it is easy to use and because it gives values which are in reasonable agreement with the experimental ones. However, because this model is based on the kinetic theory of gases, it should be used for small particles only. This model (as many others) assumes that particle charge can be described with a continuous function. Especially in the case of small particles, only the lowest charge numbers (0, 1, 2) are possible, and therefore the model—which does not take into account the discrete charge numbers—is somewhat questionable. 

Section 5.6 considers the kinetic theory of gases, the molecular model on which the ideal gas law is based. Finally, in Section 5.7 we describe the extent to which real gases deviate from the law. ... 

There is a restriction on this simple model for the C0-N02 reaction. According to the kinetic theory of gases, for a reaction mixture at 700 K and concentrations of 0.10 M, every CO molecule should collide with about 109 N02 molecules in one second. If every collision were effective, the reaction should be over in a fraction of a second. In reality, this does not happen under these conditions, the half-life is about 10 s. This implies that not every CO-N02 collision leads to reaction. 

Z, the collision frequency, which gives the number of molecular collisions occurring in unit time at unit concentrations of reactants. This quantity can be calculated quite accurately from the kinetic theory of gases, but we will not describe that calculation. 

The pressure behavior shown in Figure 4-3 is readily explained in terms of the kinetic theory of gases. There is so much space between the molecules that each behaves independently, contributing its share to the total pressure through its occasional collisions with the container walls. The water molecules in the third bulb are seldom close to each other or to molecules provided by the air. Consequently, they contribute to the pressure exactly the same amount they do in the second bulb—the pressure they would exert if the air were not present. The 0.0011 mole of water vapor contributes 20 mm of pressure whether the air is there or not. The 0.0050 mole of air contributes 93 mm of pressure whether the water vapor is there or not. Together, the two partial pressures, 20 mm and 93 mm, determine the measured total pressure. 

Ludwig Boltzmann (1844-1906) was born in Vienna. His work of importance in chemistry became of interest in plastics because of his development of the kinetic theory of gases and rules governing their viscosity and diffusion. They are known as the Boltzmann s Law and Principle, still regarded as one of the cornerstones of physical science. 

This is called Avogadro s theorem (1811) it appears here simply as a definition of molecular weighty and this is really the manner in which the relation is applied in chemistry. The kinetic theory of gases gives a new, and much deeper, significance to the statement by introducing the conception of the molecule this, however, does not concern us in thermodynamics, and since the molecular weights are purely relative numbers, the deductions made in this hook are equally strict whichever standpoint is adopted. 

The highest value of c, 1 667, which is that predicted by the kinetic theory of gases, is observed only with monatomic gases (argon, mercury). Diatomic gases have the value 1 4, triatomic 1 3, and k decreases with increasing molecular complexity (cf. Chap. XVIII.). 

It has been assumed in the deduction of (1) that the solute is an ideal gas, or at least a volatile substance. The extension of the result to solutions of substances like sugar, or metallic salts, must therefore be regarded as depending on the supposition that the distinction between volatile and non-volatile substances is one of degree rather than of kind, because a finite (possibly exceedingly small) vapour pressure may be attributed to every substance at any temperature above absolute zero. This assumption is justified by the known continuity of pleasure in measurable regions, and by the kinetic theory of gases. 

The kinetic theory of gases shows that the pressure p exerted by a gas is given by ... 

An explanation of potential energy involves an explanation of force both terms are simply another way of saying that we know nothing about the thing to be explained. A distinct advance is made when a force can be explained in terms of the kinetic energy of a system in motion, an illustration of which is afforded by the kinetic theory of gases, which replaced the supposed forces of repulsion between the molecules of gases (the existence of which is disproved by Joule s experiment, 73) by molecular impacts. 

Wherever heat is involved temperature also fulfils an important role firstly because the heat content of a body is a function of its temperature and, secondly, because temperature difference or temperature gradient determines the rate at which heat is transferred. Temperature has the dimension 9 which is independent of M,L and T, provided that no resort is made to the kinetic theory of gases in which temperature is shown to be directly proportional to the square of the velocity of the molecules. 

The molecular diffusivity D may be expressed in terms of the molecular velocity um and the mean free path of the molecules Xrn. In Chapter 12 it is shown that for conditions where the kinetic theory of gases is applicable, the molecular diffusivity is proportional to the product umXm. Thus, the higher the velocity of the molecules, the greater is the distance they travel before colliding with other molecules, and the higher is the diffusivity D. 

In the previous section, the molecular basis for the processes of momentum transfer, heat transfer and mass transfer has been discussed. It has been shown that, in a fluid in which there is a momentum gradient, a temperature gradient or a concentration gradient, the consequential momentum, heat and mass transfer processes arise as a result of the random motion of the molecules. For an ideal gas, the kinetic theory of gases is applicable and the physical properties p,/p, k/Cpp and D, which determine the transfer rates, are all seen to be proportional to the product of a molecular velocity and the mean free path of the molecules. 

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