Kinetic theories

The word kinetic comes from the Greek word which means to do with movement . 

In kinetic theory, energy in the forms of heat and movement are related higher temperature indicates greater movement. All particles are assumed to move, whether they are in solids, liquids or gases. 

1 In a container of gas, the space occupied by the gas molecules is very small compared with the total space. So when we think of the volume of a gas, we are mostly considering empty space. This explains why gases have low density and can be compressed. 

2 The molecules of gas move about continuously and rapidly, hitting each other and the walls of the container randomly. This explains why different gases mix completely. Collisions between the gas molecules and the walls account for the pressure exerted by the gas. 

5 The molecules of gas have no forces of attraction between them. The movement of the molecules is random and independent of the other molecules present. (As we shah see, this is only an approximation. If there were no attractions at all, no gas could ever be hquefied.) 

The kinetic theory of concerdrated solutions. In applying van der Waals equation 

of the solute. From the measurements of Morse and his collaborators on the osmotic pressure of cane sugar, Sackur f obtained values which are in excellent agreement with this equation. 

The constant h decreases as the temperature rises, which may be explained by a decrease in the hydration. 

These equations differ from those for dilute solutions by the term (1—6c) in the denominator. The change in the vapour pressure, and in the boiling point and freezing point, is therefore relatively greater in concentrated than in dilute solutions. This result is confirmed qualitatively by experiment in almost every case, as a review of the literature of the subject will show (see in particular the researches of Abegg, Auwers, and others). It is clear from the primitive nature of the underlying assumptions that the above equations can only be accurate for a limited range of concentration. 

A quantitative test of these relationships is not yet possible, as the heats of dilution of solutions of non-electrolytes have not been determined, and the available data on the vapour pressures of moderately concentrated solutions are too inaccurate. 

Definitely the most important theory in emulsion polymerisation is the Smith-Ewart theory. This theory was first pubhshed in 1948 (Smith Ewart, 1948) and since then has been the subject of continuing discussion and refinement. The theory is based on the Harkins mechanisms and then tries to predict the rate of reaction and its dependence upon the concentrations of the main components of the system. The rate of reaction is considered to be equal to the total rate of polymerisation in the nucleated soap micelles, which then have been converted to polymer particles. There is no polymerisation in the aqueous phase or in the monomer drops. The total rate can then be set equal to the rate in each polymer particle, multiplied by the number of particles  

Here M is the total amount of monomer in the system, kp the propagation rate constant, [M]p the concentration of monomer in the latex particles, n the average number of radicals in the particles, N the total number of particles and Na is the Avogadro s number. 

The quantitative theory is therefore centred on predicting (a) the number of particles nucleated and (b) the rate of polymerisation in each particle. The Smith-Ewart theory operates in the three intervals of the polymerisation process, and defines three cases for the kinetics. The intervals correspond to the three stages in the Harkins theory Interval I is the nucleation stage where micelles are present and the particle number increases Interval II corresponds to the stage when the particle number is constant and free monomer drops are also present Interval III is the last part of the polymerisation when the monomer drops have disappeared. Smith and Ewart developed an expression for the particle number created by nucleation in the soap micelles that is stiU considered essentially correct, within its limits (meaning that monomers, surfactants and generally conditions can be found when the Smith-Ewart theory is not correct and that our understanding today is more detailed). The expression for the particle number, N, is 

Here pi is the rate of initiation, p. is the volumetric growth rate, p = dv/dt, a is the specific surface area of the emulsifier ( soap ) and [S] is the concentration of emulsifier (also denoted as [ ]). The constant k has a value between 0.37 in the lower limit and 0.53 in 

The second part of the Smith—Ewart theory concentrates on calculating the average number of radicals per particle. As long as the monomer concentration in the particles is constant, as may often be the case in Interval 11, this number then yields the rate of polymerisation. Smith and Ewart did this by means of a recursion equation that is valid for the situation prevailing after particle formation is finished. 

Plane across which particles travel carrying mass and kinetk energy 

Note Here, 1/2 is used because it is assumed that only 1/2 of the total number of particles at each location move up from z-l plane or down from plane z + l.  

Averaging over the whole hemisphere of solid angle 2n, 

---

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 ... 

It must also be realized that this thin surface region is in a very turbulent state. Since the liquid is in equilibrium with its vapor, then, clearly, there is a two-way and balanced traffic of molecules hitting and condensing on the surface from the vapor phase and of molecules evaporating from the surface into the vapor phase. From the gas kinetic theory, the number of moles striking 1 cm of surface per second is... 

It is instructive to consider just how mobile the surface atoms of a solid might be expected to be. Following the approach in Section III-2, one may first consider the evaporation-condensation equilibrium. The number of molecules hitting a 1-cm surface per second is from kinetic theory... 

J. Frenkel, Kinetic Theory of Liquids, Clarendon Press, Oxford, 1946. 

In evaluating if a site can be regarded as a two-dimensional potential box, then the rate of adsorption will be given by the rate of molecules impinging on the site area oq- From gas kinetic theory. 

The acconunodation coefficient for Kr on a carbon filament is determined experimentally as follows. The electrically heated filament at temperature 72 is stretched down the center of a cylindrical cell containing Kr gas at 7. Gas molecules hitting the filament cool it, and to maintain its temperature a resistance heating of Q cal sec cm is needed. Derive from simple gas kinetic theory the expression... 

Mention was made in Section XVIII-2E of programmed desorption this technique gives specific information about both the adsorption and the desorption of specific molecular states, at least when applied to single-crystal surfaces. The kinetic theory involved is essentially that used in Section XVI-3A. It will be recalled that the adsorption rate was there taken to be simply the rate at which molecules from the gas phase would strike a site area times the fraction of unoccupied sites. If the adsorption is activated, the fraction of molecules hitting and sticking that can proceed to a chemisorbed state is given by exp(-E /RT). The adsorption rate constant of Eq. XVII-13 becomes... 

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. 

Born M and Green H S 1946 A general kinetic theory of liquids I. The molecular distribution functions Proc. R. Soc. A 188 10... 

Born M and Green H S 1949 A General Kinetic Theory of Liquids (Cambridge Cambridge University Press)... 

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... 

The kinetic theory of transport processes in gases rests upon three basic assumptions. 

Statistical mechanics and kinetic theory, as we have seen, are typically concerned with the average behaviour of an ensemble of similarly prepared systems. One usually hopes, and occasionally can demonstrate, that the variations of these properties from one system to another in the ensemble, or that the variation with time of the properties of any... 

Brush S 1966-1972 Kinetic Theory vols 1 -3 (New York Pergamon)... 

Resibois P and de keener M 1977 Ciassicai Kinetic Theory of Fiuids (New York Wiley)... 

Present R D 1958 Kinetic Theory of Gases (New York McGraw-Hill)... 

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)... 

Cohen E G D 1993 Fifty years of kinetic theory Physica A 194 229... 

Ernst M FI 1998 Bogoliubov-Choh-Uhlenbeck theory cradle of modern kinetic theory Progress in Statistical Physics ed W Sung et al (Singapore World Scientific)... 

The current frontiers for the subject of non-equilibrium thennodynamics are rich and active. Two areas dommate interest non-linear effects and molecular bioenergetics. The linearization step used in the near equilibrium regime is inappropriate far from equilibrium. Progress with a microscopic kinetic theory [38] for non-linear fluctuation phenomena has been made. Carefiil experiments [39] confinn this theory. Non-equilibrium long range correlations play an important role in some of the light scattering effects in fluids in far from equilibrium states [38, 39]. 

Piiiing M J and Smith i W M (eds) 1987 Modern Gas Kinetics. Theory, Experiment and Application (Qxford Biackweii) Giibert R G and Smith S C (eds) 1990 Theory of Unimolecular and Recombination Reactions (Qxford Biackweii) Fioibrook K A, Piiiing M J and Robertson S Fi (eds) 1996 Unimolecular Reactions 2nd edn (Chichester Wiiey)... 

Many additional refinements have been made, primarily to take into account more aspects of the microscopic solvent structure, within the framework of diffiision models of bimolecular chemical reactions that encompass also many-body and dynamic effects, such as, for example, treatments based on kinetic theory [35]. One should keep in mind, however, that in many cases die practical value of these advanced theoretical models for a quantitative analysis or prediction of reaction rate data in solution may be limited. 

Kapral R 1981 Kinetic theory of chemical reactions in liquids Adv. Chem. Phys. 48 71... 

Montgomery J A Jr, Chandler D and Berne B J 1979 Trajectory analysis of a kinetic theory for isomerization dynamics in condensed phases J. Chem. Phys. 70 4056... 

Collisional energy transfer in molecules is a field in itself and is of relevance for kinetic theory (chapter A3.1). gas phase kmetics (chapter A3.4). RRKM theory (chapter A3.12). the theory of unimolecular reactions in general,... 

Pilling M J and Smith 1W M (eds) 1987 Modern Gas Kinetics. Theory, Experiment and Application (Oxford Blackwell)... 

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. 

---