Kinetic theory of gases and

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. ... 

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. 

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. 

F. W. Sears, An Introduction to Thermodynamics, The Kinetic Theory of Gases, and Statistical Mechanics, 2nd edn., Addison-Wesley, Reading, Massachusetts, 1966. 

An expression for the absolute rate of condensation can be developed readily if the simple kinetic theory of gases and the ideal gas law are applied (S2) ... 

This chapter treats the descriptions of the molecular events that lead to the kinetic phenomena that one observes in the laboratory. These events are referred to as the mechanism of the reaction. The chapter begins with definitions of the various terms that are basic to the concept of reaction mechanisms, indicates how elementary events may be combined to yield a description that is consistent with observed macroscopic phenomena, and discusses some of the techniques that may be used to elucidate the mechanism of a reaction. Finally, two basic molecular theories of chemical kinetics are discussed—the kinetic theory of gases and the transition state theory. The determination of a reaction mechanism is a much more complex problem than that of obtaining an accurate rate expression, and the well-educated chemical engineer should have a knowledge of and an appreciation for some of the techniques used in such studies. 

It follows from this discussion that all of the transport properties can be derived in principle from the simple kinetic theory of gases, and their interrelationship through A, and c leads one to expect that they are all characterized by a relatively small temperature coefficient. The simple theory suggests that this should be a dependence on T1/2, but because of intermolecular forces, the experimental results usually indicate a larger temperature dependence even up to r3/2 for the case of molecular inter-diffusion. The Arrhenius equation which would involve an enthalpy of activation does not apply because no activated state is involved in the transport processes. If, however, the temperature dependence of these processes is fitted to such an expression as an algebraic approximation, then an activation enthalpy of a few kilojoules is observed. It will thus be found that when the kinetics of a gas-solid or liquid reaction depends upon the transport properties of the gas phase, the apparent activation enthalpy will be a few kilojoules only (less than 50 kJ). 

Some opportunities of such approximations are well illustrated by considering two characteristic examples. The first example will be a dusty-gas model, where porous media is considered as one of components of a gas mix of huge molecules (or particles of a dust), mobile or rigidly fixed in space [249,252,253], Such a model allows a direct application of methods and results of kinetic theory of gases and is effectively applied to the description of mass transfer processes in PS. The history of such an approach, the origins of which can be found in the works by Thomas Graham (1830 to 1840) is considered in Ref. [249], Actually, the model was first proposed by James Maxwell (1860), further it was independently reported by Deryagin and Bakanov (1957), and then also independently reported by Evans, Watson, and Mason (1961 see Refs. [249,252]). 

A series of episodes in the historical development of our view of chemical atoms are presented. Emphasis is placed on the key observations that drove chemists and physicists to conclude that atoms were real objects and to envision their stracture and properties. The kinetic theory of gases and measmements of gas transport yielded good estimates for atomic size. The discovery of the electrorr, proton and neutron strongly irtfluenced discttssion of the constitution of atoms. The observation of a massive, dertse nucleus by alpha particle scattering and the measrrrement of the nuclear charge resrrlted in an enduring model of the nuclear atom. The role of optical spectroscopy in the development of a theory of electronic stracture is presented. The actors in this story were often well rewarded for their efforts to see the atoms. 

To bridge the gap between molecular processes and empirical coefficients and between laboratory determinations of input data and an engineering approach to predictions, we want to develop the above fundamental equations in terms of the kinetic theory of gases and reaction rate theory. There are three principal candidates for the rate-controlling... 

He made important contributions to the kinetic theory of gases and published papers on what is... 

In gas mixtures which are close in their physical properties (e.g., C0-02-N2 or even CH4-02-N2), the dimensionless ratios of the material constants v/k, v/D, D/k are determined by the kinetic theory of gases, and are close to 1, changing little from one case to another. For sharply different gases, for example, in lean mixtures of hydrogen with air or oxygen, where this is not so, the ratio D/k reaches 3-4 and we observe peculiar phenomena which deserve special consideration. Ignoring these cases as well, we find... 

It is worth noting at this point that the various scientific theories that quantitatively and mathematically formulate natural phenomena are in fact mathematical models of nature. Such, for example, are the kinetic theory of gases and rubber elasticity, Bohr s atomic model, molecular theories of polymer solutions, and even the equations of transport phenomena cited earlier in this chapter. Not unlike the engineering mathematical models, they contain simplifying assumptions. For example, the transport equations involve the assumption that matter can be viewed as a continuum and that even in fast, irreversible processes, local equilibrium can be achieved. The paramount difference between a mathematical model of a natural process and that of an engineering system is the required level of accuracy and, of course, the generality of the phenomena involved. 

Chapter 5 gives a microscopic-world explanation of the second law, and uses Boltzmann s definition of entropy to derive some elementary statistical mechanics relationships. These are used to develop the kinetic theory of gases and derive formulas for thermodynamic functions based on microscopic partition functions. These formulas are apphed to ideal gases, simple polymer mechanics, and the classical approximation to rotations and vibrations of molecules. 

The time element At used in the kinetic theory of gases, and in particular in the ff-theorem, although very... 

R. D. Present, Kinetic Theory of Gases, McGraw-Hiii, New York (1958) P- P- Schram, Kinetic Theory of Gases and Plasmas, Kluwer, Norwell, MA (1991). 

L. C. Woods, An Introduction to the Kinetic Theory of Gases and Magnetoplasmas, Oxford Univ. Press, New York (1993). 

In dynamic equilibrium, the rate of desorption equals Nr" because r" Is the probability that an adsorbed molecule will desorb in one second. The rate of adsorption is determined by the available empty area a Ng- N), the pressure (i.e. by the number of molecules in the gas phase per unit volume) and by the rate at which they move. The result Is a p N N]/[2nmkT) molecules per second. The factor (27tmfcT) stems from the kinetic theory of gases and Is related to the collision frequency. Equating the two rates gives, after some rearrangements, the Langmuir equation with... 

This result could not have been obtained from thermodynamics alone. The kinetic theory of gases and Uquids, on the other hand, leads us to conclude that TJ TJxy as work must be done against the molecular forces when the volume is increased from Vg to (see the paragraph on the Joule-Thomson effect, p. 97, and see also p. 152). 

First of all, should be proportional to the area A, because doubling the area will double the number of collisions with the wall. Second, Z should be proportional to the average molecular speed, M, because molecules moving twice as fast will collide twice as often with a given wall area. Finally, the wall collision rate should be proportional to the number density, N/T because twice as many molecules in a given volume will have twice as many collisions with the wall. All of these arguments are consistent with the kinetic theory of gases and are confirmed by the full mathematical analysis. We conclude that... 

The pair of molar heat capacities Cy and Cp for an ideal monatomic gas can be calculated from the results of the kinetic theory of gases and the ideal gas equation of state. From Section 9.5, the average translational kinetic energy of n moles of an ideal gas is... 

Ludwig Boltzmann (1844-1906) gained his PhD in 1867 as a scholar of J. Stefan in Vienna. He was a physics professor in Graz, Munich, Leipzig and Vienna. His main area of work was the kinetic theory of gases and its relationship with the second law of thermodynamics. In 1877 he found the fundamental relation between the entropy of a system and the logarithm of the number of possible molecular distributions which make up the macroscopic state of the system. 

As the fundamental concepts of chemical kinetics developed, there was a strong interest in studying chemical reactions in the gas phase. At low pressures the reacting molecules in a gaseous solution are far from one another, and the theoretical description of equilibrium thermodynamic properties was well developed. Thus, the kinetic theory of gases and collision processes was applied first to construct a model for chemical reaction kinetics. This was followed by transition state theory and a more detailed understanding of elementary reactions on the basis of quantum mechanics. Eventually, these concepts were applied to reactions in liquid solutions with consideration of the role of the non-reacting medium, that is, the solvent. 

In general [10] the equation for the transition probability is always an integral equation of the same type as the Boltzmann equation in the kinetic theory of gases, and only in certain limits (such as considered here) may it be written as a partial differential equation. 

The nuclear atom is the picture of the atom as a positive nucleus surrounded by negative electrons. Although the idea of atoms in speculative philosophy goes back to at least the time of Democrims, the atom as the basis of a scientifically credible theory emerges only in nineteenth century, with the rationalization by Dalton in 1808 of the law of definite proportions. Nevertheless, atoms were regarded by many scientists of the positivist school of Ernst Mach as being at best a convenient hypothesis, despite the success of the atomistic MaxweU-Boltzmaim kinetic theory of gases and it was not until 1908, when Perrin s experiments confirmed Einstein s atomistic analysis of... 

When heat flows through a mixture initially of uniform composition, small diffusion currents are set up, with one component transported in the direction of heat flow, and the other in the opposite direction. This is known as the thermal diffusion effect. The existence of thermal diffusion was predicted theoretically in 1911 by Enskog [El, E2] from the kinetic theory of gases and confirmed experimentally by Chapman [Cl, C2] in 1916. It is not surprising that the effect was not discovered sooner, because it is very small. For example, when a mixture of 50 percent hydrogen and 50 percent nitrogen is held in a temperature gradient between 260 and 10°C, the difference in composition at steady state is only 5 percent. In isotopic mixtures the effect is even smaller. 

---